Question 1: Palette 1/145

Allen JEE (Main) 2023 MCQ

A body of mass 200g is tied to a spring of spring constant 12.5 N/m, while the other end of spring is fixed at point O. If the body moves about O in a circular path on a smooth horizontal surface with constant angular speed 5 rad/s, then the ratio of extension in the spring to its natural length will be:

A

1 : 2

B

1 : 1

C

2 : 3

D

2 : 5

Question 2: Palette 2/145

Allen JEE (Main) 2023 MCQ

Identify the correct statements from the following:
(A) Work done by a man in lifting a bucket out of a well by means of a rope tied to the bucket is negative.
(B) Work done by gravitational force in lifting a bucket out of a well by a rope tied to the bucket is negative.
(C) Work done by friction on a body sliding down an inclined plane is positive.
(D) Work done by an applied force on a body moving on a rough horizontal plane with uniform velocity in zero.
(E) Work done by the air resistance on an oscillating pendulum in negative.
Choose the correct answer from the options given below:

A

B and E only

B

A and C only

C

B, D and E only

D

B and D only

Question 3: Palette 3/145

Allen 2022 MCQ

A boy ties a stone of mass 100 g to the end of a 2 m long string and whirls it around in a horizontal plane. The string can withstand the maximum tension of 80 N. If the maximum speed with which the stone can revolve is $\frac{K}{\pi}$ rev./min. The value of K is: (Assume the string is massless and unstretchable) (JEE (Main) 2022)

A

400

B

300

C

600

D

800

Question 4: Palette 4/145

Allen JEE (Main) 2022 MCQ

Potential energy as a function of r is given by $U = \frac{A}{r^{10}} - \frac{B}{r^{5}}$ , where r is the interatomic distance, A and B are positive constants. The equilibrium distance between the two atoms will be:

A

$\left(\frac{A}{B}\right)^{\frac{1}{5}}$

B

$\left(\frac{B}{A}\right)^{\frac{1}{5}}$

C

$\left(\frac{2A}{B}\right)^{\frac{1}{5}}$

D

$\left(\frac{B}{2A}\right)^{\frac{1}{5}}$

Question 5: Palette 5/145

Allen JEE (Main) 2022 MCQ

A particle experiences a variable force $\vec{F} = (4x\hat{\imath} + 3y^2\hat{j})$ in a horizontal $x-y$ plane. Assume distance in meters and force in newton. If the particle moves from point (1, 2) to point (2, 3) in the $x-y$ plane, the Kinetic Energy changes by

A

$50.0 J$

B

$12.5 J$

C

$25.0 J$

D

$0 J$

Question 6: Palette 6/145

Allen JEE (Main) 2021 MCQ

If the Kinetic energy of a moving body becomes four times its initial Kinetic energy, then the percentage change in its momentum will be :

A

100%

B

200%

C

300%

D

400%

Question 7: Palette 7/145

Allen JEE (Main) 2021 MCQ

A porter lifts a heavy suitcase of mass 80 kg and at the destination lowers it down by a distance of 80 cm with a constant velocity. Calculate the work done by the porter in lowering the suitcase. (Take $g = 9.8 \, ms^{-2}$ )

A

-62720.0 J

B

-627.2 J

C

+627.2 J

D

784.0 J

Question 8: Palette 8/145

Allen JEE (Main) 2021 MCQ

A boy is rolling a 0.5 kg ball on the frictionless floor with the speed of $20 \, ms^{-1}$ . The ball gets deflected by an obstacle on the way. After deflection it moves with 5% of its initial kinetic energy. What is the speed of the ball now?

A

$19.0 \, ms^{-1}$

B

$4.47 \, ms^{-1}$

C

$14.41 \, ms^{-1}$

D

$1.00 \, ms^{-1}$

Question 9: Palette 9/145

Allen JEE (Main) 2021 MCQ

A body of mass 'm' dropped from a height 'h' reaches the ground with a speed of $0.8\sqrt{gh}$ . The value of work done by the air-friction is :

A

-0.68 mgh

B

mgh

C

1.64 mgh

D

0.64 mgh

Question 10: Palette 10/145

Allen JEE (Main) 2021 MCQ

Statement I: A cyclist is moving on an unbanked road with a speed of $7 \, kmh^{-1}$ and takes a sharp circular turn along a path of radius of 2m without reducing the speed. The static friction coefficient is 0.2. The cyclist will not slip and pass the curve ( $g = 9.8 \, m/s^{2}$ ) Statement II: If the road is banked at an angle of $45^{\circ}$ , cyclist can cross the curve of 2m radius with the speed of $18.5 \, kmh^{-1}$ without slipping. In the light of the above statements, choose the correct answer from the options given below.

A

Statement I is incorrect and statement II is correct

B

Statement I is correct and statement II is incorrect

C

Both statement I and statement II are false

D

Both statement I and statement II are true

Question 11: Palette 11/145

Allen JEE (Main) 2021 MCQ

The normal reaction 'N' for a vehicle of 800 kg mass, negotiating a turn on a 30° banked road at maximum possible speed without skidding is ____ × $10^{3}$ kg m/s $^{2}$ . [cos 30° = 0.87, $\mu_{s} = 0.2$ ]

A

10.2

B

7.2

C

12.4

D

6.96

Question 12: Palette 12/145

Allen JEE (Main) 2021 MCQ

A particle of mass m is suspended from a ceiling through a string of length L. The particle moves in a horizontal circle of radius r such that $r = \frac{L}{\sqrt{2}}$ . The speed of particle will be:

A

$\sqrt{rg}$

B

$\sqrt{2rg}$

C

$2\sqrt{rg}$

D

$\sqrt{\frac{rg}{2}}$

Question 13: Palette 13/145

Allen JEE (Main) 2021 MCQ

A huge circular arc of length 4.4 ly subtends an angle '4s' at the centre of the circle. How long it would take for a body to complete 4 revolution if its speed is 8 AU per second? Given: $1 \, ly = 9.46 \times 10^{15} \, m$ $1 \, AU = 1.5 \times 10^{11} \, m$

A

$4.1 \times 10^{8} \, s$

B

$4.5 \times 10^{10} \, s$

C

$3.5 \times 10^{6} \, s$

D

$7.2 \times 10^{8} \, s$

Question 14: Palette 14/145

Allen JEE (Main) 2020 MCQ

A 60 HP electric motor lifts an elevator having a maximum total load capacity of 2000 kg. If the frictional force on the elevator is 4000 N, the speed of the elevator at full load is close to: (1 HP = 746 W, g = 10 ms $^{-2}$ )

A

1.7 ms $^{-1}$

B

2.0 ms $^{-1}$

C

1.9 ms $^{-1}$

D

1.5 ms $^{-1}$

Question 15: Palette 15/145

Allen JEE (Advanced) 2020 MSQ

A student skates up a ramp that makes an angle $30^{\circ}$ with the horizontal. He/she starts (as shown in the figure) at the bottom of the ramp with speed $v_{0}$ and wants to turn around over a semicircular path xyz of radius R during which he/she reaches a maximum height h (at point y) from the ground as shown in the figure. Assume that the energy loss is negligible and the force required for this turn at the highest point is provided by his/her weight only. Then (g is the acceleration due to gravity)

A

$v_{0}^{2} - 2gh = \frac{1}{2} gR$

B

$v_{0}^{2} - 2gh = \frac{\sqrt{3}}{2} gR$

C

the centripetal force required at points x and z is zero

D

the centripetal force required is maximum at points x and z

Question 16: Palette 16/145

Allen JEE (Advanced) 2018 MSQ

A particle of mass m is initially at rest at the origin. It is subjected to a force and starts moving along the x-axis. Its kinetic energy K changes with time as $dK/dt = \gamma t$ , where $\gamma$ is a positive constant of appropriate dimensions. Which of the following statements is (are) true?

A

The force applied on the particle is constant

B

The speed of the particle is proportional to time

C

The distance of the particle from the origin increases linearly with time

D

The force is conservative

Question 17: Palette 17/145

Allen JEE (Advanced) 2016 MCQ

Comprehension Passage:

A frame of reference that is accelerated with respect to an inertial frame of reference is called a non-inertial frame of reference. A coordinate system fixed on a circular disc rotating about a fixed axis with a constant angular velocity $\omega$ is an example of a non-inertial frame of reference. The relationship between the force $\vec{F}_{rot}$ experienced by a particle of mass m moving on the rotating disc and the force $\vec{F}_{in}$ experienced by the particle in an inertial frame of reference is $\vec {F} _ {r o t} = \vec {F} _ {i n} + 2 m (\vec {v} _ {r o t} \times \vec {\omega}) + m (\vec {\omega} \times \vec {r}) \times \vec {\omega},$ where $\vec{v}_{rot}$ is the velocity of the particle in the rotating frame of reference and $o\vec{r}$ is the position vector of the particle with respect to the centre of the disc. Now consider a smooth slot along a diameter of a disc of radius R rotating counter-clockwise with a constant angular speed $\omega$ about its vertical axis through its center. assign a coordinate system with the origin at the centre of the disc, the x-axis along the slot, the y-axis perpendicular to the slot and the z-axis along the rotation axis $(\vec{\omega} = \omega\hat{k})$ . A small block of mass m is gently placed in the slot at $\vec{r} = (r/2)\hat{t}$ at t = 0 and is constrained to move only along the slot. The distance r of the block at time t is:

A

$\frac{R}{4}\left(e^{2\omega t} + e^{-2\omega t}\right)$

B

$\frac{R}{2}\cos 2\omega t$

C

$\frac{R}{2}\cos \omega t$

D

$\frac{R}{4}\left(e^{\omega t} + e^{-\omega t}\right)$

Question 18: Palette 18/145

Allen JEE (Advanced) 2016 MCQ

Comprehension Passage:

A frame of reference that is accelerated with respect to an inertial frame of reference is called a non-inertial frame of reference. A coordinate system fixed on a circular disc rotating about a fixed axis with a constant angular velocity $\omega$ is an example of a non-inertial frame of reference. The relationship between the force $\vec{F}_{rot}$ experienced by a particle of mass m moving on the rotating disc and the force $\vec{F}_{in}$ experienced by the particle in an inertial frame of reference is $\vec {F} _ {r o t} = \vec {F} _ {i n} + 2 m (\vec {v} _ {r o t} \times \vec {\omega}) + m (\vec {\omega} \times \vec {r}) \times \vec {\omega},$ where $\vec{v}_{rot}$ is the velocity of the particle in the rotating frame of reference and $o\vec{r}$ is the position vector of the particle with respect to the centre of the disc. Now consider a smooth slot along a diameter of a disc of radius R rotating counter-clockwise with a constant angular speed $\omega$ about its vertical axis through its center. assign a coordinate system with the origin at the centre of the disc, the x-axis along the slot, the y-axis perpendicular to the slot and the z-axis along the rotation axis $(\vec{\omega} = \omega\hat{k})$ . A small block of mass m is gently placed in the slot at $\vec{r} = (r/2)\hat{t}$ at t = 0 and is constrained to move only along the slot. The net reaction of the disc on the block is:

A

$-m\omega^{2}R\cos\omega t\hat{j}-mg\hat{k}$

B

$m\omega^{2}R\sin\omega t\hat{j}-mg\hat{k}$

C

$\frac{1}{2}m\omega^{2}R\left(e^{\omega t}-e^{-\omega t}\right)\hat{j}+mg\hat{k}$

D

$\frac{1}{2}m\omega^{2}R\left(e^{2\omega t}-e^{-2\omega t}\right)\hat{j}+mg\hat{k}$

Question 19: Palette 19/145

Allen JEE (Advanced) 2014 MCQ

A wire, which passes through the hole in a small bead, is bent in the form of quarter of a circle. The wire is fixed vertically on ground as shown in the figure. The bead is released from near the top of the wire and it slides along the wire without friction. As the bead moves from A to B, the force it applies on the wire is :-

A

Always radially outwards

B

Always radially inwards

C

Radially outwards initially and radially inwards later.

D

Radially inwards initially and radially outwards later.

Question 20: Palette 20/145

Allen JEE (Advanced) 2013 MCQ

The work done on a particle of mass m by a force, $K\left[\frac{x}{(x^{2}+y^{2})^{3/2}}\hat{i}+\frac{y}{(x^{2}+y^{2})^{3/2}}\hat{j}\right]$ (K being a constant of appropriate dimensions), when the particle is taken from the point (a, 0) to the point (0, a) along a circular path of radius a about the origin in the x-y plane is :-

A

$\frac{2K\pi}{a}$

B

$\frac{K\pi}{a}$

C

$\frac{K\pi}{2a}$

D

0

Question 21: Palette 21/145

Allen JEE (Advanced) 2013 MCQ

Comprehension Passage:

A small block of mass 1 kg is released from rest at the top of a rough track. The track is a circular arc of radius 40 m. The block slides along the track without toppling and a frictional force acts on it in the direction opposite to the instantaneous velocity. The work done in overcoming the friction up to the point Q, as shown in the figure below, is 150 J. $g = 1 0 \text { m s } ^ {- 2})$ The magnitude of the normal reaction that acts on the block at the point $Q$ is

A

$7.5 N$

B

$8.6 N$

C

$11.5 N$

D

$22.5 N$

Question 22: Palette 22/145

Allen JEE (Advanced) 2013 MCQ

Comprehension Passage:

A small block of mass 1 kg is released from rest at the top of a rough track. The track is a circular arc of radius 40 m. The block slides along the track without toppling and a frictional force acts on it in the direction opposite to the instantaneous velocity. The work done in overcoming the friction up to the point Q, as shown in the figure below, is 150 J. $g = 1 0 \text { m s } ^ {- 2})$ The speed of the block when it reaches the point $Q$ is

A

$5 \mathrm{~ms}^{-1}$

B

$10 \mathrm{~ms}^{-1}$

C

$10 \sqrt{3} \mathrm{~ms}^{-1}$

D

$20 \mathrm{~ms}^{-1}$

Question 23: Palette 23/145

Allen MCQ

Two moving particles P and Q are 10m apart at a certain instant. The velocity of P is 8m/s making an angle of $30^{\circ}$ with the line joining P and Q and Q is moving with 6m/s making an angle of $30^{\circ}$ with PQ as shown in figure. Then angular velocity of P with respect to Q is

A

Zero

B

0.1 rad/s

C

0.4 rad/s

D

0.7 rad/s

Question 24: Palette 24/145

Allen MCQ

The second's hand of a watch has length 6 cm. Speed of end point and magnitude of difference of velocities at two perpendicular positions will be

A

$2\pi$ and 0 mm/s

B

$2\sqrt{2}\pi$ and 4.44 mm/s

C

$2\sqrt{2}\pi$ and $2\pi$ mm/s

D

$2\pi$ and $2\sqrt{2}\pi$ mm/s

Question 25: Palette 25/145

Allen MCQ

The magnitude of displacement of a particle moving in a circle of radius $a$ with constant angular speed $\omega$ varies with time $t$ as:

A

$2a\sin \omega t$

B

$2a\sin \frac{\omega t}{2}$

C

$2a\cos \omega t$

D

$2a\cos \frac{\omega t}{2}$

Question 26: Palette 26/145

Allen MCQ

Two particles describe the same circle of radius R in the same direction with the same speed V. then at the given instant relative angular velocity of 2 with respect to 1 will be :

A

$\frac{2v\sin\frac{\theta}{2}}{R}$

B

$\frac{v}{2R\sin\frac{\theta}{2}}$

C

$\frac{v}{R}$

D

$\frac{v\cos\frac{\theta}{2}}{R}$

Question 27: Palette 27/145

Allen MCQ

An insect trapped in a circular groove of some radius moves along the groove steadily and completes 10 revolutions in 100s. Choose the correct statement:-

A

The angular velocity of the insect is $10 \, \text{rad/s}$

B

The angular velocity of the insect is $\frac{\pi}{5} \, \text{rad/s}$

C

The angular velocity of the insect is $\frac{2\pi}{5} \, \text{rad/s}$

D

The angular velocity of the insect can't be defined as radius is unknown.

Question 28: Palette 28/145

Allen MCQ

At $t = 0$ a flywheel is rotating with angular velocity $\omega_0$ . It then undergoes uniform angular acceleration for a time $t_1$ , at the end of which the angular velocity is $\omega_1$ . How many revolutions did the flywheel make during this time interval?

A

$\frac{1}{2} (\omega_0 + \omega_1)t_1$

B

$\frac{\omega_0t_1}{2\pi}$

C

$\frac{\omega_1t_1}{2\pi}$

D

$\frac{(\omega_0 + \omega_1)t_1}{4\pi}$

Question 29: Palette 29/145

Allen MCQ

Which vector in the figures best represents the acceleration of a pendulum mass at the intermediate point in its swing?

A

B

C

D

Question 30: Palette 30/145

Allen MCQ

A particle $A$ moves along a circle of radius $R = 50 \, \text{cm}$ so that its radius vector $r$ relative to the point $O$ (figure) rotates with the constant angular velocity $\omega = 0.40 \, \text{rad/s}$ . Then modulus of the velocity of the particle, and the modulus of its total acceleration will be

A

v = 0.4 m/s, $a = 0.4 \, m/s^{2}$

B

$v = 0.32 \, m/s, a = 0.32 \, m/s^2$

C

v = 0.32 m/s, $a = 0.4 \, m/s^{2}$

D

$v = 0.4m / s, a = 0.32m / s^2$

Question 31: Palette 31/145

Allen MCQ

A particle is projected with initial velocity 50m/sec making an angle of $53^{\circ}$ with horizontal. Find the radius of curvature of particle when velocity of particle is perpendicular to the direction of its initial velocity:

A

175.8 m

B

200.3 m

C

165.6 m

D

170.8 m

Question 32: Palette 32/145

Allen MCQ

A particle starts from rest and performing circular motion of constant radius with speed given by $v = \alpha \sqrt{x}$ where $\alpha$ is a constant and x is the distance covered. The correct graph of magnitude of its tangential acceleration ( $a_{t}$ ) and centripetal acceleration ( $a_{c}$ ) versus t will be:

A

B

C

D

Question 33: Palette 33/145

Allen MCQ

A particle is moving in a circular path. The acceleration and momentum of the particle at a certain moment are $\vec{a}=(4\hat{i}+3\hat{j})m/s^{2}$ and $\vec{p}=(8\hat{i}-6\hat{j})kg-m/s$ . The motion of the particle is

A

uniform circular motion

B

accelerated circular motion

C

de-accelerated circular motion

D

can not say anything with $\vec{a}$ and $\vec{p}$ only

Question 34: Palette 34/145

Allen MCQ

A point P moves in counter clockwise direction on a circular path as shown in the figure. The movement of 'P' is such that it sweeps out a length $s = t^{2} + 5$ , where s is in metres and t is in seconds. The radius of the path is 20 m. The acceleration of 'P' when $t = 5\sqrt{\frac{3}{10}}$ seconds is nearly:

A

$2m / s^2$

B

$1.5m / s^2$

C

$2.5m / s^2$

D

$3m / s^2$

Question 35: Palette 35/145

Allen MCQ

A particle is acted upon by a force of constant magnitude which is always perpendicular to the velocity of the particle. The motion of the particle takes place in a plane, it follows that-

A

Its velocity is constant

B

Its acceleration is constant

C

Its kinetic energy is constant

D

It moves in a straight line

Question 36: Palette 36/145

Allen MCQ

A particle of mass m is tied to a light string and rotated with a speed v along a circular path of radius r. If T = tension in the string and mg = gravitational force on the particle then the actual forces acting on the particle are :

A

mg and T only

B

mg, T and an additional force of $\frac{mv^{2}}{r}$ directed inwards.

C

$mg, T$ and an additional force of $\frac{mv^2}{r}$ directed outwards.

D

only a force $\frac{mv^2}{r}$ directed outwards.

Question 37: Palette 37/145

Allen MCQ

An unbanked circular highway curve on level ground makes a turn of $90^{\circ}$ . The highway carries traffic at $108 km-hr^{-1}$ , and the centripetal force on a vehicle is not to exceed 1/10 of its weight. What is the minimum length of the curve?

A

1.4 km

B

1 km

C

0.6 km

D

1.7 km

Question 38: Palette 38/145

Allen MCQ

A long horizontal rod has a bead which can slide along its length and is initially placed at a distance L from one end A of the rod. The rod is set in angular motion about A with a constant angular acceleration, $\alpha$ . If the coefficient of friction between the rod and bead is $\mu$ , and gravity is neglected, then the time after which the bead starts slipping is:-

A

$\sqrt{\frac{\mu}{\alpha}}$

B

$\frac{\mu}{\sqrt{\alpha}}$

C

$\frac{1}{\sqrt{\mu\alpha}}$

D

Infinitesimal

Question 39: Palette 39/145

Allen MCQ

A car is moving along a circular track with tangential acceleration of magnitude $a_{0}$ . It just start to slip at speed $v_{0}$ then find radius of circle. (Coefficient of friction is $\mu$ )

A

$\frac{v_{0}^{2}}{\sqrt{(\mu g)^{2} + a_{0}^{2}}}$

B

$\frac{v_{0}^{2}}{\sqrt{(\mu g)^{2} - a_{0}^{2}}}$

C

$\frac{v_{0}^{2}}{\sqrt{(\mu g)^{2} - 2a_{0}^{2}}}$

D

$\frac{v_{0}^{2}}{\sqrt{(\mu g)^{2} + 2a_{0}^{2}}}$

Question 40: Palette 40/145

Allen MCQ

A bead is constrained to move on rod in gravity free space as shown in figure. The rod is rotating with angular velocity $\omega$ and angular acceleration $\alpha$ about its end. If $\mu$ is coefficient of friction. Mark the correct option. (Rod rotates in the plane of paper)

A

If $\mu = \frac{\omega^2}{\alpha}$ friction on bead is static in nature

B

If $\mu > \frac{\omega^2}{\alpha}$ friction on bead is kinetic in nature

C

If $\mu < \frac{\omega^2}{\alpha}$ friction is static

D

If bead does not slide relative to rod. Friction will not exist between bead and rod.

Question 41: Palette 41/145

Allen MCQ

A horizontal turn table is initially at rest. It starts to rotate in horizontal plane with constant angular acceleration $3 \, rad/s^{2}$ . A block of mass $1 \, kg$ is placed 1 metre away from the axis of rotation on the turn table. A horizontal force $F = 5t \, N$ where t is in seconds, is acting on the block always towards the center of rotation. The block begins to slip at t = 1s. The coefficient of friction between block and turn table is

A

0.2

B

0.3

C

0.4

D

0.5

Question 42: Palette 42/145

Allen MCQ

A cricket player throws a ball straight upward in the air. It rises and falls back to the hands of the player. During the catch, his hands are displaced downward a few inches as the ball slows down, then.

A

During the catch while the ball is in hand and is moving downward, the work done by the player on the ball is positive.

B

During the throw work done by gravity is positive.

C

During the throw while ball is in hand and moving upward, work done by ball on hand is negative.

D

During the catch while the ball is in hand and moving downward, work done by hand plus gravity on the ball is positive.

Question 43: Palette 43/145

Allen MCQ

A force of magnitude of 30 N acting along $\hat{i} + \hat{j} + \hat{k}$ , displaces a particle from point (2, 4, 1) m to (3, 5, 2) m. The work done during this displacement is :-

A

90 J

B

30 J

C

$30\sqrt{3}J$

D

$30/\sqrt{3}J$ .

Question 44: Palette 44/145

Allen MCQ

A rope is used to lower vertically a block of mass M by a distance x with a constant downward acceleration g/2. The work done by the rope on the block is:

A

Mgx

B

$\frac{1}{2}$ Mgx $^{2}$

C

$-\frac{1}{2}$ Mgx

D

Mgx $^{2}$

Question 45: Palette 45/145

Allen MCQ

A block of mass 10 kg is released from the top of smooth wedge (whose inclination at θ = 37°) which is moving with a constant velocity 10 m/s. The work done by the force applied by wedge on the block in 2 sec will be (in joules) :-

A

1600

B

960

C

Zero

D

720

Question 46: Palette 46/145

Allen MCQ

In the figure shown find work done by tension in string on 3 kg block in first two second, from beginning of motion. The system is set free at t = 0 ( $g = 10 \, m/s^{2}$ ) (Assume pulley and string is ideal)

A

-96 Joule

B

-48 Joule

C

+24 Joule

D

96 Joule

Question 47: Palette 47/145

Allen MCQ

Force acting on a particle is $(2\hat{i} + 3\hat{j})$ N. Work done by this force is zero, when a particle is moved on the line $3y + kx = 5$ . Here value of k is

A

2

B

4

C

6

D

8

Question 48: Palette 48/145

Allen MCQ

A particle is moved from $(0,0)$ to $(a,a)$ under a force from $\vec{F}=\left(3\hat{i}+4\hat{j}\right)$ two paths. Path 1 is OP and path 2 is OQP. Let $W_{1}$ and $W_{2}$ be the work done by this force in these two paths. Then:

A

$W_{1} = W_{2}$

B

$W _ {1} = 2 W _ {2}$

C

$W _ {2} = 2 W _ {1}$

D

$W_{2} = 4W_{1}$

Question 49: Palette 49/145

Allen MCQ

The components of a force acting on a particle are varying according to the graphs shown. To reach at point B (8, 20, 0) from point A(0, 5, 12) the particle moves on paths parallel to x-axis then y-axis and then z-axis, then work done by this force is :-

A

$192J$

B

$58J$

C

250 J

D

125 J

Question 50: Palette 50/145

Allen MCQ

The work done by the force $\vec{F} = x^{2}\hat{i} + y^{2}\hat{j}$ around the path shown in the figure is

A

$\frac{2}{3} a^{3}$

B

zero

C

$a^{3}$

D

$\frac{4}{3} a^{3}$

Question 51: Palette 51/145

Allen MCQ

By applying a force $\vec{F}=(3xy-5z)\hat{j}-4\hat{k}$ a particle is moved along the path $y=x^{2}$ from point $(0,0,0)$ to the point $(2,4,0)$ . The work done by the F on the particle is :-

A

$\frac{280}{5}$

B

$\frac{140}{5}$

C

$\frac{232}{5}$

D

$\frac{192}{5}$

Question 52: Palette 52/145

Allen MCQ

A man who is running has half the kinetic energy of the boy of half his mass. The man speeds up by 1 m/s and then has the same kinetic energy as the boy. The original speed of the man was

A

$\sqrt{2}$ m/s

B

$(\sqrt{2}-1)$ m/s

C

2 m/s

D

$(\sqrt{2}+1)$ m/s

Question 53: Palette 53/145

Allen MCQ

A body of mass 0.5 kg travels in a straight line with velocity $v = ax^{3/2}$ where $a = 5 m^{-1/2}s^{-1}$ . The work done by the net force during its displacement from x = 0 to x = 2 m is

A

1.5 J

B

50 J

C

10 J

D

100 J

Question 54: Palette 54/145

Allen MCQ

In a shotput event an athlete throws the shotput of mass 10 kg with an initial speed of 1m/s at $45^{\circ}$ from a height 1.5 m above ground. Assuming air resistance to be negligible and acceleration due to gravity to be $10\ m/s^{2}$ , the kinetic energy of the shotput when it just reaches the ground will be

A

2.5 J

B

5.0 J

C

52.5 J

D

155.0 J

Question 55: Palette 55/145

Allen MCQ

In the figure, a block slides along a track from one level to a higher level, by moving through an intermediate valley. The track is frictionless until the block reaches the higher level. There a frictional force stops the block in a distance d. The block's initial speed $v_{0}$ is 6 m/s, the height difference h is 1.1m and the coefficient of kinetic friction $\mu$ is 0.6. The value of d is

A

1.17 m

B

1.71 m

C

7.11 m

D

11.7 m

Question 56: Palette 56/145

Allen MCQ

A small object with mass $m = 2 \, kg$ is attached to the free end of an ideal spring with $k = 10 \, Nm^{-1}$ . The other end of the spring is connected to a fixed frictionless pivot located at the origin $O$ as shown in the Figure. The relaxed length of the spring is $1 \, m$ . An unspecified force $F$ carries the object at rest from point $A$ to point $B$ . At point $B$ , the object has the speed $5 \, ms^{-1}$ . The work done by the force $F$ is (Neglect gravity):-

A

$130 \, J$

B

$80 \, J$

C

$-105 \, J$

D

$-80 \, J$

Question 57: Palette 57/145

Allen MCQ

In the arrangement shown a block of mass 2kg is pressed against a spring which is compressed by 10 cm and is applying a force of 80 N to the block. The spring is not attached to the block. The coefficient of friction is 0.5. When the block is released it moves up the plane by some distance and then comes back down. The kinetic energy of the block just before it starts compressing the spring again is

A

$2J$

B

$1.6J$

C

$0.8J$

D

$0.4J$

Question 58: Palette 58/145

Allen MCQ

A block of mass 10kg is attached to a relaxed spring. Other end is attached to ceiling and block is released keeping the spring in vertical position. The spring is such that it has elongation of 10 cm under a force of 100 N. The maximum velocity of block in subsequent motion is:-

A

1 m/s

B

1.5 m/s

C

2.0 m/s

D

2.5 m/s

Question 59: Palette 59/145

Allen MCQ

A light spring of length 20 cm and force constant 2 N/cm is placed vertically on a table. A small block of mass 1 kg falls on it. The length h from the surface of the table at which the block will have the maximum velocity is:

A

20 cm

B

15 cm

C

10 cm

D

5cm

Question 60: Palette 60/145

Allen MCQ

In the figure shown, the system is released from rest. Find the velocity of block A when block B has fallen a distance ' $\ell'$ . Assume all pulleys to be massless and frictionless.

A

$\sqrt{\frac{g\ell}{5}}$

B

$\sqrt{g\ell}$

C

$\sqrt{5g\ell}$

D

None of these

Question 61: Palette 61/145

Allen MCQ

A radial force $\vec{F} = \frac{A}{r^2}\hat{r}$ acts on a particle. The work done by the force in displacing a particle from (1, 0) to (3, 4) is:

A

$\frac{A}{5}$

B

$\frac{A}{2}$

C

$\frac{4A}{5}$

D

$\frac{4A}{3}$

Question 62: Palette 62/145

Allen MCQ

When a conservative force does positive work on a body

A

the potential energy increases

B

the potential energy decreases

C

total energy increases

D

total energy decreases

Question 63: Palette 63/145

Allen MCQ

A wedge of mass M fitted with a spring of stiffness 'k' is kept on a smooth horizontal surface. A rod of mass m is kept on the wedge as shown in the figure. System is in equilibrium. Assuming that all surfaces are smooth, the potential energy stored in the spring is:

A

$\frac{mg^2\tan^2\theta}{2k}$

B

$\frac{m^2g\tan^2\theta}{2k}$

C

$\frac{m^2g^2\tan^2\theta}{2k}$

D

$\frac{m^2g^2\tan^2\theta}{k}$

Question 64: Palette 64/145

Allen MCQ

A body with mass 2 kg moves in one direction in the presence of a force which is described by the potential energy graph. If the body is released from rest at x = 2m, then its speed when it crosses x = 5 m is

A

zero

B

$1ms^{-1}$

C

$2ms^{-1}$

D

$3ms^{-1}$

Question 65: Palette 65/145

Allen MCQ

The potential energy between two atoms in a molecule is given by $U = ax^{3} - bx^{2}$ where a and b are positive constants and x is the distance between the atoms. The atom is in stable equilibrium when x is equal to :-

A

0

B

$\frac{2b}{3a}$

C

$\frac{3a}{2b}$

D

$\frac{3b}{2a}$

Question 66: Palette 66/145

Allen MCQ

A conservative force $\vec{F}=(4x-8)\hat{i}N$ acts on a particle that moves on the x-axis. The potential energy associated with this force is assigned a value of 8J at x=0. Which of the following best represents variation in potential energy with position coordinate x?

A

B

C

D

Question 67: Palette 67/145

Allen MCQ

Potential energy curve U of a particle as function of the position of a particle is shown. The particle has total mechanical energy E of 3.0 joules.

A

It can never be present at x = 0 m.

B

It can never be present at x = 5 m

C

At x = 2 its kinetic energy is 0 J

D

At x = 1 its kinetic energy 3 J

Question 68: Palette 68/145

Allen MCQ

The potential energy $U(x)$ of a particle moving along x-axis is shown in figure. Which one of the following graphs represents the variation of kinetic energy of this particle if its total mechanical energy is $E_{0}$ ?

A

B

C

D

Question 69: Palette 69/145

Allen MCQ

A stone is tied to a string of length 2 m is whirled in a vertical circle with the other end of the string at the centre. At a certain instant of time, the stone is at its lowest position and has a speed 10 m/s. The magnitude of the change in its velocity as it reaches a position where the string is horizontal is

A

$\sqrt{60}$

B

$\sqrt{40}$

C

$\sqrt{80}$

D

$\sqrt{160}$

Question 70: Palette 70/145

Allen MCQ

A small block slides down from rest at point A on the surface of a smooth circular cylinder, as shown. At point B, the block falls off (leaves) the cylinder. The equation relating the angles $\theta_{1}$ and $\theta_{2}$ is given by

A

$\sin \theta_{2} = \frac{2}{3}\sin \theta_{1}$

B

$\sin \theta_{2} = \frac{3}{2}\sin \theta_{1}$

C

$\cos \theta_{2} = \frac{2}{3}\cos \theta_{1}$

D

$\cos \theta_{2} = \frac{3}{2}\cos \theta_{1}$

Question 71: Palette 71/145

Allen MCQ

A particle of mass m is connected from a light inextensible string of length $\ell$ such that it behaves as a simple pendulum. Now string is pulled to point A making an angle $\theta_{1}$ with the vertical and it is released from the point A then.

A

Speed of the particle when string makes an angle $\theta_{2}$ with vertical, is $\sqrt{2g\ell(cos\theta_{2}-cos\theta_{1})}$

B

Speed of the particle when string makes an angle $\theta_{2}$ with vertical, is $\sqrt{g\ell(cos\theta_2 - cos\theta_1)}$

C

The tension in the string when string makes an angle $\theta_{2}$ with vertical is $mg$ ( $4\cos\theta_{2}-2\cos\theta_{1}$ )

D

The tension in the string when string makes an angle $\theta_{2}$ with vertical is $mg\left(\cos\theta_{2}-\cos\theta_{1}\right)$

Question 72: Palette 72/145

Allen MCQ

A stone of mass m at the end of a string of length L is whirled in a vertical circle at a constant speed. The tension in the string will be maximum when the stone is

A

At the top of the circle

B

Half way down from the top

C

Quarter way down from the top

D

At the bottom of the circle

Question 73: Palette 73/145

Allen MCQ

Two bodies of mass m and 4m are attached to a string as shown in the figure. The body of mass m hanging from string is executing oscillation with angular amplitude $60^{\circ}$ while other body is at rest on a horizontal surface. The minimum coefficient of friction between the mass 4m and the horizontal surface is :

A

$1 / 4$

B

$3 / 4$

C

$1 / 2$

D

$1 / 8$

Question 74: Palette 74/145

Allen MCQ

As shown in figure BEF is a fixed vertical circular tube. A block of mass m starts moving in the tube at point B with velocity V towards E. It is just able to complete the vertical circle, then choose the INCORRECT option :-

A

velocity at B must be $\sqrt{3Rg}$

B

velocity at F must be $\sqrt{2Rg}$

C

normal reaction at point F is 2mg

D

the normal reaction at point E is 6 mg

Question 75: Palette 75/145

Allen MCQ

A body of mass m accelerates uniformly from rest to $v_{1}$ in time $t_{1}$ . The instantaneous power delivered to the body as a function of time t is-

A

$\frac{mv_{1}t}{t_{1}}$

B

$\frac{mv_{1}^{2}t}{t_{1}^{2}}$

C

$\frac{mv_{1}t^{2}}{t_{1}}$

D

$\frac{mv_{1}^{2}t}{t_{1}}$

Question 76: Palette 76/145

Allen MCQ

Assume the aerodynamic drag force on a car is proportional to its speed. If the power output from the engine is doubled, then the maximum speed of the car.

A

is unchanged

B

increases by a factor of $\sqrt{2}$

C

is also doubled

D

increases by a factor of four

Question 77: Palette 77/145

Allen MCQ

A particle (bob) of mass 1 kg is performing vertical circular motion. Then :

A

Power delivered by tension at point 4) is negative.

B

Power delivered by tension at point 2) is positive.

C

Power delivered by gravitational force at point 3) is zero

D

Power delivered by gravitational force at point 4) is negative.

Question 78: Palette 78/145

Allen MSQ

A particle is moving along a circular path as shown in figure below. The instantaneous velocity of the particle is $\vec{v}=(4m/s)\hat{i}-(3\mathrm{~m/s})\hat{j}$ . The particle may be moving through

A

first quadrant

B

second quadrant

C

third quadrant

D

fourth quadrant

Question 79: Palette 79/145

Allen MSQ

The second-hand of a tower clock is 1.00 m long. Which of the following is/are CORRECT statement(s).

A

Linear velocity of the tip of the hand is $\frac{10\pi}{3} cm / s$ .

B

Angular velocity of the hand is $\frac{\pi}{30} rad / s$ .

C

Average acceleration of the tip of the hand in 15 minutes is zero.

D

Instantaneous acceleration of the tip of the hand is $\frac{\pi^2}{9} cm / s^2$ .

Question 80: Palette 80/145

Allen MSQ

Let $\vec{v}(t)$ be the velocity of a particle at time t. Then :

A

$|d\vec{v}(t) / dt|$ and $d|\vec{v}(t)| / dt$ are always equal

B

$|d\vec{v}(t)/dt|$ and $d|\vec{v}(t)|/dt$ may be equal

C

$d|\vec{v}(t)| / dt$ can be zero while $|d\vec{v}(t) / dt|$ is not zero

D

$d| \vec{v}(t)| / dt \neq 0$ implies $|d\vec{v}(t) / dt| \neq 0$

Question 81: Palette 81/145

Allen MSQ

A particle is in motion on the x-axis. The variation of its velocity with position is as shown. The graph is circle and its equation is $x^{2} + v^{2} = 1$ , where x is in m and v in m/s. The CORRECT statement(s) is/are :-

A

When x is positive, acceleration is negative.

B

When x is negative, acceleration is positive.

C

At Q, acceleration has magnitude $\frac{1}{\sqrt{2}}$ m/s $^{2}$

D

At S, acceleration is infinite.

Question 82: Palette 82/145

Allen MSQ

Two bodies A and B are moving with same constant speed v in clockwise direction in a horizontal circle of radius R and are initially diametrically opposite. The particle B now achieves a tangential acceleration $a \, m/s^{2}$ . Then which option is/are correct?

A

they collide after time $\sqrt{\frac{\pi R}{a}}$

B

they collide after time $\sqrt{\frac{2\pi R}{a}}$

C

relative velocity just before collision is $\sqrt{\pi a R}$

D

relative velocity just before collision is $\sqrt{2\pi aR}$

Question 83: Palette 83/145

Allen MSQ

An ant travels along a long rod with a constant velocity relative to the rod starting from the origin. The rod is kept initially along the positive x-axis. At t = 0, the rod also starts rotating with an angular velocity $\omega$ (anticlockwise) in x-y plane about origin. Then

A

the position of the ant at any time t is $\vec{r}=ut\left[\cos\omega t\hat{i}+\sin\omega t\hat{j}\right]$

B

the speed of the ant at any time t is $u\sqrt{1+\omega^{2}t^{2}}$

C

the magnitude of the tangential acceleration of the ant at any time t is $\frac{\omega^{2}tu}{\sqrt{1+\omega^{2}t^{2}}}$

D

the speed of the ant at any time t is $\sqrt{1+2\omega^{2}t^{2}u}$

Question 84: Palette 84/145

Allen MSQ

In the shown figure inside a fixed hollow cylinder with vertical axis a pendulum is rotating in a conical path with its axis same as that of the cylinder with uniform angular velocity. Radius of cylinder is 30 cm, length of string is 50 cm and mass of bob is 400 gm. The bob makes contact with the inner frictionless wall of the cylinder while moving :-

A

The minimum value of angular velocity of the bob so that it does not leave contact is 5 rad/s

B

Tension in the string is 5 N for all values of angular velocity

C

For angular velocity of 10 rad/s the bob pushes the cylinder with a force of 9 N

D

For angular velocity of 10 rad/s, tension in the string is 20N

Question 85: Palette 85/145

Allen MSQ

A particle P of mass m is attached to a vertical axis by two strings AP and BP of length l each. The separation AB = l. P rotates around the axis with an angular velocity $\omega$ . The tensions in the two strings are $T_{1}$ and $T_{2}$

A

$T_{1} = T_{2}$

B

$T_{1} + T_{2} = m\omega^{2}l$

C

$T_{1} - T_{2} = 2mg$

D

$BP$ will remain taut only if $\omega \geq \sqrt{\frac{2g}{\ell}}$

Question 86: Palette 86/145

Allen MSQ

On a circular turn table rotating about its center horizontally with uniform angular velocity $\omega$ rad/s placed two blocks of mass 1 kg and 2 kg, on a diameter symmetrically about center. Their separation is 1m and friction is sufficient to avoid slipping. The spring between them as shown is stretched and applied force of 5N. If $f_{1}$ and $f_{2}$ are values of friction on 1 kg and 2kg block respectively:-

A

For $\omega = 2\mathrm{rad} / s, f_{1} = 3N$ and $f_{2} = 1N$

B

For $\omega = 3\mathrm{rad} / s, f_{1} = 0.5N$ and $f_{2} = 4N$

C

For $\omega = \sqrt{10}$ rad/s, $f_{1} = 0$ and $f_{2} = 5N$

D

For $\omega = \sqrt{10}$ rad/s, $f_{1} = 0$ and $f_{2} = 0N$

Question 87: Palette 87/145

Allen MSQ

A heavy particle is tied to the end $A$ of a string of length $1.6 \, m$ . Its other end $O$ is fixed. It revolves as a conical pendulum with the string making $60^\circ$ with the vertical. Then

A

its period of revolution is $\frac{4\pi}{7}$ sec

B

the tension in the string is double the weight of the particle

C

the velocity of the particle = $2.8\sqrt{3}$ m/s

D

the centripetal acceleration of the particle is $9.8\sqrt{3}$ m/s $^{2}$

Question 88: Palette 88/145

Allen MSQ

A car is moving with constant speed on a rough banked road. Figure shows the free body diagram of car in three situation A, B and C respectively:- A) B) C)

A

Car in A has more speed than car in C

B

Car in A has less speed than car in B

C

FBD for car in A is not possible

D

If $\mu > \tan \theta$ the $FBD$ for car $C$ is not possible

Question 89: Palette 89/145

Allen MSQ

Which of the following statements is TRUE for a system comprising of two bodies in contact exerting frictional force on each other :

A

total work done by static friction on whole system is always zero.

B

work done by static friction on a body is always zero

C

work done by kinetic friction on a body is always negative

D

total work done by internal kinetic friction on whole system is always negative

Question 90: Palette 90/145

Allen MSQ

Select the correct alternatives :-

A

Work energy theorem can be applied in non-inertial frame of reference also.

B

Work done by any force can be positive, negative or zero depends on choice of reference frame.

C

Kinetic friction force always acts opposite to direction of velocity.

D

Work done by a force is independent of the choice of reference frame.

Question 91: Palette 91/145

Allen MSQ

A body of mass 'm' is moving slowly up the rough hill from point A to point B as shown in figure by a force which is acting tangential to surface at each point on the hill. Work done by this force is :

A

Independent of shape of trajectory.

B

Independent of horizontal component of displacement.

C

Depends on coefficient of friction between the surfaces.

D

Depends on the mass of the body.

Question 92: Palette 92/145

Allen MSQ

In figure, a block of mass m is released from rest when spring was in its natural length. The pulley also has mass m but it is frictionless. Suppose the value of m is such that finally it is just able to lift the block of mass 2kg up after being released. Mark the correct statement (s) :-

A

Mass of m required to just lift 2kg block is 2kg.

B

The tension in the rod, when m has zero acceleration is 20N.

C

The normal force acting on 2 kg block when m has zero acceleration is 10N.

D

The tension in the string when displacement of m is maximum possible is 20N.

Question 93: Palette 93/145

Allen MSQ

Block A of mass 1 kg is kept on block B of mass 2kg which is kept on smooth surface as shown in figure. At t = 0 both blocks are at rest and a force of 6N is applied on block B. Friction coefficient between A and B is 0.5. Which of the following is/are correct :-

A

Acceleration of both the blocks is $2m/s^{2}$

B

Work done by friction (in ground frame) on A is zero from t = 0 to t = 2 sec

C

Friction force on $A$ is $2N$

D

Kinetic energy of A at t = 2 sec is 8J

Question 94: Palette 94/145

Allen MSQ

Which of the following is/are conservative force(s)?

A

$\vec{F}=2r^{3}\hat{r}$

B

$\vec{F}=-\frac{5}{r}\hat{r}$

C

$\vec{F}=\frac{3(x\hat{i}+y\hat{j})}{(x^{2}+y^{2})^{3/2}}$

D

$\vec{F}=\frac{3(yi+x\hat{j})}{(x^{2}+y^{2})^{3/2}}$

Question 95: Palette 95/145

Allen MSQ

If one of the forces acting on a particle is conservative then :

A

Work done by this force is zero when the particle moves exactly once around any closed path.

B

Work done by this force equals the change in the kinetic energy of the particle.

C

It obeys Newton's second law.

D

Work done by this force depends on the end points of the motion, not on the path in between.

Question 96: Palette 96/145

Allen MSQ

A particle of mass 5 kg moving in the X - Y plane has its potential energy given by $U = (-7x + 24y)J$ . The particle is initially at origin and has a velocity $\vec{u} = (14.4\hat{i} + 4.2\hat{j})m/s$ .

A

The particle has speed 25 m/s at t = 4 sec.

B

The particle has an acceleration $25 \, m/s^{2}$

C

The acceleration of particle is normal to its initial velocity

D

All above are correct

Question 97: Palette 97/145

Allen MSQ

A particle is given a velocity at the bottom most position in a smooth spherical shell of radius 2m. It just complete a vertical circular motion. Then

A

acceleration of particle when the velocity of the particle is in vertically upward direction is $g\sqrt{10}$

B

acceleration of particle when the velocity of the particle is in vertically downward direction is $g\sqrt{10}$

C

acceleration of the particle at the top most point of path is g

D

acceleration of the particle at the bottom most point is 5 g

Question 98: Palette 98/145

Allen MSQ

A particle of mass 2 kg is projected with an initial speed u = 10 m/sec at an angle $\theta = 30^{\circ}$ with the horizontal:

A

Total work done on the particle during the first half of the total time of flight of the particle is $(-25) J$ .

B

Total work done on the particle during the total time of flight of the particle is 0 J.

C

Average power delivered to the particle during the first half of the flight is (-50) watt.

D

The radius of curvature of the trajectory of the particle at the highest point of the projectile is 7.5m.

Question 99: Palette 99/145

Allen MSQ

A particle is moving from point (1, 2, 3) to (4, 6, 8) with initial speed $5\sqrt{2} m/s$ on a straight line. It experiences a force of $\left(3\hat{i}+4\hat{j}\right)N$ throughout its journey. Choose the CORRECT statement(s):

A

Initial velocity vector of the particle is $\left(3\hat{i} + 4\hat{j} + 5\hat{k}\right)m / s$

B

Work done by the given force in the journey is $25J\left(W = \vec{F}.\vec{S}\right)$

C

Initial power delivered by the given force is 25 watt ( $P = \vec{F}.\vec{V}$ , P is power and V is velocity)

D

If velocity of the particle doesn't change there must be some other force acting on the particle.

Question 100: Palette 100/145

Allen MSQ

A particle is subjected to a conservative force as seen in the graphs, which of the following are correct.

A

Particle is in stable equilibrium at point 3 and b.

B

Particle is in neutral equilibrium at point b and 2.

C

No power is delivered by the force to the particle at 1, 3, and b.

D

Particle has maximum kinetic energy at position b.

Question 101: Palette 101/145

Allen MCQ

Comprehension Passage:

A block of mass m is kept in an elevator which starts moving downward with an acceleration $a_{0}$ as shown in figure. The block is observed by two observers A and B for a time interval $t_{0}$ . The observer B finds that the work done by gravity is

A

$\frac{1}{2}mg^{2}t_{0}^{2}$

B

$-\frac{1}{2}mg^{2}t_{0}^{2}$

C

$\frac{1}{2}mga_{0}t_{0}^{2}$

D

$-\frac{1}{2}mga_{0}t_{0}^{2}$

Question 102: Palette 102/145

Allen MCQ

Comprehension Passage:

A block of mass m is kept in an elevator which starts moving downward with an acceleration $a_{0}$ as shown in figure. The block is observed by two observers A and B for a time interval $t_{0}$ . The observer B finds that work done by normal reaction N is :-

A

zero

B

$-Na_{0}t_{0}^{2}$

C

$\frac{Na_{0}t^{2}}{2}$

D

None of these

Question 103: Palette 103/145

Allen MCQ

Comprehension Passage:

A block of mass m is kept in an elevator which starts moving downward with an acceleration $a_{0}$ as shown in figure. The block is observed by two observers A and B for a time interval $t_{0}$ . According to observer B, the net work done on the block is

A

$-\frac{1}{2}ma_{0}t_{0}^{2}$

B

$\frac{1}{2}ma_{0}^{2}t_{0}^{2}$

C

$\frac{1}{2}mga_{0}t_{0}^{2}$

D

$-\frac{1}{2}mga_{0}t_{0}^{2}$

Question 104: Palette 104/145

Allen MCQ

Comprehension Passage:

A block of mass m is kept in an elevator which starts moving downward with an acceleration $a_{0}$ as shown in figure. The block is observed by two observers A and B for a time interval $t_{0}$ . According to the observer A

A

the work done by gravity is zero

B

the work done by normal reaction is zero

C

the work done by pseudo force is zero

D

all the above

Question 105: Palette 105/145

Allen MCQ

Comprehension Passage:

In the figure the variation of potential energy of a particle of mass m = 2kg is represented w.r.t. its x-coordinate. The particle moves under the effect of this conservative force along the x-axis. If the particle is released at the origin then:

A

It will move towards positive x-axis.

B

It will move towards negative x-axis.

C

It will remain stationary at the origin.

D

Its subsequent motion cannot be decided due to lack of information.

Question 106: Palette 106/145

Allen MCQ

Comprehension Passage:

In the figure the variation of potential energy of a particle of mass m = 2kg is represented w.r.t. its x-coordinate. The particle moves under the effect of this conservative force along the x-axis. If the particle is released at $x = 2 + \Delta$ where $\Delta \rightarrow 0$ (it is positive) then its maximum speed in subsequent motion will be-

A

$\sqrt{22}$ m/s

B

$\sqrt{25}$ m/s

C

$\sqrt{24}$ m/s

D

$\sqrt{23}$ m/s

Question 107: Palette 107/145

Allen MCQ

Comprehension Passage:

In the figure the variation of potential energy of a particle of mass m = 2kg is represented w.r.t. its x-coordinate. The particle moves under the effect of this conservative force along the x-axis. x = -5 m and x = 10 m positions of the particle are respectively of-

A

Neutral and stable equilibrium

B

Neutral and unstable equilibrium

C

Unstable and stable equilibrium

D

Stable and unstable equilibrium.

Question 108: Palette 108/145

Allen MCQ

Comprehension Passage: PARAGRAPH FOR QUESTION NOS. 94 AND 95

A particle is moving on given curve path. At certain instant its velocity and acceleration shown in figure: Rate of change of speed of particle at P is

A

$5\sqrt{3}m/s^{2}$

B

$10m/s^{2}$

C

$5m/s^{2}$

D

None of these

Question 109: Palette 109/145

Allen MCQ

Comprehension Passage: PARAGRAPH FOR QUESTION NOS. 94 AND 95

A particle is moving on given curve path. At certain instant its velocity and acceleration shown in figure: Radius of curvature at point P is:

A

$5\sqrt{3}m$

B

10 m

C

5 m

D

None of these

Question 110: Palette 110/145

Allen MSQ

As shown in figure ‘AB’ represents an infinite wall tangential to a horizontal semi circular track. ‘O’ is a point source of light on the ground at the centre of the circle. A block moves along the circular track with speed ‘v’ starting from the point where the wall touches the circle. If the velocity and acceleration of shadow along the length of the wall is respectively ‘V’ and ‘a’ then;

A

$V = v \cos \frac{vt}{R}$

B

$V = v \sec^{2} \left( \frac{vt}{R} \right)$

C

$a = \frac{v^{2}}{R} \sec^{2} \left( \frac{vt}{R} \right) \tan \left( \frac{vt}{R} \right)$

D

$a = \frac{2v^{2}}{R} \sec^{2} \left( \frac{vt}{R} \right) \tan \left( \frac{vt}{R} \right)$

Question 111: Palette 111/145

Allen MCQ

A small coin of mass 80g is placed on the horizontal surface of a rotating disc. The disc starts from rest and is given a constant angular acceleration $\alpha = 2rad/s^{2}$ . The coefficient of static friction between the coin and the disc is $\mu_{s} = 3/4$ and coefficient of kinetic friction is $\mu_{k} = 0.5$ . The coin is placed at a distance r = 1m from the centre of the disc. The magnitude of the resultant force on the coin exerted by the disc just before it starts slipping on the disc is:

A

0.2 N

B

0.3 N

C

0.4 N

D

1 N

Question 112: Palette 112/145

Allen MCQ

A block of mass m is being pulled up the rough inclined by an agent delivering constant power P. The coefficient of friction between the block and the inclined is $\mu$ . The maximum speed of the block during the course of ascent is:

A

$v=\frac{P}{mg\sin\theta+\mu mg\cos\theta}$

B

$v=\frac{P}{mg\sin\theta-\mu mg\cos\theta}$

C

$v = \frac{2P}{mg\sin\theta - \mu mg\cos\theta}$

D

$v=\frac{3P}{mg\sin\theta-\mu mg\cos\theta}$

Question 113: Palette 113/145

Allen MCQ

A ring ‘A’ of mass ‘m’ is attached to a stretched spring of force constant K, which is fixed at C on a smooth vertical circular track of radius R. Points A and C are diametrically opposite. When the ring slips from rest on the track to point B, making an angle of $30^{\circ}$ with AC. ( $\angle ACB = 30^{\circ}$ ) spring becomes unstretched. Find the velocity of the ring at B.

A

$\left[\frac{KR^2}{2m} (2 - \sqrt{3})^2 +gR\sqrt{3}\right]^{1 / 2}$

B

$\left[\frac{KR^2}{m} (2 - \sqrt{3})^2 +gR\right]^{1 / 2}$

C

$\left[\frac{2KR^2}{m} (2 - \sqrt{3})^2 +gR\sqrt{3}\right]^{1 / 2}$

D

$\left[\frac{KR^2}{2m} (\sqrt{2} -1)^2 +gR\right]^{1 / 2}$

Question 114: Palette 114/145

Allen MCQ

Two particles move on a circular path (one just inside and the other just outside) with angular velocities $\omega$ and $5\omega$ starting from the same point. Then which of the following is incorrect?

A

they cross each other at regular intervals of time $\frac{2\pi}{4\omega}$ when their angular velocities are oppositely directed.

B

they cross each other at points on the path subtending an angle of $60^{\circ}$ at the centre if their angular velocities are oppositely directed.

C

they cross at intervals of time $\frac{\pi}{3\omega}$ if their angular velocities are oppositely directed.

D

they cross each other at points on the path subtending $90^{\circ}$ at the centre if their angular velocities are in the same sense.

Question 115: Palette 115/145

Allen MCQ

A point mass m is suspended from a light thread of length $\ell$ , fixed at O, is whirled in a horizontal circle at constant speed as shown. From your point of view, stationary with respect to the mass, the forces on the mass are :

A

B

C

D

Question 116: Palette 116/145

Allen MCQ

A ball of mass ‘m’ is rotating in a circle of radius ‘r’ with speed v inside a smooth cone as shown in figure. Let N be the normal reaction on the ball by the cone, then choose the correct option.

A

$N = mg\cos \theta$

B

$g\sin \theta = \frac{v^2}{r^2}\cos \theta$

C

$N\sin \theta -\frac{mv^2}{r} = 0$

D

None of these

Question 117: Palette 117/145

Allen MCQ

An aircraft executes a horizontal loop at a speed of $720 \, km/h$ with its wings banked at $15^{\circ}$ . What is the radius of the loop (in Km)?

A

12 Km

B

15 Km

C

20 Km

D

21 Km

Question 118: Palette 118/145

Allen MCQ

A stone is launched upward at $45^{\circ}$ with speed $v_{0}$ . A bee follows the trajectory of the stone at a constant speed equal to the initial speed of the stone. Find the radius of curvature at the top point of the trajectory.

A

$\frac{v_{0}}{g}$

B

$\frac{v_{0}^{2}}{g}$

C

$\frac{v_{0}}{2g^{2}}$

D

$\frac{V_{0}^{2}}{2g}$

Question 119: Palette 119/145

Allen MCQ

A traffic policeman standing at the intersection sees 2 cars A and B turning at angles $53^{\circ}$ and $90^{\circ}$ respectively (as shown in the figure). Their velocities are $V_{A} = 20 \, m/s$ , $V_{B} = 10 \, m/s$ . Then, which car appears to be moving faster to the traffic policeman :-

A

A

B

B

C

Both equally fast

D

Insufficient info

Question 120: Palette 120/145

Allen MCQ

A spring of force constant 800 N/m has an extension of 5 cm. The work done in extending it from 5cm to 15 cm is-

A

16 J

B

8 J

C

32 J

D

24 J

Question 121: Palette 121/145

Allen MCQ

In the figure shown all the surfaces are frictionless, and mass of the block, $m = 1 \, kg$ . The block and wedge are held initially at rest. Now wedge is given a horizontal acceleration of $10 \, m/s^2$ by applying a force on the wedge, so that the block does not slip on the wedge. Then work done by the normal force in ground frame on the block in $\sqrt{3}$ seconds is:

A

30 J

B

60 J

C

150 J

D

$100 \sqrt {3} J$

Question 122: Palette 122/145

Allen MCQ

In a region of only gravitational field of mass 'M' a particle is shifted from A to B via three different paths in the figure. The work done in different paths are $W_{1}$ , $W_{2}$ , $W_{3}$ respectively then

A

$W_{1} = W_{2} = W_{3}$

B

$W_{1} = W_{2} > W_{3}$

C

$W_{1} > W_{2} > W_{3}$

D

$W_{1} < W_{2} < W_{3}$

Question 123: Palette 123/145

Allen MCQ

A mass of 5 kg is moving along a circular path of radius 1 m. If the mass moves with 300 revolutions per minute, its kinetic energy would be

A

250 π² joule

B

100π² joule

C

5π² joule

D

0 joule

Question 124: Palette 124/145

Allen MCQ

A particle moves on a rough horizontal ground with some initial velocity say $v_{0}$ . If 3/4 of its kinetic energy is lost in friction in time $t_{0}$ then coefficient of friction between the particle and the ground is :-

A

$\frac{v_{0}}{2gt_{0}}$

B

$\frac{v_{0}}{4gt_{0}}$

C

$\frac{3v_{0}}{4gt_{0}}$

D

$\frac{v_{0}}{gt_{0}}$

Question 125: Palette 125/145

Allen MCQ

A block of mass m is hung vertically from an elastic thread of force constant mg/a. Initially the thread was at its natural length and the block is allowed to fall freely. The kinetic energy of the block when it passes through the equilibrium position will be:

A

mga

B

mga/2

C

zero

D

2mga

Question 126: Palette 126/145

Allen MCQ

A block of mass m is attached with a massless spring of force constant k. The block is placed over a rough inclined surface for which the coefficient of friction is $\mu = \frac{3}{4}$ . The minimum value of M required to move the block up the plane is: (Neglect mass of string and pulley and friction in pulley)

A

$\frac{3}{5}m$

B

$\frac{4}{5}m$

C

2 m

D

$\frac{3}{2}m$

Question 127: Palette 127/145

Allen MCQ

A particle moves in a straight line with retardation proportional to its displacement. Its loss of kinetic energy for any displacement x is proportional to-

A

$x^{2}$

B

$e^{x}$

C

x

D

$\log_{e}x$

Question 128: Palette 128/145

Allen MCQ

Two inclined frictionless tracks, one gradual and the other steep meet at A from where two stones are allowed to slide down from rest, one on each track as shown in figure. Which of the following statement is correct?

A

Both the stones reach the bottom at the same time but not with the same speed.

B

Both the stones reach the bottom with the same speed and stone I reaches the bottom earlier than stone II.

C

Both the stones reach the bottom with the same speed and stone II reaches the bottom earlier than stone I.

D

Both the stones reach the bottom at different times and with different speeds.

Question 129: Palette 129/145

Allen MCQ

Track OABCD (as shown is figure) is smooth and fixed in vertical plane. What minimum speed has to be given to a particle lying at point A, so that it can reach point C?

A

60 m/s

B

100 m/s

C

$70 m / s$

D

80 m/s

Question 130: Palette 130/145

Allen MCQ

A particle is placed at the origin and a force F = kx is acting on it (where k is a positive constant). If $U(0) = 0$ , the graph of $U(x)$ versus x will be (where U is the potential energy function)

A

B

C

D

Question 131: Palette 131/145

Allen MCQ

In the figure shown the potential energy $(U)$ of a particle is plotted against its position 'x' from origin. Then which of the following statement is correct. A particle at:

A

$x_{1}$ is in stable equilibrium

B

$x_{2}$ is in stable equilibrium

C

$x_{3}$ is in stable equilibrium

D

None of these

Question 132: Palette 132/145

Allen MCQ

As a particle moves along the x-axis it is acted upon by a conservative force. The potential energy is shown below as a function of the coordinate x of the particle. Rank the labelled regions according to the magnitude of the force, least to greatest.

A

AB, BC, CD

B

AB, CD, BC

C

BC, CD, AB

D

BC, AB, CD

Question 133: Palette 133/145

Allen MCQ

One end of a string of length $\ell=\frac{14}{9}m$ is fixed and a mass of 1 kg is tied to the other end. The ball is given a velocity $2\sqrt{g\ell}$ at the bottom most point as shown in figure. The string is cut when the ball becomes horizontal. Find the distance (in m) travelled till it stop for the 1st time (Take $\pi=\frac{22}{7}$ ).

A

1

B

3

C

4

D

6

Question 134: Palette 134/145

Allen MCQ

A small ball can move in a vertical plane along a semi-circle of radius r without friction. At what speed is the ball to launch from point A so that its acceleration is 3g at point B?

A

$(3gr)^{1/2}$

B

$(2gr)^{1/2}$

C

$(gr)^{1/2}$

D

$2(gr)^{1/2}$

Question 135: Palette 135/145

Allen MCQ

A particle 'A' of mass $\frac{10}{7}$ kg is moving in the positive x-direction. Its initial position is x = 0 and initial velocity is 1 m/s. The velocity at x = 10m is (Use the graph given)

A

4 m/s

B

2 m/s

C

$3\sqrt{2}$ m/s

D

$\frac{100}{3}$ m/s

Question 136: Palette 136/145

Allen MCQ

A block of mass m moves with speed v against a smooth, fixed vertical circular groove of radius r kept on smooth horizontal surface. Find net force on the block.

A

mg

B

$\frac{mv^{2}}{r}$

C

$\sqrt{(mg)^{2}+\left(\frac{mV^{2}}{r}\right)^{2}}$

D

$\sqrt{(mg)+\left(\frac{mV^{2}}{r}\right)}$

Question 137: Palette 137/145

Allen MSQ

Three particles A, B, C are located at the corners of an equilateral triangle as shown in figure. Each of the particle is moving with velocity v. Then at the instant shown, the relative angular velocity of

A

A wrt B is $\frac{v \cos 30^{\circ}}{a}$ in z - direction

B

B wrt C is $\frac{v \cos 30^{\circ}}{a}$ in z-direction

C

A wrt C is $\frac{v \cos 30^{\circ}}{a}$ in z - direction

D

B wrt A is $\frac{v \cos 30^{\circ}}{a}$ in z-direction

Question 138: Palette 138/145

Allen MSQ

A particle of mass m is at rest in a train moving with constant velocity with respect to ground. Now the particle is accelerated by a constant force $F_{0}$ acting along the direction of motion of train for time $t_{0}$ . A girl in the train and a boy on the ground measure the work done by this force. Which of the following are INCORRECT?

A

Both will measure the same work

B

Boy will measure higher value than the girl

C

Girl will measure higher value than the boy

D

Data are insufficient for the measurement of work done by the force $F_{0}$

Question 139: Palette 139/145

Allen MSQ

A smooth track in the form of a quarter circle of radius 6 m lies in the vertical plane. A particle moves from $P_{1}$ to $P_{2}$ under the action of forces $\vec{F}_{1}, \vec{F}_{2}$ and $\vec{F}_{3}$ . Force $\vec{F}_{1}$ is always toward $P_{2}$ and is always 20N in magnitude. $\vec{F}_{2}$ force always acts horizontally and is always 30 N in magnitude. Force $\vec{F}_{3}$ always acts tangentially to the track and is of magnitude 15 N. Select the correct alternative(s)

A

work done by $\vec{F}_{1}$ is 120 J

B

work done by $\vec{F}_{2}$ is 180 J

C

work done by $\vec{F}_{3}$ is $45 \pi$

D

$\vec{F}_{1}$ is conservative in nature

Question 140: Palette 140/145

Allen MSQ

A particle is shifted from $A$ to $B$ and then from $B$ to $C$ where $A, B$ and $C$ are the midpoints of the corresponding faces of a cube of side $2m$ . If a force $\vec{F} = (3\hat{i} + 4\hat{j} - 5\hat{k})N$ is continuously acting on the particle, then select correct alternative :-

A

work done from $A$ to $B$ is 7 $J$

B

work done from $B$ to $C$ is $1J$

C

work done $A$ to $C$ is 8 $J$

D

$\vec{F}$ is a conservative force

Question 141: Palette 141/145

Allen MCQ

Comprehension Passage: PARAGRAPH FOR QUESTION 8 AND 9

An object of mass M is gently placed on a horizontal conveyor belt, which is moving with uniform velocity $v_{o}$ as shown in the figure. The coefficient of static friction is $\mu_{s}$ , the coefficient of kinetic friction is $\mu_{k}$ , and the acceleration of gravity is g. Initially the object slips for a while but finally moves without slipping together with the belt. How far the conveyor belt moves while the object is slipping?

A

$\frac{v_{o}^{2}}{\mu_{k}g}$

B

$\frac{v_{o}^{2}}{2\mu_{k}g}$

C

$\frac{v_{o}^{2}}{\mu_{s}g}$

D

$\frac{v_{o}^{2}}{2\mu_{s}g}$

Question 142: Palette 142/145

Allen MCQ

Comprehension Passage: PARAGRAPH FOR QUESTION 8 AND 9

An object of mass M is gently placed on a horizontal conveyor belt, which is moving with uniform velocity $v_{o}$ as shown in the figure. The coefficient of static friction is $\mu_{s}$ , the coefficient of kinetic friction is $\mu_{k}$ , and the acceleration of gravity is g. Initially the object slips for a while but finally moves without slipping together with the belt. Work done on the object by friction force relative to the reference frame moving with the conveyor belt is

A

$\frac{1}{2}Mv_{o}^{2}$

B

$-\frac{1}{2}Mv_{o}^{2}$

C

$Mv_{o}^{2}$

D

zero

Question 143: Palette 143/145

Allen MCQ

Comprehension Passage: PARAGRAPH FOR QUESTION NO. 10 TO 12

A point like object of mass m starts from point K as shown in the figure. It slides inside along the full length of the smooth track of radius R, and then moves freely and travels to point C. [The track is kept in vertical plane] Determine the vertical initial velocity of the point like object. Determine the vertical initial velocity of the point like object.

A

$v_{0} = \sqrt{2g(h + R) + \frac{d^{2}g}{R}}$

B

$v_{0} = \sqrt{2g(h - R) + \frac{d^{2}g}{2R}}$

C

$v_{0} = \sqrt{2g(h + R) + \frac{d^{2}g}{2R}}$

D

$v_{0} = \sqrt{2g(h + R) + \frac{dg}{2R}}$

Question 144: Palette 144/145

Allen MCQ

Comprehension Passage: PARAGRAPH FOR QUESTION NO. 10 TO 12

A point like object of mass m starts from point K as shown in the figure. It slides inside along the full length of the smooth track of radius R, and then moves freely and travels to point C. [The track is kept in vertical plane] What is the minimum possible distance OC = d, necessary for the object to slide along the entire length of the track?

A

$d_{\min} = R \sqrt{2}$

B

$d_{\min} = \sqrt{3} R$

C

$d_{\min} = \frac{R}{\sqrt{2}}$

D

$d_{\min} = \frac{R}{\sqrt{3}}$

Question 145: Palette 145/145

Allen MCQ

Comprehension Passage: PARAGRAPH FOR QUESTION NO. 10 TO 12

A point like object of mass m starts from point K as shown in the figure. It slides inside along the full length of the smooth track of radius R, and then moves freely and travels to point C. [The track is kept in vertical plane] Find the normal force exerted by the track at point A.

A

$F_{A} = mg \left( \frac{d^{2}}{2R^{2}} - 2 \right)$

B

$F_{A} = mg \left( \frac{d^{2}}{2R^{2}} + 2 \right)$

C

$F_{A} = mg \left( \frac{3d^{2}}{R^{2}} + 2 \right)$

D

$F_{A} = mg \left( \frac{d}{3R^{2}} + 2 \right)$