Work, Power & Energy
769 Questions
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Q501
Allen
JEE Main Practice Paper
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.
Q502
Allen
JEE Main Practice Paper
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
Q503
Allen
JEE Main Practice Paper
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.
Q504
Allen
JEE Main Practice Paper
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
Q505
Allen
JEE Main Practice Paper
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
Q506
Allen
JEE Main Practice Paper
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
Q507
Allen
JEE Main Practice Paper
Numerical
A particle starts moving in a non-uniform circular motion, has angular acceleration as shown in figure. The angular velocity at the end of 4 radian is given by $\omega$ rad/s then find the value of $\omega$ .
Correct Answer: 6
Explanation:
Ans. (6)
$\alpha = \frac {\omega d \omega}{d \theta}$
$\int \omega d \omega = \int \alpha d \theta$
${\frac {\omega^ {2}}{2}} = \mathrm{Areaunder} \alpha \mathrm{vs} \theta \mathrm{graph} = {\frac {1}{2}} (9 \times 4)$
$\omega = \sqrt {3 6} = 6 \mathrm{rad/s}$
$\alpha = \frac {\omega d \omega}{d \theta}$
$\int \omega d \omega = \int \alpha d \theta$
${\frac {\omega^ {2}}{2}} = \mathrm{Areaunder} \alpha \mathrm{vs} \theta \mathrm{graph} = {\frac {1}{2}} (9 \times 4)$
$\omega = \sqrt {3 6} = 6 \mathrm{rad/s}$
Q508
Allen
JEE Main Practice Paper
Numerical
A stone is thrown horizontally with the velocity 15 m/s. If the ratio of normal acceleration to the tangential acceleration stone in 1 second after it begins to move is x, then value of 2x will be
Correct Answer: 3
Explanation:
Ans. (3)
At $t = 1$ sec
$\vec {v} = 1 5 \hat {\imath} - 1 0 \hat {j}$
$\vec {a} = - 1 0 \hat {j}$
$a _ {t} = \frac {\vec {a} \cdot \vec {v}}{v} = \frac {2 0}{\sqrt {1 3}}$
$a _ {N} = \sqrt {a ^ {2} - a _ {t} ^ {2}} = \frac {3 0}{\sqrt {1 3}}$
$\frac {a _ {N}}{a _ {T}} = \frac {3 0}{2 0} = 1. 5 = x$
At $t = 1$ sec
$\vec {v} = 1 5 \hat {\imath} - 1 0 \hat {j}$
$\vec {a} = - 1 0 \hat {j}$
$a _ {t} = \frac {\vec {a} \cdot \vec {v}}{v} = \frac {2 0}{\sqrt {1 3}}$
$a _ {N} = \sqrt {a ^ {2} - a _ {t} ^ {2}} = \frac {3 0}{\sqrt {1 3}}$
$\frac {a _ {N}}{a _ {T}} = \frac {3 0}{2 0} = 1. 5 = x$
Q509
Allen
JEE Main Practice Paper
Numerical
A body constrained to move along the z-axis of a coordinate system is subject to a constant force F given by
$F = - \hat {\imath} + 2 \hat {j} + 3 \hat {k} N$
where $\hat{i},\hat{j},\hat{k}$ are unit vectors along the x, y and z-axis of the system respectively. What is the work done (in J) by this force in moving the body a distance of 4 m along the z-axis?
Correct Answer: 12
Explanation:
Ans. (12) $W = F_{\mathrm{Z}} \cdot \Delta Z = 3 \times 4 = 12J$
Q510
Allen
JEE Main Practice Paper
Numerical
A point mass of 0.5 kg is moving along x-axis as $x = t^{2} + 2t$ , where x is in meters and t is in seconds. Find the work done (in J) by all the forces acting on the body during the time interval [0, 2s].
Correct Answer: 8
Explanation:
Ans. (8)
Method I: $W_{\text{all}} = \Delta KE = \frac{1}{2} m \left( v_{2}^{2} - v_{0}^{2} \right)$ $v = \frac{dx}{dt} = 2t + 2 = 2(t + 1)$ $v_{2} = 6 \, m/s, v_{0} = 2 \, m/s$ $W_{\text{all}} = \frac{1}{2} \left( \frac{1}{2} \right) (6^{2} - 2^{2}) = 8J$ Method II: $x = t^{2} + 2t$ $a = \frac{d^{2}x}{dt^{2}} = 2 \, m/s^{2} = constant$ so F = m(A) = 1 N = constant
so, W = F.S = (1) (8-0) = 8J
Method I: $W_{\text{all}} = \Delta KE = \frac{1}{2} m \left( v_{2}^{2} - v_{0}^{2} \right)$ $v = \frac{dx}{dt} = 2t + 2 = 2(t + 1)$ $v_{2} = 6 \, m/s, v_{0} = 2 \, m/s$ $W_{\text{all}} = \frac{1}{2} \left( \frac{1}{2} \right) (6^{2} - 2^{2}) = 8J$ Method II: $x = t^{2} + 2t$ $a = \frac{d^{2}x}{dt^{2}} = 2 \, m/s^{2} = constant$ so F = m(A) = 1 N = constant
so, W = F.S = (1) (8-0) = 8J
Q511
Allen
JEE Main Practice Paper
Numerical
A block is released from rest from top of a rough curved track as shown in figure. It comes to rest at some point on the horizontal part. If its mass is 200 gm, calculate negative of work by friction in joules.
Correct Answer: 4
Explanation:
Ans. (4)
$W _ {\mathrm{f}} + W _ {\mathrm{g}} = 0 \Rightarrow W _ {\mathrm{f}} = - W _ {\mathrm{g}} = - m g h = - 0. 2 \times 1 0 \times 2 = - 4 J$
$W _ {\mathrm{f}} + W _ {\mathrm{g}} = 0 \Rightarrow W _ {\mathrm{f}} = - W _ {\mathrm{g}} = - m g h = - 0. 2 \times 1 0 \times 2 = - 4 J$
Q512
Allen
JEE Main Practice Paper
Numerical
The potential energy (in joules) function of a particle in a region of space is given as :
$U = (2 x ^ {2} + 3 y ^ {3} + 2 z)$
Here x, y and z are in metres. Find the magnitude of x component of force (in newton) acting on the particle at point P (1m, 2m, 3m).
Correct Answer: 4
Explanation:
Ans. (4)
$F _ {x} = - \frac {d U}{d x} = - 4 x$
$F _ {x} = - \frac {d U}{d x} = - 4 x$
Q513
Allen
JEE Main Practice Paper
Numerical
Initially spring is relaxed when $m_{1}$ is released from rest. Calculate for what minimum value of $m_{1}$ the block of mass m will just leave the contact with surface? Give answer in terms of $\frac{m}{m_{1}}$ .
Correct Answer: 2
Explanation:
Ans. (2) $ky = mg$ $m_{1}gy = \frac{1}{2} ky^{2}$ $2\mathrm{m}_{1}\mathrm{g} = mg$ $m_{1} = \frac{m}{2}$
Q514
Allen
JEE Main Practice Paper
Numerical
The potential energy of a 2 kg particle moving along the x axis is given by
$\mathrm{U} (x) = (4.0 \mathrm{J} / m ^ {2}) x ^ {2} + (1.0 \mathrm{J} / m ^ {4}) x ^ {4}$
When the particle is at x = 1.0m, find the magnitude (in $m/s^{2}$ ) of its acceleration.
[Only conservative forces are acting]
Correct Answer: 6
Explanation:
Ans. (6)
$U = 4 x ^ {2} + x ^ {4}$
$F = - \frac {d U}{d x} = - 8 x - 4 x ^ {3}$
$\mathrm{At} x = 1, F = - 8 - 4 = - 1 2 N$
$\Rightarrow a = \frac {F}{m} = \frac {- 1 2}{2} = - 6 m / s ^ {2}$
$U = 4 x ^ {2} + x ^ {4}$
$F = - \frac {d U}{d x} = - 8 x - 4 x ^ {3}$
$\mathrm{At} x = 1, F = - 8 - 4 = - 1 2 N$
$\Rightarrow a = \frac {F}{m} = \frac {- 1 2}{2} = - 6 m / s ^ {2}$
Q515
Allen
JEE Main Practice Paper
Numerical
A particle is given a certain velocity v at point P as shown on a hemispherical smooth surface. Find the value of v (in m/s), such that when particle reaches Q, the normal reaction of surface becomes equal to particle's weight. $R = 1.6\ m, g = 10\ m/s^{2}$
Correct Answer: 4
Explanation:
Ans. (4) $N = mg \Rightarrow v_{\mathrm{f}} = 0$ $mg\frac{R}{2} = \frac{1}{2} mv_i^2 \Rightarrow v = \sqrt{gR} = 4m / s$
Q516
Allen
JEE Main Practice Paper
Numerical
A skier starts from rest at the top of a hill. The skier coasts down the hill and up a second hill, as the drawing illustrates. The crest of the second hill is circular, with a radius of r = 36 m. Neglect friction and air resistance. What must be the height h (in m) of the first hill so that the skier just loses contact with the snow at the crest of the second hill?
Correct Answer: 18
Explanation:
Ans. (18) $mg = \frac{mv^2}{r}$ ...(i) $v^2 = rg = 360$ From energy conservation $mgh = \frac{1}{2} mv^2$
Q517
Allen
JEE Advanced Practice Paper
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.
$2(gr)^{1/2}$
Q518
Allen
JEE Advanced Practice Paper
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
Q519
Allen
JEE Advanced Practice Paper
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.
$\sqrt{(mg)+\left(\frac{mV^{2}}{r}\right)}$
Q520
Allen
JEE Advanced Practice Paper
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
Q521
Allen
JEE Advanced Practice Paper
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 $F_{0}$
Q522
Allen
JEE Advanced Practice Paper
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
Q523
Allen
JEE Advanced Practice Paper
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
Q524
Allen
JEE Advanced Practice Paper
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?
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.
$\frac{v_{o}^{2}}{2\mu_{s}g}$
Q525
Allen
JEE Advanced Practice Paper
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
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