Rotational Motion
270 Questions
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Q251
Allen
Rotational Motion
Numerical
Two discs $A$ and $B$ touch each other as shown in figure. A rope tightly wound on $A$ is pulled down at $2 m / s^{2}$ . Find the friction (in $N$ ) force between $A$ and $B$ if slipping is absent
Correct Answer: 2
Explanation:
Ans. (2)
$\text { Now for } (2) f \times 2 R = I _ {2} \alpha_ {2}$
$I _ {2} = \frac {2 \times 4 R ^ {2}}{2} = 4 R ^ {2}$
Now $\alpha_{2}2R = 2$ (a is same)
$f ^ {2} R = 4 R ^ {2} \frac {1}{R}$
$\therefore f = 2$
$\text { Now for } (2) f \times 2 R = I _ {2} \alpha_ {2}$
$I _ {2} = \frac {2 \times 4 R ^ {2}}{2} = 4 R ^ {2}$
Now $\alpha_{2}2R = 2$ (a is same)
$f ^ {2} R = 4 R ^ {2} \frac {1}{R}$
$\therefore f = 2$
Q252
Allen
JEE Advanced PYQ
MCQ
A uniform wooden stick of mass 1.6 kg and length $\ell$ rests in an inclined manner on a smooth, vertical wall of height h (< $\ell$ ) such that a small portion of the stick extends beyond the wall. The reaction force of the wall on the stick is perpendicular to the stick. The stick makes an angle of $30^{\circ}$ with the wall and the bottom of the stick is on a rough floor. The reaction of the wall on the stick is equal in magnitude to the reaction of the floor on the stick. The ratio h/ $\ell$ and the frictional force f at the bottom of the stick are: (g = 10 ms $^{-2}$ )
[A] $\frac{h}{\ell} = \frac{\sqrt{3}}{16}$ , $f = \frac{16\sqrt{3}}{3}N$ (B) $\frac{h}{\ell} = \frac{3}{16}$ , $f = \frac{16\sqrt{3}}{3}N$ (C) $\frac{h}{\ell} = \frac{3\sqrt{3}}{16}$ , $f = \frac{8\sqrt{3}}{3}N$ (D) $\frac{h}{\ell} = \frac{3\sqrt{3}}{16}$ , $f = \frac{8\sqrt{3}}{3}N$
Correct Answer: D
Explanation:
Ans. (ACD)
A ring is rolling on the horizontal surface.
$\omega = \frac {u}{R}$
Point of contact of ring with ground surface is instantaneous centre of rotation. So we can say that body is performing fixed axis rotation about point A at that instant
$0 \leq V \leq 2 u$
Line CP. Makes an angle. Of $30^{\circ}$ with horizontal.
A ring is rolling on the horizontal surface.
$\omega = \frac {u}{R}$
Point of contact of ring with ground surface is instantaneous centre of rotation. So we can say that body is performing fixed axis rotation about point A at that instant
$0 \leq V \leq 2 u$
Line CP. Makes an angle. Of $30^{\circ}$ with horizontal.
Q253
Allen
JEE Advanced Practice Paper
MCQ
A small sphere $A$ of mass $m$ and radius $r$ rolls without slipping inside a large fixed hemispherical bowl of radius $R (>> r)$ as shown in figure. If the sphere starts from rest at the top point of the hemisphere find the normal force exerted by the small sphere on the hemisphere when it is at the bottom $B$ of the hemisphere.
$$
\frac {7}{5} m g
$$
$$
\frac {7}{5} m g
$$
A.
$\frac{10}{7} mg$
B.
$\frac{17}{7} mg$
C.
$\frac{5}{7} mg$
D.
$\frac{7}{5} m$
Q254
Allen
JEE Advanced Practice Paper
MCQ
A straight rod of length L is released on a frictionless horizontal floor in a vertical position. As it falls + slips, the distance of a point on the rod from the lower end, which follows a quarter circular locus is :-
A.
L/2
B.
L/4
C.
L/8
D.
None
Q255
Allen
JEE Advanced Practice Paper
MSQ
A ring rolls without slipping on the ground. Its centre C moves with a constant speed u. P is any point on the ring. The speed of P with respect to the ground is v.
A.
$0 \leq n \leq 2u$
B.
v = u, if CP is horizontal
C.
v = u, if CP makes an angle of $30^{\circ}$ with the horizontal and P is below the horizontal level of C.
D.
$v = \sqrt{2} u$ , if $CP$ is horizontal
Q256
Allen
JEE Advanced Practice Paper
MSQ
A disc of circumference s is at rest at a point A on a horizontal surface when a constant horizontal force begins to act on its centre. Between A and B there is sufficient friction to prevent slipping, and the surface is smooth to the right of B. AB = s. The disc moves from A to B in time T. To the right of B,
A.
the angular acceleration of the disc will disappear, linear acceleration will remain unchanged.
B.
linear acceleration of the disc will increase.
C.
the disc will make one rotation in time T/2.
D.
the disc will cover a distance greater than s in further time T.
Q257
Allen
JEE Advanced Practice Paper
MSQ
A body is in equilibrium under the influence of a number of forces. Each force has a different line of action. The minimum number of forces required is
A.
2, if their lines of action pass through the centre of mass of the body.
B.
3, if their lines of action are not parallel.
C.
3, if their lines of action are parallel.
D.
4, if their lines of action are parallel and all the forces have the same magnitude.
Q258
Allen
JEE Advanced Practice Paper
MCQ
Comprehension Passage: PARAGRAPH FOR QUESTION NO. 6 TO 8
In figure, the winch is mounted on an axle, and the 6-sided nut is welded to the winch. By turning the nut with a wrench, a person can rotate the winch. For instance, turning the nut clockwise lifts the block off the ground, because more and more rope gets wrapped around the winch.
Three students agree that using a longer wrench makes it easier to turn the winch. But they disagree about why. All three students are talking about the case where the winch is used, over a 10 s time interval, to lift the block one metre off the ground.
Student 1: By using a longer wrench, the person decreases the average force he must exert on the wrench, in order to lift the block one metre in 10s.
Student 2: Using a longer wrench reduces the work done by the person as he uses the winch to lift the block 1m in 10s.
Student 3 : Using a longer wrench reduces the power that the person must exert to lift the block 1m in 10s.
Student 1 is :-
Student 1 is :-
A.
correct, because the torque that the wrench must exert to lift the block doesn't depend on the wrench's length.
B.
correct, because using a longer wrench decreases the torque it must exert on the winch.
C.
incorrect, because the torque that the wrench must exert to lift the block doesn't depend on the wrench's length.
D.
Incorrect, because using a longer wrench decreases the torque it must exert on the winch.
Q259
Allen
JEE Advanced Practice Paper
MCQ
Comprehension Passage: PARAGRAPH FOR QUESTION NO. 6 TO 8
In figure, the winch is mounted on an axle, and the 6-sided nut is welded to the winch. By turning the nut with a wrench, a person can rotate the winch. For instance, turning the nut clockwise lifts the block off the ground, because more and more rope gets wrapped around the winch.
Three students agree that using a longer wrench makes it easier to turn the winch. But they disagree about why. All three students are talking about the case where the winch is used, over a 10 s time interval, to lift the block one metre off the ground.
Student 1: By using a longer wrench, the person decreases the average force he must exert on the wrench, in order to lift the block one metre in 10s.
Student 2: Using a longer wrench reduces the work done by the person as he uses the winch to lift the block 1m in 10s.
Student 3 : Using a longer wrench reduces the power that the person must exert to lift the block 1m in 10s.
Which of the following is true about student 2 and 3 :
Which of the following is true about student 2 and 3 :
A.
Student 2 and 3 are both correct
B.
Student 2 is correct, but student 3 is incorrect
C.
Student 3 is correct, but student 2 is incorrect
D.
Student 2 and 3 are both incorrect
Q260
Allen
JEE Advanced Practice Paper
MCQ
Comprehension Passage: PARAGRAPH FOR QUESTION NO. 6 TO 8
In figure, the winch is mounted on an axle, and the 6-sided nut is welded to the winch. By turning the nut with a wrench, a person can rotate the winch. For instance, turning the nut clockwise lifts the block off the ground, because more and more rope gets wrapped around the winch.
Three students agree that using a longer wrench makes it easier to turn the winch. But they disagree about why. All three students are talking about the case where the winch is used, over a 10 s time interval, to lift the block one metre off the ground.
Student 1: By using a longer wrench, the person decreases the average force he must exert on the wrench, in order to lift the block one metre in 10s.
Student 2: Using a longer wrench reduces the work done by the person as he uses the winch to lift the block 1m in 10s.
Student 3 : Using a longer wrench reduces the power that the person must exert to lift the block 1m in 10s.
If several wrenches all apply the same torque to a nut, which graph best expresses the relationship between the force the person must apply to the wrench, and the length of the wrench :-
If several wrenches all apply the same torque to a nut, which graph best expresses the relationship between the force the person must apply to the wrench, and the length of the wrench :-
A.
B.
C.
D.
Q261
Allen
JEE Advanced Practice Paper
MCQ
Comprehension Passage: PARAGRAPH FOR QUESTION NO. 9 AND 10
A uniform rod is fixed to a rotating turntable so that its lower end is on the axis of the turntable and it makes an angle of $20^{\circ}$ to the vertical. (The rod is thus rotating with uniform angular velocity about a vertical axis passing through one end.) If the turntable is rotating clockwise as seen from above.
What is the direction of the rod's angular momentum vector (calculated about its lower end)?
What is the direction of the rod's angular momentum vector (calculated about its lower end)?
A.
vertically downwards
B.
down at $20^{\circ}$ to the horizontal
C.
up at $20^{\circ}$ to the horizontal
D.
vertically upwards
Q262
Allen
JEE Advanced Practice Paper
MCQ
Comprehension Passage: PARAGRAPH FOR QUESTION NO. 9 AND 10
A uniform rod is fixed to a rotating turntable so that its lower end is on the axis of the turntable and it makes an angle of $20^{\circ}$ to the vertical. (The rod is thus rotating with uniform angular velocity about a vertical axis passing through one end.) If the turntable is rotating clockwise as seen from above.
Is there a torque acting on it, and if so in what direction?
Is there a torque acting on it, and if so in what direction?
A.
yes, vertically
B.
yes, horizontally
C.
yes at $20^{\circ}$ to the horizontal
D.
no
Q263
Allen
JEE Advanced Practice Paper
MCQ
Comprehension Passage: PARAGRAPH FOR QUESTION NO. 11 TO 14
A spring having initial unstretched length $\ell_{0}$ is lying on a smooth table. It's one end is fixed and the other one is fastened to a small particle of mass m. The particle is imparted an initial speed $v_{0}$ horizontally in a direction perpendicular to the spring. In the course of the motion in horizontal plane, the maximum elongation of the spring is $\Delta\ell = \ell_{0}/10$ . (Given: m = 0.1 kg, $\ell_{0} = 1m$ , $v_{0} = 11m/s$ ).
In the course of motion, which of the quantities relating to spring block system are conserved?
In the course of motion, which of the quantities relating to spring block system are conserved?
A.
kinetic energy
B.
momentum
C.
angular momentum
D.
potential energy
Q264
Allen
JEE Advanced Practice Paper
MCQ
Comprehension Passage: PARAGRAPH FOR QUESTION NO. 11 TO 14
A spring having initial unstretched length $\ell_{0}$ is lying on a smooth table. It's one end is fixed and the other one is fastened to a small particle of mass m. The particle is imparted an initial speed $v_{0}$ horizontally in a direction perpendicular to the spring. In the course of the motion in horizontal plane, the maximum elongation of the spring is $\Delta\ell = \ell_{0}/10$ . (Given: m = 0.1 kg, $\ell_{0} = 1m$ , $v_{0} = 11m/s$ ).
Which of the following is correct about initial situation and situation at maximum elongation?
Which of the following is correct about initial situation and situation at maximum elongation?
A.
the orientation of spring in both positions should be perpendicular to each other.
B.
the velocity at maximum extension should be zero.
C.
the velocity at maximum extension as well as at initial position should be perpendicular to spring.
D.
Acceleration should be zero at the maximum extension as well as at initial position.
Q265
Allen
JEE Advanced Practice Paper
MCQ
Comprehension Passage: PARAGRAPH FOR QUESTION NO. 11 TO 14
A spring having initial unstretched length $\ell_{0}$ is lying on a smooth table. It's one end is fixed and the other one is fastened to a small particle of mass m. The particle is imparted an initial speed $v_{0}$ horizontally in a direction perpendicular to the spring. In the course of the motion in horizontal plane, the maximum elongation of the spring is $\Delta\ell = \ell_{0}/10$ . (Given: m = 0.1 kg, $\ell_{0} = 1m$ , $v_{0} = 11m/s$ ).
Student-A: At maximum extension $mv'(\ell_{0} + \Delta\ell) = mv_{0}\ell_{0}$ .
Student-B: at maximum extension $k\Delta\ell = \frac{mv'^{2}}{\left(\ell_{0} + \Delta\ell\right)}$ .
Student-C: $\frac{1}{2} mv_0^2 = \frac{1}{2} mv'^2 + \frac{1}{2} k\Delta \ell^2$ .
Where v' is velocity at instant of maximum extension:
Student-A: At maximum extension $mv'(\ell_{0} + \Delta\ell) = mv_{0}\ell_{0}$ .
Student-B: at maximum extension $k\Delta\ell = \frac{mv'^{2}}{\left(\ell_{0} + \Delta\ell\right)}$ .
Student-C: $\frac{1}{2} mv_0^2 = \frac{1}{2} mv'^2 + \frac{1}{2} k\Delta \ell^2$ .
Where v' is velocity at instant of maximum extension:
A.
Only Student-A and B are correct
B.
Only Student-A and C are correct
C.
Only Student-B and C are correct
D.
All are correct
Q266
Allen
JEE Advanced Practice Paper
MCQ
Comprehension Passage: PARAGRAPH FOR QUESTION NO. 11 TO 14
A spring having initial unstretched length $\ell_{0}$ is lying on a smooth table. It's one end is fixed and the other one is fastened to a small particle of mass m. The particle is imparted an initial speed $v_{0}$ horizontally in a direction perpendicular to the spring. In the course of the motion in horizontal plane, the maximum elongation of the spring is $\Delta\ell = \ell_{0}/10$ . (Given: m = 0.1 kg, $\ell_{0} = 1m$ , $v_{0} = 11m/s$ ).
The value of spring constant (in N/m) is:
The value of spring constant (in N/m) is:
A.
100
B.
210
C.
420
D.
105
Q267
Allen
JEE Advanced Practice Paper
Numerical
In the figure, A and B are two blocks of mass 4 kg and 2 kg respectively attached to the two ends of a light string passing over a disc C of mass 40 kg and radius 0.1 m. The disc is free to rotate about a fixed horizontal axis, coinciding with its own axis. The system is released from rest and the string does not slip over the disc. The number of revolutions made by the disc at the end of 10 sec. from the start is $n \times 10^{3}/26\pi$ . Find n.
Correct Answer: 5
Explanation:
Ans. (5)
$4 0 - T _ {A} = 4 a\tag{... (i}$
$T _ {B} - 2 0 = 2 a$
$(T _ {A} - T _ {B}) \times 0. 1 \frac {M (0 . 1) ^ {2}}{2} \alpha\tag{... (ii}$
... (iii)
$T _ {A} - T _ {B} = 2 \alpha\tag{... (vi}$
$a = \alpha R \Rightarrow a = \frac {\alpha}{1 0}\tag{... (iv}$
$\text { Now } (1) + (2) \Rightarrow T _ {B} - T _ {A} + 2 0 = 6 a$
$\therefore - 2 \alpha + 2 0 = 6 a$
$\therefore \quad - 2 \alpha + 2 0 = \frac {6 \alpha}{1 0} \Rightarrow 2 0 \frac {2 6 \alpha}{1 0}$
$\therefore \alpha = \frac {2 0 0}{2 6}$
$\therefore \alpha = \frac {1 0 0}{1 3}$
$\therefore \quad a = \frac {\alpha}{1 0} = \frac {1 0}{1 3} m / s ^ {2}$
$\frac {1}{2} \alpha t ^ {2} = \frac {1}{2} \times \frac {1 0 0}{1 3} \times \frac {1 0 0}{2 \pi} = \frac {5 0 0 0}{2 6 \pi} r \ell v$
$4 0 - T _ {A} = 4 a\tag{... (i}$
$T _ {B} - 2 0 = 2 a$
$(T _ {A} - T _ {B}) \times 0. 1 \frac {M (0 . 1) ^ {2}}{2} \alpha\tag{... (ii}$
... (iii)
$T _ {A} - T _ {B} = 2 \alpha\tag{... (vi}$
$a = \alpha R \Rightarrow a = \frac {\alpha}{1 0}\tag{... (iv}$
$\text { Now } (1) + (2) \Rightarrow T _ {B} - T _ {A} + 2 0 = 6 a$
$\therefore - 2 \alpha + 2 0 = 6 a$
$\therefore \quad - 2 \alpha + 2 0 = \frac {6 \alpha}{1 0} \Rightarrow 2 0 \frac {2 6 \alpha}{1 0}$
$\therefore \alpha = \frac {2 0 0}{2 6}$
$\therefore \alpha = \frac {1 0 0}{1 3}$
$\therefore \quad a = \frac {\alpha}{1 0} = \frac {1 0}{1 3} m / s ^ {2}$
$\frac {1}{2} \alpha t ^ {2} = \frac {1}{2} \times \frac {1 0 0}{1 3} \times \frac {1 0 0}{2 \pi} = \frac {5 0 0 0}{2 6 \pi} r \ell v$
Q268
Allen
JEE Advanced Practice Paper
Numerical
A 20 kg cabinet is mounted on small casters that allow it to move freely ( $\mu = 0$ ) on the floor. If a 100 N force is applied as shown. The maximum value of h for which cabinet will not tip is $0.5 \times n$ . Find the value of n.
Correct Answer: 3
Explanation:
Ans. (3) $F = ma \Rightarrow a = 5 \, m/s^{2}$ $F(h - 0.9) = mg \times 0.3$ ∴ $100(h - 0.9) = 200 \times 0.3$ ∴ h = 1.5
and $100(0.9 - h) = 200 \times 3$ $\Rightarrow$ h = 0.3
∴ 0.3 < h < 1.5
and $100(0.9 - h) = 200 \times 3$ $\Rightarrow$ h = 0.3
∴ 0.3 < h < 1.5
Q269
Allen
JEE Advanced Practice Paper
Numerical
A uniform disc of mass m and radius R rotates about a fixed vertical axis passing through its centre with angular velocity $\omega$ . A particle of same mass m and having velocity $2\ \omega R$ towards centre of the disc collides with the disc moving horizontally and sticks to its rim the angular velocity of the disc is $\omega/n$ . Find n.
Correct Answer: 3
Explanation:
Ans. (3) $\frac{mR^2}{2}\omega = \left(\frac{mR^2}{2} + mR^2\right)\omega'$ $\therefore \omega' = \frac{\omega}{3}$
Q270
Allen
JEE Advanced Practice Paper
Match the Columns
A rigid cylinder is kept on a smooth horizontal surface as shown. If Column-I indicates velocities of various points (3-centre of cylinder, 2- top point, 4-bottom point, 1- on the level of 3 at the rim) on it shown, choose correct state of motion from Column-II.
| Column-I | Column-II | ||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
|
|
Correct Answer: (A-R), (B-S), (C-P), (D-Q)
Explanation:
Ans. (A-R), (B-S), (C-P), (D-Q)
(A) $V_{cm} = l$ and $\omega R = 1$ (B) $v_{rel\ 1/3} \neq 0$ ∴ Not a rigid body
(C) $v_{2} = \hat{i}, v_{3} = 0$ i.e. rotating at same position only (p).
(D) $V_{i} = -\hat{i} - \hat{j}$ $V_{4} = 0$ (Lowest point) so (Q).
## Important Notes
The image contains only background horizontal lines, which are stylistic or background elements and not text placeholders. According to the correction guidelines, such lines must be ignored. Therefore, the correct OCR output is empty.
(no text)
## Important Notes
The image contains only background horizontal lines, which are stylistic or background elements and not text placeholders. According to the correction guidelines, such lines must be ignored. Therefore, the correct OCR output is empty.
(no text)
(A) $V_{cm} = l$ and $\omega R = 1$ (B) $v_{rel\ 1/3} \neq 0$ ∴ Not a rigid body
(C) $v_{2} = \hat{i}, v_{3} = 0$ i.e. rotating at same position only (p).
(D) $V_{i} = -\hat{i} - \hat{j}$ $V_{4} = 0$ (Lowest point) so (Q).
## Important Notes
The image contains only background horizontal lines, which are stylistic or background elements and not text placeholders. According to the correction guidelines, such lines must be ignored. Therefore, the correct OCR output is empty.
(no text)
## Important Notes
The image contains only background horizontal lines, which are stylistic or background elements and not text placeholders. According to the correction guidelines, such lines must be ignored. Therefore, the correct OCR output is empty.
(no text)