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Rotational Motion

270 Questions Start Allen Test
All Exams Allen Advance Errorless
Q126 Allen Impulse & Angular Impulse MCQ
Two particles each of mass M are connected by a massless rod of length l. The rod is lying on the smooth surface. If one of the particle is given an impulse MV as shown in the figure then angular velocity of the rod would be
A.
v/l
B.
$2v / l$
C.
v/2l
D.
None
Q127 Allen Impulse & Angular Impulse MCQ
A uniform rod AB of mass m and length l is at rest on a smooth horizontal surface. An impulse J is applied to the end B, perpendicular to the rod in the horizontal direction. Speed of particle P at a distance $\frac{l}{6}$ from the centre towards A of the rod after time $t = \frac{\pi m l}{12J}$ is
A.
$2 \frac{J}{m}$
B.
$\frac{J}{\sqrt{2}m}$
C.
$\frac{J}{m}$
D.
$\sqrt{2}\frac{J}{m}$
Q128 Allen Miscellaneous MCQ
A sphere is placed rotating with its centre initially at rest in a corner as shown in figure (a) and (b). Coefficient of friction between all surfaces and the sphere is 1/3. Find the ratio of the frictional force $\frac{f_a}{f_b}$ by ground in situations (a) and (b).
(a)

(b)
A.
1
B.
$\frac{9}{10}$
C.
$\frac{10}{9}$
D.
none
Q129 Allen Miscellaneous MCQ
A yo-yo is resting on a perfectly rough horizontal table. Forces $F_{1}$ , $F_{2}$ and $F_{3}$ are applied separately as shown. The correct statement is :-
A.
when $F_{3}$ is applied the centre of mass will move to the right.
B.
when $F_{2}$ is applied the centre of mass will move to the left.
C.
when $F_{1}$ is applied the centre of mass will move to the right.
D.
when $F_{2}$ is applied the centre of mass will move to the right.
Q130 Allen Miscellaneous MCQ
One ice skater of mass m moves with speed 2v to the right, while another of the same mass m moves with speed v toward the left, as shown in figure I. Their paths are separated by a distance b. At t = 0, when they are both at x = 0, they grasp a pole of length b and negligible mass. For t > 0, consider the system as a rigid body of two masses m separated by distance b, as shown in figure II. Which of the following is the correct formula for the motion after t = 0 of the skater initially at y = b/2?
A.
$x = 2vt, y = b / 2$
B.
$x = vt + 0.5b\sin (3vt / b),y = 0.5b\cos (3vt / b)$
C.
$x = 0.5vt + 0.5b \sin(3vt/b), y = 0.5b \cos(3vt/b)$
D.
$x = 0.5vt + 0.5b \sin(6vt/b), y = 0.5b \cos(6vt/b)$
Q131 Allen MULTIPLE CORRECT TYPE QUESTIONS MSQ
A uniform rod of mass M = 2 kg and length L is suspended by two smooth hinges 1 and 2 as shown in the figure. A force F = 4N is applied downward at a distance from the hinge 2. Due to the application of force F, the hinge 2 breaks. At this instant, applied force F is also removed. The rod starts to rotate downward about hinge 1. $(g = 10 \, m/s^{2})$ . Choose the correct statement(s) :-
A.
The reaction at hinge 1, before hinge 2 breaks, is 11 N
B.
The reaction at hinge 1, before hinge 2 breaks, is 24 N
C.
The reaction at hinge 1, just after breaking of hinge 2 is 5N
D.
The reaction at hinge 1, just after breaking of hinge 2 is 8N
Q132 Allen MULTIPLE CORRECT TYPE QUESTIONS MSQ
A thin ring is free to rotate in the vertical plane about O. Ring is released from rest at $\theta = 0$ , when ring rotates through angle $\theta$ :
A.
Angular acceleration of ring is $\frac{g}{2}\frac{\cos\theta}{R}$
B.
Centripetal acceleration of ring is $2g \sin\theta$
C.
Net force by hinge is $\frac{mg}{2}\sqrt{15\sin^{2}\theta+1}$
D.
Net force by hinge is $mg\sqrt{4\sin^{2}\theta+\frac{3}{4}\cos^{2}\theta}$
Q133 Allen MULTIPLE CORRECT TYPE QUESTIONS MSQ
Two particles each of mass m are attached at end points of a massless rod AB of length $\ell$ . Rod is hinged at point C as shown. Rod is released from rest from horizontal position. At the instant when rod reaches its vertical position as shown, which of the following is/are CORRECT :-
A.
Speed of the particle at B is thrice the speed of particle at A
B.
Net force on particle B is $\frac{6mg}{5}$
C.
Angular acceleration of the system is zero
D.
Both $x \& y$ components of hinge force are nonzero
Q134 Allen MULTIPLE CORRECT TYPE QUESTIONS MSQ
A rod is connected with the (rigid body) disc at centre of disc point 'P'. The disc can freely rotate about point P where it is hinged with rod and rod can rotate freely about fixed hinge Q. Now whole arrangement is released from horizontal position as shown in the diagram.
A.
Initial angular acceleration of (rigid body) disc is zero.
B.
Initial angular acceleration of (rigid body) disc is $\frac{9g}{8\ell}$ .
C.
Angular speed of (rigid body) disc when rod becomes vertical is zero.
D.
Angular speed of (rigid body) disc when rod becomes vertical is $\frac{3}{2}\sqrt{\frac{g}{\ell}}$ rad/s.
Q135 Allen MULTIPLE CORRECT TYPE QUESTIONS MSQ
ABCD is a square plate with centre O. The moments of inertia of the plate about the perpendicular axis through O is I and about the axes 1, 2, 3 and 4 are $I_{1}$ , $I_{2}$ , $I_{3}$ and $I_{4}$ respectively. It follows that:
A.
$I_{2} = I_{3}$
B.
$I = I_{1} + I_{4}$
C.
$I = I_{2} + I_{4}$
D.
$I_{1} = I_{3}$
Q136 Allen MULTIPLE CORRECT TYPE QUESTIONS MSQ
A block with a square base measuring 'a' × 'a' and height h, is placed on an inclined plane. The coefficient of friction is μ. The angle of inclination (θ) of the plane is gradually increased. The block will
A.
topple before sliding if $\mu > \frac{a}{h}$
B.
topple before sliding if $\mu < \frac{a}{h}$
C.
slide before toppling if $\mu > \frac{a}{h}$
D.
slide before toppling if $\mu < \frac{a}{h}$
Q137 Allen MULTIPLE CORRECT TYPE QUESTIONS MSQ
A particle falls freely near the surface of the earth. Consider a fixed point O (not vertically below the particle) on the ground.
A.
Angular momentum of the particle about O is increasing.
B.
Torque of the gravitational force on the particle about O is decreasing.
C.
The moment of inertia of the particle about O is decreasing.
D.
The angular velocity of the particle about O is increasing.
Q138 Allen MULTIPLE CORRECT TYPE QUESTIONS MSQ
A small ball of mass m suspended from the ceiling at a point O by a thread of length $\ell$ moves along a horizontal circle with a constant angular velocity $\omega$ .
A.
angular momentum is constant about O
B.
angular momentum is constant about C
C.
vertical component of angular momentum about O is constant
D.
Magnitude of angular momentum about O is constant
Q139 Allen MULTIPLE CORRECT TYPE QUESTIONS MSQ
A block of mass m moves on a horizontal rough surface with initial velocity v. The height of the centre of mass of the block is h from the surface. Consider a point A on the surface.
A.
angular momentum about A is mvh initially
B.
the velocity of the block decreases at time passes.
C.
torque of the forces acting on block is zero about A
D.
angular momentum is not conserved about A.
Q140 Allen MULTIPLE CORRECT TYPE QUESTIONS MSQ
A satellite in outer-space is at an initial rotation speed of $\omega = 4$ rad/sec at t = 0. As it deploys its solar panels, it changes its rotational inertia as shown in graph below. The correct plot of magnitude of angular momentum and angular velocity as functions of time :
A.
B.
C.
D.
Q141 Allen MULTIPLE CORRECT TYPE QUESTIONS MSQ
In making a science-fiction movie about an asteroid that collides with Earth, a studio constructs a model Earth pivoted at a support fixed at the south Pole and spinning rapidly about the (vertical) south-to-north axis with angular speed $\omega$ pointing out of the North Pole. A Model asteroid approaches horizontally and strikes a grazing blow with the model Earth just at the North Pole. Describe the new motion of the Model Earth.
A.
The earth now rotates about an axis which is initially in YZ plane but rotates about the Z axis.
B.
The earth now rotates about an axis which is initially in XZ plane but rotates about the Z axis.
C.
The angular velocity of earth in Z direction remains the same
D.
The angular momentum of earth in Z direction remains the same.
Q142 Allen MULTIPLE CORRECT TYPE QUESTIONS MSQ
A circular disc of radius R rolls without slipping on a rough horizontal surface. At the instant shown its linear velocity is V, linear acceleration a, angular velocity $\omega$ and angular acceleration a. Four points A, B, C and D lie on its circumference such that the diameter AC is vertical and BD horizontal then choose the correct option(s).
A.
$V_{B} = \sqrt{V^{2} + (R\omega)^{2}}$
B.
$V_{C} = V + R\omega$
C.
$a_{A} = \sqrt{(a - R\alpha)^{2} + (\omega^{2}R)^{2}}$
D.
$a_{D} = \sqrt{(a + \omega^{2}R)^{2} + (R\alpha)^{2}}$
Q143 Allen MULTIPLE CORRECT TYPE QUESTIONS MSQ
The disc of radius r is confined to roll without slipping at A and B. If the plates have the velocities shown, then
A.
linear velocity $v_{0} = v$
B.
angular velocity of disc is $\frac{3v}{2r}$
C.
angular velocity of disc is $\frac{2v}{r}$
D.
None of these
Q144 Allen MULTIPLE CORRECT TYPE QUESTIONS MSQ
A massless rod has a massless hollow sphere attached to it. This sphere can be fully filled either with a liquid (non viscous) or with a solid (rigidly fitted into sphere) of same mass. System is released from rest from initial position (as shown). When it reaches final position which of the following is/are true for the system.
A.
Kinetic energy in case of liquid will be more than in case of solid.
B.
Velocity of centre (V) in case of liquid will be more than in case of solid.
C.
About C angular momentum in case of liquid will be more than in case of solid.
D.
About C angular momentum in case of liquid will be less than in case of solid.
Q145 Allen MULTIPLE CORRECT TYPE QUESTIONS MSQ
A disc of radius a is rigidly attached at its circumference by two rods each of length 3a, two rods are always perpendicular to each other. The combination is suspended vertically from the free end of one rod (as shown in figure). It is swinging in the plane of the disc such that the centre of disc have velocity v. If mass of disc and each rod is m.
A.
Kinetic energy of the disc $\frac{1}{2} mV^2 +\frac{1}{2}\left(\frac{ma^2}{2}\right)\left(\frac{V}{4a}\right)^2$
B.
Kinetic energy of the rod 2 is $\frac{1}{2} mV^2 +\frac{1}{2}\times \left(\frac{m(3a)^2}{12}\right)\times \left(\frac{V}{4a}\right)^2$
C.
Kinetic energy of the rod 1 is $\frac{1}{2} \times \left( \frac{m(3a)^2}{3} \right) \times \left( \frac{V}{4a} \right)^2$
D.
Kinetic energy of the system of (disc and both rod) is
$$<br /> \frac {1}{2} \left[ \left(\frac {m (3 a) ^ {2}}{3}\right) + \left(\frac {m a ^ {2}}{2} + m (4 a) ^ {2}\right) + \left(\frac {m (3 a) ^ {2}}{1 2} + m \frac {8 9}{4} a ^ {2}\right) \right] \left(\frac {V}{4 a}\right) ^ {2}<br /> $$
Q146 Allen MULTIPLE CORRECT TYPE QUESTIONS MSQ
A disc of mass ‘m’ and radius ‘R’ is rotating with angular speed $\omega_{0}$ and speed zero as shown and then placed on a rough surface (coefficient of friction= $\mu$ ) and left. Then
A.
angular momentum of disc is conserved about point Q.
B.
angular momentum of disc is conserved about point P on the ground.
C.
initially the force of friction is $\mu mg$ leftward.
D.
initially the force of friction is $\mu$ mg rightward.
Q147 Allen MULTIPLE CORRECT TYPE QUESTIONS MSQ
A billiard ball initially at rest is given a sharp blow by a cue stick. The impulse is horizontal and is applied at a distance 2R/3 below the centreline, as shown in figure. The initial speed of the ball is $v_{0}$ , and the coefficient of kinetic friction is $\mu_{k}$ .
A.
Initially kinetic friction acts in $-\hat{i}$ direction
B.
Initially kinetic friction acts in $\hat{i}$ direction.
C.
Ball instantaneously starts pure rolling.
D.
Initial angular velocity of ball is $\frac{5v_{0}}{3R}$
Q148 Allen MULTIPLE CORRECT TYPE QUESTIONS MSQ
A solid sphere of mass 10 kg is placed on a rough horizontal surface with coefficient of friction $\mu = 0.1$ . If a force of 28 N is applied at the top of the sphere then,
A.
friction force will be 10 N in forward direction.
B.
friction force will be less than 10N in backward direction.
C.
Acceleration of centre of mass will be $\frac{29}{7}$ m/s $^{2}$ .
D.
Acceleration of centre of mass will be $3.8m/s^{2}$ .
Q149 Allen MULTIPLE CORRECT TYPE QUESTIONS MSQ
If a cylinder is rolling down the incline with sliding.
A.
after some time it may start pure rolling
B.
after sometime it will start pure rolling
C.
it may be possible that it will never start pure rolling
D.
none of these
Q150 Allen MULTIPLE CORRECT TYPE QUESTIONS MSQ
A plank with a uniform sphere placed on it rests on a smooth horizontal plane. Plank is pulled to right by a constant force F. If sphere does not slip over the plank. Which of the following is correct.
A.
Acceleration of the centre of sphere is less than that of the plank.
B.
Work done by friction acting on the sphere is equal to its total kinetic energy.
C.
Total kinetic energy of the system is equal to work done by the force F.
D.
None of the above