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Rotational Motion

270 Questions Start Allen Test
All Exams Allen Advance Errorless
Q176 Allen Impulse & Angular Impulse MCQ

Comprehension Passage: PARAGRAPH FOR QUESTION NO. 129 TO 131

A uniform rod of mass 4m and length L lies on a smooth horizontal table. A particle of mass m moving on the table with speed $v_{0}$ strikes the rod perpendicularly at an end and stops. Angular velocity of the rod after collision is
A.
$\frac{3v_0}{2L}$
B.
$\frac{3v_0}{4L}$
C.
$\frac{v_0}{2L}$
D.
$\frac{v_0}{4L}$
Q177 Allen Impulse & Angular Impulse MCQ

Comprehension Passage: PARAGRAPH FOR QUESTION NO. 129 TO 131

A uniform rod of mass 4m and length L lies on a smooth horizontal table. A particle of mass m moving on the table with speed $v_{0}$ strikes the rod perpendicularly at an end and stops. Distance travelled by the centre of rod by the time it turns through one revolution is
A.
$\pi L$
B.
$2\pi L$
C.
$\frac{\pi L}{12}$
D.
$\frac{\pi L}{3}$
Q178 Allen Miscellaneous MSQ

Comprehension Passage: PARAGRAPH FOR QUESTION NO. 132 TO 135

A ring of mass M and radius R sliding with a velocity $v_{0}$ suddenly enters into rough surface where the coefficient of friction is $\mu$ , as shown in figure. Choose the correct statement(s)
A.
As the ring enters on the rough surface, the limiting friction force acts on it.
B.
The direction of friction is opposite to the direction of motion.
C.
The friction force accelerates the ring in the clockwise sense about its centre of mass.
D.
As the ring enters on the rough surface it starts rolling.
Q179 Allen Miscellaneous MCQ

Comprehension Passage: PARAGRAPH FOR QUESTION NO. 132 TO 135

A ring of mass M and radius R sliding with a velocity $v_{0}$ suddenly enters into rough surface where the coefficient of friction is $\mu$ , as shown in figure. Choose the correct statement(s)
A.
The momentum of the ring is conserved
B.
The angular momentum of the ring is conserved about its centre of mass
C.
The angular momentum of the ring conserved about any point on the horizontal surface in line of friction.
D.
The mechanical energy of the ring is conserved
Q180 Allen Miscellaneous MSQ

Comprehension Passage: PARAGRAPH FOR QUESTION NO. 132 TO 135

A ring of mass M and radius R sliding with a velocity $v_{0}$ suddenly enters into rough surface where the coefficient of friction is $\mu$ , as shown in figure. Choose the correct statement(s):
A.
The ring starts its rolling motion when the centre of mas stationary.
B.
The ring starts rolling motion when the point of contact becomes stationary.
C.
The time after which the ring starts rolling is $\frac{v_{0}}{2\mu g}$ .
D.
The rolling velocity is $\frac{v_{0}}{2}$ .
Q181 Allen Miscellaneous MSQ

Comprehension Passage: PARAGRAPH FOR QUESTION NO. 132 TO 135

A ring of mass M and radius R sliding with a velocity $v_{0}$ suddenly enters into rough surface where the coefficient of friction is $\mu$ , as shown in figure. Choose the correct alternative(s) :-
A.
The linear distance moved by the centre of mass before the ring starts rolling is $\frac{3v_{0}^{2}}{8\mu g}$ .
B.
The net work done by friction force is $-\frac{3}{8} mv_0^2$ .
C.
The loss is kinetic energy of the ring is $\frac{mv_0^2}{4}$ .
D.
The gain in rotational kinetic energy is $+\frac{mv_0^2}{8}$ .
Q182 Allen CRTM Kinematics Match the Columns
Consider a body rolling on a horizontal surface as shown in figure (Symbols have their usual meaning)

Column-IColumn-II
(A)Velocity is zero
(B)Speed in maximum
(C)$0 < \mathrm{speed} < v_{cm}$
(D)$1.3v_{\mathrm{cm}} < \mathrm{speed} < 2v_{\mathrm{cm}}$
(P)Point $A$
(Q)Point $B$
(R)Point $C$
(S)Point $D$
(T)Point E
Q183 Allen Miscellaneous Match the Columns
depicts various situations where some sudden events are taking place. describes changes in various parameters of systems immediately after the events taking place in .
Column-I
Column-IColumn-II
(A)Joker is standing on revolving platform and batman throws the ball and joker catches the ball while it was moving horizontally.

Joker, ball and platform is system.
(B)Joker throws the ball horizontally and perpendicular to his motion while standing on the revolving platform.

Joker, ball and platform is system.
(C)Joker jumps horizontally towards right from the cart which is moving at speed v on smooth horizontal floor.

Joker and cart is the system
(D)Joker drops himself vertically from the moving cart with no horizontal velocity relative to cart.

Joker and cart is the system
(P)Linear momentum remains conserved.
(Q)Mechanical energy is conserved
(R)Mechanical energy increases.
(S)Mechanical energy decreases.
(T)$v$ or $\omega$ changes
Q184 Allen Concept of $w, \alpha, v, a$ Numerical
A uniform bar AB of mass m and a ball of the same mass are released from rest from the same horizontal position. The bar is hinged at end A. There is gravity downwards. What is the distance of the point from point B that has the same acceleration as that of ball, immediately after release?
Q185 Allen Concept of $w, \alpha, v, a$ Numerical
A uniform beam of length L and mass m is supported as shown. If the cable at B suddenly breaks, determine; (a) the acceleration of end B. (b) the reaction at the pin support.
Q186 Allen Concept of $w, \alpha, v, a$ Numerical
A rod of mass m and length $\ell$ is hinged at it's upper end. It can rotate in vertical plane. It is given angular velocity $\omega$ so that it can complete vertical circle. Find (a) $\omega$ (b) Tension at centre of rod at initial moment.
Q187 Allen Moment of Inertia Numerical
Three equal masses m are rigidly connected to each other by massless rods of length l forming an equilateral triangle, as shown in the figure. What is the ratio of the moment of inertia of the assembly for an axis through B compared with that for an axis through A (centroid). Both the axis are perpendicular to the plane of triangle.
Q188 Allen Moment of Inertia Numerical
Two rods of equal mass m and length l lie along the x axis and y axis with their centres at origin. What is the moment of inertia of the system about the line x = y
Q189 Allen Moment of Inertia Numerical
Find the MI of a rod about an axis through its centre of mass and perpendicular to the length whose linear density varies as $\lambda = ax$ where a is a constant and x is the position of an element of the rod relative to its left end. The length of the rod is $\ell$ .
Q190 Allen Moment of Inertia Numerical
A hole of radius $R / 2$ is cut from a solid sphere of radius $R$ . If the mass of the remaining part of sphere is $M$ , then find moment of inertia of the body about an axis through $O$ .
Q191 Allen Equilibrium Numerical
In an experiment with a massless beam balance an unknown mass m is balanced by two known masses of 16kg and 4 kg as shown in figure. Find the value of the unknown mass m.
Q192 Allen Equilibrium Numerical
Find the force F required to keep the system in equilibrium. The dimensions of the system are d = 0.3 m and a = 0.2 m. Assume the rods to be massless.
Q193 Allen Equilibrium Numerical
A 3.0 kg bobbin consists of a central cylinder of radius 5.0 cm and two end plates each of radius 6.0 cm. It is placed on a slotted incline, where friction is sufficient to prevent sliding. A block of mass 4.5 kg is suspended from a cord wound around the bobbin and passing through the slot under the incline. If the bobbin is in static equilibrium, what is the angle of tilt of the incline?
Q194 Allen $\tau = I\alpha$ Torque Numerical
A force $\vec{F}=2\hat{i}+3\hat{j}N$ is applied to an object that is pivoted about a fixed axle aligned along the z coordinate axis. If the force is applied at the point $\vec{r}=4\hat{i}+5\hat{j}m$ , find (a) the magnitude of the net torque about the z-axis and (b) the direction of the torque vector $\tau$ .
Q195 Allen $\tau = I\alpha$ Torque Numerical
In the following figure $r_{1}$ and $r_{2}$ are 5 cm and 30 cm respectively. If the moment of inertia of the wheel is $5100 kg \cdot m^{2}$ about the axis passing through its centre and perpendicular to the plane of wheel, then what will be its angular acceleration?
Q196 Allen $\tau = I\alpha$ Torque Numerical
A turning moment of 100 Nm is applied to a shaft which passes through a wheel of radius R = 50cm. The minimum force with which brakes must be applied to the wheel perpendicular to the surface (see figure) so that the wheel does not rotate is..... Ɨ $10^{2}$ N. The coefficient of friction equals 0.25.
Q197 Allen Angular Momentum & its Conservation Numerical
A uniform ring is rotating about vertical axis with angular velocity $\omega$ initially. A point insect (S) having the same mass as that of the ring starts walking from the lowest point $P_{1}$ and finally reaches the point $P_{2}$ (as shown in figure). What is the final angular velocity of the ring?
Q198 Allen Kinetic Energy Numerical
A ring of mass m and radius R has three particles attached to the ring as shown in the figure. The centre of the ring has a speed $v_{0}$ . Find the kinetic energy of the system. (Slipping is absent)
Q199 Allen Kinetic Energy Numerical
A small sphere of mass m and radius r is released from rest at A and rolls without sliding on the curved surface to point B where it leaves the surface with a horizontal velocity. Knowing that a = 1.5 m and b = 1.2 m, determine; (A) the speed of the sphere as it strikes the ground at C. (B) the corresponding distance $c$ .
Q200 Allen Kinetic Energy Numerical
A rectangular rigid fixed block has a long horizontal edge. A solid homogeneous cylinder of radius R is placed horizontally at rest with its length parallel to the edge such that the axis of the cylinder and the edge of the block are in the same vertical plane as shown in figure. there is sufficient friction present at the edge so that a very small displacement cause the cylinder to roll of the edge without slipping. Determine: (A) the angle $\theta_{c}$ through which the cylinder rotates before it leaves contact with the edge, (B) the speed of the centre of mass of the cylinder before leaving contact with the edge, and (C) the ratio of the translational to rotational kinetic energies of the cylinder when its centre of mass is in horizontal line with the edge.