Newton's Second Law: Rotatory Version

Q.

An insect is at the bottom of a hemispherical ditch of radius $1m$. It crawls up the ditch but starts slipping after it is at height $h$ from the bottom. If the coefficient of friction between the ground and the insect is $0.75$, then $h$ is: ($g=10m{s}^{-2}$)

1. $0.45m$

2. $0.60m$

3. $0.20m$

4. $0.80m$

Q. A rod is pivoted about its center. A $5\text{N}$ force is applied $4\text{m}$ from the pivot and another $5\text{N}$ force is applied $2$ $\text{m}$ from the pivot, as shown. The magnitude of the total torque about the pivot (in $\text{N-m}$) is :

1. $0$
2. $5$
3. $15$
4. $8.7$

Q.

A body is rolling down an inclined plane without slipping. How does the acceleration of the rolling body depend on its radius?

Q. A solid cylinder of mass $m=4\text{kg}$ and radius $R=10\text{cm}$ has a string wrapped around it as shown in figure.The cylinder is held horizontal by fixing the two free ends of the string to the hook on the ceiling such that both the strings are exactly vertical. The cylinder is released to fall under gravity. Find the linear acceleration $\left(a\right)$ of the cylinder.

1. $\frac{2g}{3}$
2. $\frac{g}{3}$
3. $\frac{g}{2}$
4. $g$

Q. The pulley shown in figure has moment of inertia $I=5{\text{kg-m}}^2$ about its axis and radius $R=1\text{m}$. Find the acceleration of the blocks if the masses of the blocks are $M=3\text{kg}$ and $m=2\text{kg}$. Assume that the string is light and does not slip on the pulley and $g=10{\text{m/s}}^2$.

1. $3{\text{m/s}}^2$
2. $1{\text{m/s}}^2$
3. $2{\text{m/s}}^2$
4. $\frac{1}{2}{\text{m/s}}^2$

Q. A rod $PQ$ of mass $m$ and length $L$ is hinged at its one end $P$. The rod is kept horizontal by a massless string tied to end $Q$. When the string is cut, initial angular acceleration of the rod is

1. $\frac{3g}{2L}$
2. $\frac{g}{L}$
3. $\frac{2g}{L}$
4. $\frac{2g}{3L}$

Q. A certain bicycle can go up on a gentle incline with a constant speed. If the frictional force of the ground pushing the rear wheel is $F_2=4\text{N}$, then what pulling force $F_1$ must the chain apply on the sprocket wheel if $R_1=30\text{m}$ and $R_2=5\text{m}$?

1. $4\text{N}$
2. $24\text{N}$
3. $140\text{N}$
4. $\frac{35}{4}\text{N}$

Q. A fly wheel of moment of inertia $3\times {10}^{2}kg{m}^2$ is rotating with uniform angular speed of 4.6 $rad{s}^{-1}.$ If a torque of $6.9\times {10}^2$ Nm retards the wheel, then the time in which the wheel comes to rest is

1. 1.5 s
2. 2 s
3. 0.5 s
4. 1 s
5. 2.5 s

Q. A circular disc of mass ${}^{\prime }2m^{\prime }$ and radius ${}^{\prime }{R}^{\prime }$, is hinged at centre. A long string is wound over this disc and a body of mass ${}^{\prime }{m}^{\prime }$ is attached at its free end. Now the body is released. Find acceleration of this body.

1. $\frac{g}{2}$
2. $g$
3. $2g$
4. $\frac{g}{4}$

Q. If the net force acting on a rigid body is zero it implies that body can't have angular acceleration.

1. False
2. True

Q. A rod $PQ$ of mass $M$ and length $L$ is hinged at end $P$. The rod is kept horizontal by a massless string tied to point $Q$ as shown in figure. When string is cut, the initial angular acceleration of the rod is

1. $g/L$
2. $2g/L$
3. $\frac{2g}{3L}$
4. $\frac{3g}{2L}$

Q.

If a body is rotating about a given Axis and a Force is applied at any point on that body along any random direction, what can we say about the torque of this force about any point lying on its Axis of rotation?

1. The torque will be different about different points along the axis of rotation.

2. The torque will be same about different point along the axis of rotation

3. The magnitude of torque will remain the same about different points but the direction would vary.

4. It depends upon the orientation of the body and therefore not predictable

Q. A uniform rod of length $L$ and mass $M$ is pivoted freely at one end and placed in vertical position. What is the angular acceleration of the rod when it is at an angle $\theta$ with the vertical?

1. $\frac{2g\sin \theta }{L}$
2. $\frac{3g\sin \theta }{2L}$
3. $\frac{3g\sin \theta }{L}$
4. $\frac{g\sin \theta }{2L}$

Q. A thin uniform rod of length $3\text{m}$ and mass $M$ is held horizontally by two vertical strings attached to the two ends, as shown in figure. One of the strings is cut. Find the angular acceleration of the rod at the moment it is cut. (Take $g=10{\text{m/s}}^2$).

1. $15{\text{rad/s}}^2$
2. $10{\text{rad/s}}^2$
3. $5{\text{rad/s}}^2$
4. $7.5{\text{rad/s}}^2$

Q.

If the line of action of the applied force intersects the Axis of rotation, then the torque of this force about the axis of rotation is

1. Always zero

2. Always non-zero perpendicular to the axis of rotation

3. Sometime non-zero

4. Non-zero and along the axis of rotation

Q.

A hollow sphere of radius 'R' rests on a horizontal surface of finite coefficient of friction. A point object of mass 'm' moved horizontally and hits the sphere at a height of 'R/2' above its center. The collision is instantaneous and completely inelastic. Which of the following is/are correct ?

1. Total linear momentum of the system is not conserved
2. Total angular momentum about center of mass of the system remains conserved

3. The sphere gets finite angular velocity immediately after collision

4. The sphere moves with finite speed immediately after collision

Q. A cylinder of mass $m=1\text{kg}$ is suspended through two strings wrapped around it as shown in figure. Find the acceleration of $COM$ of the cylinder.

1. $\frac{2g}{5}$
2. $\frac{2g}{3}$
3. $g$
4. $\frac{g}{4}$

Q. A thread is wound around cylinder of mass $M$ and radius $R$. It is allowed to fall as shown. Find its acceleration.

1. $\frac{g}{3}$
2. $\frac{4g}{3}$
3. $\frac{2g}{3}$
4. None

Q.

A uniform solid cylinder of mass 'M' and radius 'R' is resting on a horizontal platform (which is parallel to X-Y plane) with its axis along the Y-axis and free to roll on the platform. The platform is given a motion in X-direction given by $x=A\cos \omega t$. There is no slipping between the cylinder and the platform. The maximum torque acting on the cylinder as measured about its centre of mass is

1. $\frac{1}{2}MRA{\omega }^2$
2. $MRA{\omega }^2$
3. $2mRA{\omega }^2$
4. $mRA\omega A^2{\cos }^2\omega t$

Q. A thin uniform rod, pivoted at O, is rotating in the horizontal plane with constant angular speed $\omega$, as shown int he figure. At time t = 0, a small insect starts from O and moves with constant speed v with respect to the rod towards the other end. It reaches the end of the rod at t = T and stops. The angular speed of the system remains $\omega$ throughout. The magnitude of the torque $\left(\overrightarrow{\tau }\right)$ on the system about O, as a function of time is best represented by which plot?

1. 
2. 
3. 
4. 

Q. A uniform circular disc of radius 50 cm at rest is free to turn about an axis which is perpendicular to its plane and passes through its centre. It is subjected to a torque which produces a constant angular acceleration of 2.0 rad $s^{-2}.$ Its net acceleration in $m{s}^{-2}$ at the end of 2.0 s is approximately :

1. 3
2. 8
3. 7
4. 6

Q. A ring of radius 1m has moment of inertia of $10kg-m^2$ about an axis through its centre and perpendicular to its plane. If a constant force of 50N is applied to it tangentially as shown, what will be its angular acceleration?

1. $5rad/s^2$
2. $10rad/s^2$
3. $2.5rad/s^2$
4. $\pi rad/s^2$

Q. A thread is wound around cylinder of mass $M$ and radius $R$. It is allowed to fall as shown. Find its acceleration.

1. $\frac{g}{3}$
2. $\frac{4g}{3}$
3. $\frac{2g}{3}$
4. None

Q.

If net external force on a body adds up to zero, which of the following statements are not necessarily true?

1. Linear acceleration of the centre of mass is zero

2. Angular acceleration of the body has to be zero

3. Net External torque about any point need not be zero

4. Angular momentum about any axis may not be conserved

Q. A pulley of radius 2 m is rotated about its axis by a force $F=(20t-5r^2)$ newton (where t is measured in seconds) applied tangentially. If the moment of inertia of the pulley about its axis fo rotation is 10 $kg{m}^2,$ the number of rotations made by the pulley before its direction of motion is reversed, is:

1. less than 3
2. more than 3 but less than 6
3. more than 6 but less than 9
4. more than 9

Q. Uniform rod $AB$ of mass $3\text{kg}$ and length $2\text{m}$ is hinged at end $A$ in horizontal position as shown in the figure. The other end is connected to a block of mass $1\text{kg}$ through a massless string. The pulley is smooth and massless. Find the angular acceleration of the rod about the hinge just after release. (Take anti-clockwise rotation positive)

1. $\frac{g}{8}$
2. $\frac{g}{4}$
3. $-\frac{g}{8}$
4. $-\frac{g}{2}$

Q. A ring of radius 1m has moment of inertia of $10kg-m^2$ about an axis through its centre and perpendicular to its plane. If a constant force of 50N is applied to it tangentially as shown, what will be its angular acceleration?

1. $5rad/s^2$
2. $10rad/s^2$
3. $2.5rad/s^2$
4. $\pi rad/s^2$

Q. The net torque on a rigid body due to its internal forces is always?

1. Non-Zero
2. Zero
3. Maybe Zero.
4. None of these

Q. A certain bicycle can go up on a gentle incline with a constant speed. If the frictional force of the ground pushing the rear wheel is $F_2=4\text{N}$, then what pulling force $F_1$ must the chain apply on the sprocket wheel if $R_1=30\text{m}$ and $R_2=5\text{m}$?

1. $4\text{N}$
2. $24\text{N}$
3. $140\text{N}$
4. $\frac{35}{4}\text{N}$

Q. In given figure, pulley having radius, $r=1\text{m}$ and moment of inertia $I=2{\text{kgm}}^2$ about its axis of rotation. If $T_1=4\text{N}$ and $T_2=2\text{N}$ are the tension in the strings. Find the angular acceleration of pulley. (Take anti-clockwise rotation +ve)

1. $2{\text{rad/s}}^2$
2. $1{\text{rad/s}}^2$
3. $4{\text{rad/s}}^2$
4. $3{\text{rad/s}}^2$