Rotation + Translation

Q.

A rigid body of mass 'm' and radius 'r' rolls without slipping on a rough horizontal surface. A force is acting on a point 'x' distance from the center as shown in figure. Assuming the moment of inertia of the solid body about an axis through its COM be $I_{CM}$. Find the value of x so that static friction is zero.

1. $\frac{I_{CM}}{r}$

2. $\frac{I_{CM}}{mr}$

3. $\frac{I_{CM}}{m{r}^2}$

4. $\frac{I_{CM}}{m^{2}r}$

Q. A metre stick is held vertically with one end on a rough horizontal floor. It is gently allowed to fall on the floor. Assuming that the end at the floor does not slip, find the angular speed of the rod when it hits the floor

1. 4.32 rad/s

2. 5.42 rad/s

3. 6.35 rad/s

4. Insufficient data

Q.

Two point A and B Lie on two extremes of a diameter AB as shown in figure. The sphere rotates with constant angular velocity $\omega$ about an axis as shown. What is the angular velocity of Q with respect to P?

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3. 2

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Q. A metal ball of mass 'm' is put at the point A of a loop track and the vertical distance of A from the lower most point of track is 8 times the radius 'R' of the circular part. The linear velocity of ball when it rolls of the point B to a height 'R' in the circular track will be

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Q.

A smooth sphere A is moving on a frictionless horizontal plane with angular speed $\omega$ and centre of mass is moving translationally with velocity 'v'. It collides elastically and head-on with an identical sphere B which is at rest. All surfaces are frictionless. After the collision, their angular speeds are ${\omega }_A$ and ${\omega }_B$, respectively. Then,

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Q.

Two blocks of masses 400 g and 200 g are connected through a light string going over a pulley which is free to rotate about its axis. The pulley has a moment of inertia $1.6\times {10}^{-4}kg-m^2$ and a radius 2.0 cm. Find 'x', which is the kinetic energy of the system as the 400 g block falls through 50 cm and also find 'y', the speed of the blocks at this instant.

1. x = 10.5 J, y = 2.6 m/s

2. x = 9.8 J, y = 2.6 m/s

3. x = 9.8 J, y = 1.4 m/s

4. x = 0.98 J, y = 1.4 m/s

Q. A uniform rod of mass m and length l is kept vertical with the lower end clamped. It is slightly pushed to let it fall down under gravity. Find its angular speed when the rod is passing through its lowest position. Neglect any friction at the clamp. What will be the linear speed of the free end at this instant?

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Q. A solid sphere of mass $m=80\text{kg}$ and radius $r=0.2\text{m}$ is released from height $h=\frac{5}{4}$ meter. Sphere is initially rotating about horizontal axis passing through its centre of mass. It hits with a stationary cart of mass $M=200\text{kg}$ exactly at the centre of cart. The cart can move smoothly on the horizontal surface.
The collision between sphere and cart occurs in such a way that sphere reaches at same vertical displacement after collision and falls back onto it again. It is found that sphere starts pure rolling at the end of first collision. The coefficient of friction between sphere and cart is $\mathrm{0.1.}$
Match the statement given in List-I to the values(in SI units) given in List-II.

$\begin{array}{cccc}& \text{Column-I}& & \text{Column-II}\\ \text{P.}& \text{The minimum length (in meter) of cart to occur second collision with the sphere.}& 1.& 172\\ \text{Q.}& \text{Initial angular velocity}{}^{\prime }{\omega }^{\prime }\left(\text{in}\frac{\text{rad}}{\text{sec}}\right)\text{of sphere on the cart during the process.}& 2.& 2.8\\ \text{R.}& \text{Magnitude of work done (in Joule) by sphere on the cart during the process.}& 3.& 156\\ \text{S.}& \text{Magnitude of work done (in Joule) by cart on the sphere during the process}& 4.& 19.5\\ & & 5.& 16\end{array}$
Which of the following option has the correct combination considering column –I and column – II

1. $S\to 3$
2. $Q\to 4$
3. $R\to 2$
4. $P\to 1$

Q. A small body slides over the curved surface of a semicircular fixed cylinder of radius 'r', kept horizontally on the ground as shown in the figure. At what height from the ground would the body lose contact with the surface

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Q. A cord is wound round the circumference of wheel of radius 'r'. The axis of the wheel is horizontal and moment of inertia about it is 'I'. A weight 'mg' is attached to the end of the cord and falls from rest. After falling through a distance h, the angular velocity of the wheel will be

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