Race of Rollers

Q. A body is rolling without slipping on a horizontal plane. The radius of body is $R$. If the ratio of rotational to translational kinetic energy is $\frac{9}{25}$, then the radius of gyration of the body is

1. $\frac{5R}{3}$
2. $\frac{9R}{25}$
3. $\frac{3R}{5}$
4. $\frac{25R}{9}$

Q. A uniform heavy plank of length L is resting on two rods of identical length and area of cross-section, made up of different materials. It is observed that, to keep the planks horizontal, rod A has to be at the end of the plank and rod B has to be at a distance $\frac{L}{4}$ from centre of the plank as shown in figure.

In another scenario, the rods are kept between two rigid walls such that the separation between the walls is equal to the natural length of the rod. Now the temperature of both the rods is gradually increased in the same manner, and it is observed that thermal stress developed in rod A is always thrice the thermal stress developed in rod B. Ratio of coefficient of linear expansion of rod A to coefficient of linear expansion of rod B is. (Answer upto two digits after the decimal point)

Q. From a uniform circular disc of radius $R$ and mass $9M,$ a small disc of radius $\frac{R}{3}$ is removed as shown in the figure. The moment of inertia of the remaining disc about an axis perpendicular to the plane of the disc and passing through centre of disc is

1. $10{MR}^2$
2. $\frac{37}{9}{MR}^2$
3. ${4MR}^2$
4. $\frac{40}{9}{MR}^2$

Q.

Seven identical circular planar disks, each of mass $M$ and radius $R$ are welded symmetrically as shown. The moment of inertia of the arrangement about the axis normal to the plane and passing through the point P is:

1. $\begin{array}{l}\frac{73}{2}M{R}^2\end{array}$

2. $\begin{array}{l}\frac{181}{2}M{R}^2\end{array}$

3. $\begin{array}{l}\frac{19}{2}M{R}^2\end{array}$

4. $\begin{array}{l}\frac{55}{2}M{R}^2\end{array}$

Q. a small sphere is charged to a potential of 50V and a big hollow sphere is charged to a potential of 100V electricity will flow from the smaller sphere to bigger when

Q.

Find the moment of inertia of a uniform square plate of mass m and edge a about one of its diagonals.

Q.

A solid sphere of mass m and radius r is placed inside a hollow thin spherical shell of mass M and radius R as shown in figure (11-E1). A particle of mass m is placed on the line joining the two centres at a distance x from the point o contact of the sphere and the shell. Find the magnitude of the resultant gravitational force on this particle due to the sphere and the shell if (a) r< x< 2r,
(b) 2r < x < 2R and (c) x> 2R.

Q.

What is a centripetal and centrifugal forces with examples?

Q. An ice skater is spinning about vertical axis with arms fully extended. When he folds his arms, then angular speed ____, angular momentum ____ and moment of inertia ____.

1. Remains constant, increases, decreases
2. Decreases, remains constant, increases
3. Increases, remains constant, decreases
4. None of the above

Q. The figure shows a system consisting of (i) a ring of outer radius 3R rolling clockwise without slipping on a horizontal surface with angular speed $\omega$ and (ii) an inner disc of radius 2R rotating anti-clockwise with angular speed $\omega /2.$ The ring and disc are separated by frictionless ball bearings. The system is in the x-z plane. The point P on the inner disc is at distance R from the origin, where OP makes an angle of ${30}^o$ with the horizontal. Then with respect to the horizontal surface,

1. The point P has a linear velocity $\frac{11}{4}R\omega \hat{i}+\frac{\sqrt{3}}{4}R\omega \hat{k}$
2. The point P has a linear velocity $\frac{13}{4}R\omega \hat{i}+\frac{\sqrt{3}}{4}R\omega \hat{k}$
3. The point P has a linear velocity $(3-\frac{\sqrt{3}}{4})R\omega \hat{i}+\frac{1}{4}R\omega \hat{k}$
4. The point O has a linear velocity $3R\omega \hat{i}$

Q. A body is projected vertically upwards from the surface of earth with a velocity sufficient to carry it to infinity. The time taken by it to reach height $h$ is

1. $\sqrt{\frac{R_e}{2g}}\lfloor {(1+\frac{h}{R_e})}^{3/2}-1\rfloor$
   
2. $\sqrt{\frac{2R_e}{g}}\lfloor {(1+\frac{h}{R_e})}^{3/2}-1\rfloor$
3. $\frac{1}{3}\sqrt{\frac{2R_e}{g}}\lfloor {(1+\frac{h}{R_e})}^{3/2}-1\rfloor$
4. $\frac{1}{3}\sqrt{\frac{R_e}{2g}}\lfloor {(1+\frac{h}{R_e})}^{3/2}-1\rfloor$

Q. A ring and a disc are initially at rest, side by side, at the top of an inclined plane which makes an angle ${60}^o$ with the horizontal. They start to roll without slipping at the same instant of time along the shortest path. If the time difference between their reaching the ground is $\frac{(2-\sqrt{3})}{\sqrt{10}s}$, then the height of the top of the inclined plane, in meters. is. Take $g=10m{s}^{-2}$.

Q. When a uniform solid sphere and a disc of the same mass and of the same radius rolls down without slipping on a smooth inclined plane from rest to the same distance along the inclined plane, then the ratio of the time taken by solid sphere and the disc to cover the distance is

1. $15:14$
2. ${15}^2:{14}^2$
3. $\sqrt{14}:\sqrt{15}$
4. $14:15$

Q. A thin hoop of weight $500N$ and radius $1m$ rests on a rough inclined plane as shown in the figure. The minimum coefficient of friction needed for this configuration is

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

Q.

What is pure rolling?

Q. A perfectly reflecting body (sphere in shape) is placed on a path of parallel light beam of intensity $I$ shown in figure. One end of a spring is attached to the centre of the sphere and the other end to a rigid wall as shown in the figure. Sphere is in equilibrium. Spring constant of spring is $K$. Then compression in spring is
(Radius of sphere is $R$, $c$ is the velocity of light and neglect any thermal effect and friction )

1. $\frac{2I}{Kc}\pi R^2$
2. $\frac{I}{Kc}\pi R^2$
3. $\frac{4I\pi R^2}{Kc}$
4. $\frac{I\pi R^2}{2Kc}$

Q. derive the formulas for potential of solid sphere (non conducting) (1)when point lies outside the surface (2)when point lies at the surface (3)inside the sphe

Q.

Two objects start their motion at some time. One is sliding down from rest on smooth fixed slope and other being thrown from point R at an angle $\theta$ if both get to point P at same instant and at same speed.than if $si{n}^2\theta =\frac{4n+\sqrt{2m}}{2L}$ then, $(n\times L)+m$

Q. A ring of mass $m$ and radius $r$ is allowed to roll down without slipping on an inclined plane having inclination ${\sin }^{-1}\left(\frac{1}{10}\right)$ from the horizontal. The magnitude of acceleration of it's centre of mass is

1. $9.8{\text{ms}}^{-2}$
2. $4.9{\text{ms}}^{-2}$
3. $0.49{\text{ms}}^{-2}$
4. $0.98{\text{ms}}^{-2}$

Q. A uniform constant magnetic field $\overrightarrow{B}$ is directed at an angle of ${45}^{\circ }$ to the $\text{x-axis}$ in the $\text{x-y}$ plane. $\text{PQRS}$ is a rigid, square wire frame carrying a steady current $I_o\left(\text{anticlockwise}\right)$, with its centre at origin $O$. At time $t=0$, the frame is at rest in the position shown in the figure with its sides parallel to the $x$ and $y$ axes. Each side of the frame is of mass $M$ and length $L$.

Find the angle by which the frame rotates under the action of this torque in a short interval of time $\Delta t$. $(\Delta t$ is short, so that any variation in the torque during this interval may be neglected$)$.

Given: The moment of inertia of the frame about an axis through its centre perpendicular to its plane is $\frac{4}{3}M{L}^2$.

1. $\frac{3}{2}\frac{I_{0}B}{M}(\Delta t{)}^2$
2. $\frac{3}{2}\frac{I_{0}B}{M}\left(\Delta t\right)$
3. $\frac{3}{4}\frac{I_{0}B}{M}\left(\Delta t\right)$
4. $\frac{3}{4}\frac{I_{0}B}{M}(\Delta t{)}^2$

Q.

A hollow metal sphere of radius 20 cm is charged such that the potential on its surface is 120 Volt. The potential at the centre of the sphere is

1. Zero

2. 120 V

3. 80 V

4. 6 V

Q. How to calculate gravitational potential graph ?

Q.

A car is speeding up on a horizontal road with an acceleration a. Consider the following situations in the car. (i) A ball is suspended from the ceiling through a string and is maintaining a constant angle with the vertical and does not slip on the incline. Find the angle of the incline with the horizontal.

Q. 3. What is the dimensional formula of potential gradient

Q. A rectangular coil is placed in a uniform magnetic field directed in its plane and parallel to one of its sides. The graph of $\tau vs\theta$ will be:
$(\theta$ is angular displacement of coil$)$

1. 
2. 
3. 
4. 

Q. From a circular disc of radius R and mass 9M, a small disc of radius $\frac{R}{3}$ is removed. The moment of inertia of the remaining disc about an axis perpendicular to the plane of the disc and passing through O is:

1. $4M{R}^2$
2. $\frac{40}{9}M{R}^2$
3. $10M{R}^2$
4. $\frac{37}{9}M{R}^2$

Q. a metal sphere of radius 10 cm is charged to potential of 24 V . it is joined to another identical uncharged metal sphere and potential fallas to 6 V. radius of second spher

Q.

A motor bike, initially moving at 18 km/h, accelerates at a rate of 5 m/s² and the distance covered by it from start is 87.5 m. Calculate the final velocity?

Q.

A missile launched from the ground, making an angle of ${30}^o$ with horizontal, reaches the target at a distance of $500m$. What is the initial velocity of the missile?

1. $\sqrt{\frac{{10}^5}{\sqrt{3}}}$

2. $\sqrt{\frac{{10}^4}{\sqrt{3}}}$

3. $\sqrt{\frac{100}{\sqrt{2}}}$

4. $\sqrt{\frac{{10}^4}{3}}$

Q. In identical mercury droplets charged to the same potential V coalesce to form a single bigger drop. The potential of new drop will be