Newton's Second Law: Rotatory Version

Q.

The maximum height of projectile formula is ______.

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.

What force is required to stretch a steel wire $1c{m}^2$ in cross section to double its length? (Youngs modulus for steel is $2\times {10}^{11}Pa$)

1. $3\times {10}^{7}N$
   
2. $3\times {10}^{11}N$
   
3. $2\times {10}^{11}N$
   
4. $2\times {10}^{7}N$
   

Q. A constant torque of 500Nm turns a wheel of moment of inertia 100kg m^2 about an axis through its centre. Find the gain in angular velocity in 2 seconds.

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.

Derive the second equation of motion by graphical method.

Q. Four solid spheres each of diameter $\sqrt{5}\text{cm}$ and mass $0.5\text{kg}$ are placed with their centres at the corners of a square of side $4\text{cm}$. The moment of inertia of the system about the diagonal of the square is $N\times {10}^{-4}{\text{kg m}}^2$, then $N$ is

1. $7$
2. $8$
3. $9$
4. $10$

Q. A body is in equilibrium under the action of three force vectors A, B and C simultaneously. Show that A×B=B×C=C×A.

Q. A pendulum bob of mass $m$ carrying a charge $q$ is at rest with its string making an angle $\theta$ with the vertical in a uniform horizontal electric field $E$. The tension in the string is

1. $\frac{mg}{\sin \theta }$
2. $mg$
3. $\frac{qE}{\sin \theta }$
4. $\frac{qE}{\cos \theta }$

Q. The radius of gyration of rod of length ${}^{\prime }{L}^{\prime }$ and mass ${}^{\prime }{M}^{\prime }$ about an axis perpendicular to its length and passing through a point at a distance $L/3$ from one of its ends is

1. $\frac{\sqrt{7}}{6}L$
2. $\frac{L^2}{9}$
3. $\frac{L}{3}$
4. $\frac{\sqrt{5}}{2}L$

Q. A stationary horizontal disc is free to rotate about its axis. When a torque is applied on it, its kinetic energy as a function of $\theta$, where $\theta$ is the angle by which it has rotated, is given as $k{\theta }^2$ (where $k$ is a constant). If its moment of inertia is $I$, then the angular acceleration of the disc is

1. $\frac{k}{I}\theta$
2. $\frac{k}{2I}\theta$
3. $\frac{k}{4I}\theta$
4. $\frac{2k}{I}\theta$

Q. A uniform rod of length $l$ and mass $m$ is free to rotate in a vertical plane about $A$. The rod initially in horizontal position is released. The initial angular acceleration of the rod is (Moment of inertia of rod about $A$ is $\frac{m{l}^2}{3}$)

1. $\frac{3g}{2l}$
2. $\frac{2l}{3g}$
3. $\frac{3g}{2l^2}$
4. $mg\frac{l}{2}$

Q.

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

1. Angular acceleration of the body has to be zero

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

3. Angular momentum about any axis may not be conserved

4. Linear acceleration of the centre of mass is zero

Q. A system consists of two identical spheres each of mass $1.5\text{kg}$ and radius $50\text{cm}$ at the ends of a light rod. The distance between the centres of the two spheres is $5\text{m}$. What will be the moment of inertia of the system about an axis perpendicular to the rod passing through its midpoint ?

1. $18.75{\text{kgm}}^2$
2. $19.05{\text{kgm}}^2$
3. $18.75{\text{kgm}}^2$
4. $1.905{\text{kgm}}^2$

Q. A rod of length $2.4\text{m}$ and mass $6\text{kg}$ is bent to form a regular hexagon. Find the moment of inertia of the hexagon about the axis perpendicular to the plane of hexagon passing through its centre.

1. $0.3{\text{kg-m}}^2$
2. $0.5{\text{kg-m}}^2$
3. $0.8{\text{kg-m}}^2$
4. $0.6{\text{kg-m}}^2$

Q.

Uniform rod $AB$ is hinged at end $A$ in horizontal position as shown in the figure. The other end is connected to a block of mass $m$ through a massless string as shown. The pulley is smooth and massless. Mass of block and rod is same and is equal to $m$. Then find the acceleration of the block just after release.

1. None
2. $\frac{6g}{13}$
3. $\frac{g}{4}$
4. $\frac{3g}{8}$

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. The solid cylinder of length $80~\text{cm}$ and mass M has a radius of $20~\text{cm}$. Calculate the density of the material used if the moment of inertia of the cylinder about an axis CD parallel to AB as shown in figure is \(2.7 ~\text{kg}~ \text m^2\).

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. The moment of inertia of a uniform thin rod of mass $m$ and length $l$ about the axis $PQ$ and $RS$ passing through center of rod $C$ and in the plane of the rod are $I_{PQ}$ and $I_{RS}$ respectively. Then $I_{PQ}+I_{RS}$ is equal to

1. $\frac{m{l}^2}{3}$
2. $\frac{m{l}^2}{2}$
3. $\frac{m{l}^2}{4}$
4. $\frac{m{l}^2}{12}$
   

Q.

Assertion: A Helicopter must necessarily have two propellers.

Reason: Two propellers are provided in a helicopter in order to conserve Linear Momentum.

1. Both assertion and reason are true.

2. Both assertion and reason are false.

3. Assertion is true and reason is false.

4. Assertion is true and reason is true.

Q. Three rings, each of mass P and radius Q are arranged as shown in the figure. The moment of inertia of the arrangement about yy’ axis will be-

1. $7/2P{Q}^2$
2. $2/7P{Q}^2$
3. $2/5P{Q}^2$
4. $5/2P{Q}^2$

Q. A horizontal solid cylinder (of mass $m$) is pivoted about its longitudinal axis. To the end of a thread wrapped on the cylinder a block (of mass $2m$) is attached, as shown. If the system is left free, acceleration of the block (in ${\text{ms}}^{-2}$) is
(string is massless and there is no slipping anywhere)

Q. A uniform disc of mass $M$ and radius $R$ is free to rotate about its fixed horizontal axis without friction. There is sufficient friction between the inextensible light string and the disc to prevent slipping of the string over the disc. At the shown instant, extension in the light spring is $\frac{3mg}{K}$, where $m$ is the mass of the block, $g$ is acceleration due to gravity and $K$ is spring constant. Then, the acceleration of block just after it is released is

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

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 small block of mass m is lying at rest at point P of a wedge having a smooth semi circular track of radius R. The minimum value of horizontal acceleration $a_o$ of wedge so that mass can just reach the point Q?

1. 

2. 
   

3. g

4. not possible

Q. A uniform rod $ABC$ of mass $M$ is placed vertically on a rough horizontal surface. The coefficient of kinetic friction between the rod and the surface is $\mu$. A force $F(>\mu mg)$ is applied on the rod at point $B$ at distance $l/3$ below centre of the rod as shown in figure. The initial acceleration of point $A$ is

1. $\mu g-\frac{F}{M}$
2. $\frac{F}{M}$
3. $4\mu g$
4. $2\mu g-\frac{F}{M}$

Q. A particle is projected such that the horizontal range and vertical height are the same. What is the angle of projection?

Q.

A small bead of mass $m=1\text{kg}$ is carried by a circular hoop having centre at $C$ and radius $r=1\text{m}$ which rotates about a fixed vertical axis (as shown). The coefficient of friction between bead and hoop is $\mu =0.5$. The maximum angular speed of the hoop in $\text{rad/s}$ for which the bead does not have relative motion with respect to hoop: initial position of bead is shown in figure. Answer upto 2 decimal places.

Q. A light thread is wound on a disk of mass $m$ and other end of thread is connected to a block of mass $m$, which is placed on a rough ground as shown in diagram. Find the minimum value of coefficient of friction for which block remain at rest.

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