Genius Physics (Class 11) - For IIT-JEE and CBSE

Pradeep Kshetrapal

Chapter 7

Rotational Motion - all with Video Answers

Educators

Chapter Questions

Problem 1

The distance between the carbon atom and the oxygen atom in a carbon monoxide molecule is $1.1$ $\AA$. Given, mass of carbon atom is 12 a.m.u. and mass of oxygen atom is 16 a.m.u., calculate the position of the center of mass of the carbon monoxide molecule
(a) $6.3 \AA$ from the carbon atom
(b) $1 \AA$ from the oxygen atom
(c) $0.63 \AA$ from the carbon atom
(d) $0.12 \AA$ from the oxygen atom\text

Hunza Gilgit

Hunza Gilgit

Numerade Educator

Problem 2

The velocities of three particles of masses $20 g, 30 g$ and $50 g$ are $10 \vec{i}, 10 \vec{j}$, and $10 \vec{k}$ respectively. The velocity of the centre of mass of the three particles is [EAMCET 2001]
(a) $2 \vec{i}+3 \vec{j}+5 \vec{k}$
(b) $10(\vec{i}+\vec{j}+\vec{k})$
(c) $20 \vec{i}+30 \vec{j}+5 \vec{k}$
(d) $2 \vec{i}+30 \vec{j}+50 \vec{k}$

Hunza Gilgit

Numerade Educator

Problem 3

Masses $8,2,4,2 \mathrm{~kg}$ are placed at the corners $A, B, C, D$ respectively of a square $A B C D$ of diagonal $80 \mathrm{~cm}$. The distance of centre of mass from $A$ will be
(a) $20 \mathrm{~cm}$
(b) $30 \mathrm{~cm}$
(c) $40 \mathrm{~cm}$
(d) $60 \mathrm{~cm}$

Hunza Gilgit

Numerade Educator

Problem 4

The coordinates of the positions of particles of mass 7,4 and $10 \mathrm{gm}$ are $(1,5,-3),(2,5,7)$ and $(3,3,-1) \mathrm{cm}$ respectively. The position of the centre of mass of the system would be
(a) $\left(-\frac{15}{7}, \frac{85}{17}, \frac{1}{7}\right) \mathrm{cm}$
(b) $\left(\frac{15}{7},-\frac{85}{17}, \frac{1}{7}\right) \mathrm{cm}$
(c) $\left(\frac{15}{7}, \frac{85}{21},-\frac{1}{7}\right) \mathrm{cm}$
(d) $\left(\frac{15}{7}, \frac{85}{21}, \frac{7}{3}\right) \mathrm{cm}$

Hunza Gilgit

Numerade Educator

Problem 5

The angular velocity of seconds hand of a watch will be
(a) $\frac{\pi}{60} \mathrm{rad} / \mathrm{sec}$
(b) $\frac{\pi}{30} \mathrm{rad} / \mathrm{sec}$
(c) $60 \pi \mathrm{rad} / \mathrm{sec}$
(d) $30 \pi \mathrm{rad} / \mathrm{sec}$

Hunza Gilgit

Numerade Educator

Problem 6

The wheel of a car is rotating at the rate of 1200 revolutions per minute. On pressing the accelerator for 10 sec it starts rotating at 4500 revolutions per minute. The angular acceleration of the wheel is [MP PET 2001]
(a) 30 radians $/ \mathrm{sec}^{2}$
(b) 188 o degrees $/ \mathrm{sec}^{2}$
(c) 40 radians $/ \mathrm{sec}^{2}$
(d) 1980

Hunza Gilgit

Numerade Educator

Problem 7

Angular displacement $(\theta)$ of a flywheel varies with time as $\theta=a t+b t^{2}+c t^{3}$ then angular acceleration is given by
(a) $a+2 b t-3 c t^{2}$
(b) $2 b-6 t$
(c) $a+2 b-6 t$
(d) $2 b+6 c t$

Hunza Gilgit

Numerade Educator

Problem 8

Angular displacement $(\theta)$ of a flywheel varies with time as $\theta=a t+b t^{2}+c t^{3}$ then angular acceleration is given by
(a) $a+2 b t-3 c t^{2}$
(b) $2 b-6 t$
(c) $a+2 b-6 t$
(d) $2 b+6 c t$

Hunza Gilgit

Numerade Educator

Problem 9

A wheel is at rest. Its angular velocity increases uniformly and becomes $60 \mathrm{rad} / \mathrm{sec}$ after $5 \mathrm{sec}$. The total angular displacement is
(a) $600 \mathrm{rad}$
(b) $75 \mathrm{rad}$
(c) $300 \mathrm{rad}$
(d) $150 \mathrm{rad}$

Hunza Gilgit

Numerade Educator

Problem 10

A wheel initially at rest, is rotated with a uniform angular acceleration. The wheel rotates through an angle $\theta_{1}$ in first one second and through an additional angle $\theta_{2}$ in the next one second. The
ratio $\frac{\theta_{2}}{\theta_{1}}$ is
(a) 4
(b) 2
(c) 3
(d) 1

Hunza Gilgit

Numerade Educator

Problem 11

As a part of a maintenance inspection the compressor of a jet engine is made to spin according to the graph as shown. The number of revolutions made by the compressor during the test is
(a) 9000
(b) 16570
(c) 12750
(d) 11250

Hunza Gilgit

Numerade Educator

Problem 12

Figure shows a small wheel fixed coaxially on a bigger one of double the radius. The system rotates about the common axis. The strings supporting $A$ and $B$ do not slip on the wheels. If $x$ and $y$ be the distances travelled by $A$ and $B$ in the same time interval, then
(a) $x=2 y$
(b) $x=y$
(c) $y=2 x$
(d) None of these

Hunza Gilgit

Numerade Educator

Problem 13

If the position vector of a particle is $\vec{r}=(\hat{i}+4 \hat{j})$ meter and its angular velocity is $\vec{\omega}=(\hat{j}+2 \hat{k})$ $\mathrm{rad} / \mathrm{sec}$ then its linear velocity is (in $\mathrm{m} / \mathrm{s}$ )
(a) $(8 \hat{i}-6 \hat{j}+3 \hat{k})$
(b) $(3 \hat{i}+6 \hat{j}+8 \hat{k})$
(c) $-(3 \hat{i}+6 \hat{j}+6 \hat{k})$
(d) $(6 \hat{i}+8 \hat{j}+3 \hat{k})$

Hunza Gilgit

Numerade Educator

Problem 14

Five particles of mass $=2 \mathrm{~kg}$ are attached to the rim of a circular disc of radius $0.1 \mathrm{~m}$ and negligible mass. Moment of inertia of the system about the axis passing through the centre of the disc and perpendicular to its plane is
(a) $1 \mathrm{~kg} \mathrm{~m}^{2}$
(b) $0.1 \mathrm{~kg} \mathrm{~m}^{2}$
(c) $2 \mathrm{~kg} \mathrm{~m}^{2}$
(d) $0.2 \mathrm{~kg} \mathrm{~m}^{2}$

Hunza Gilgit

Numerade Educator

Problem 15

Five particles of mass $=2 \mathrm{~kg}$ are attached to the rim of a circular disc of radius $0.1 \mathrm{~m}$ and negligible mass. Moment of inertia of the system about the axis passing through the centre of the disc and perpendicular to its plane is
(a) $1 \mathrm{~kg} \mathrm{~m}^{2}$
(b) $0.1 \mathrm{~kg} \mathrm{~m}^{2}$
(c) $2 \mathrm{~kg} \mathrm{~m}^{2}$
(d) $0.2 \mathrm{~kg} \mathrm{~m}^{2}$

Hunza Gilgit

Numerade Educator

Problem 16

Moment of inertia of a uniform circular disc about a diameter is $I$. Its moment of inertia about an axis perpendicular to its plane and passing through a point on its rim will be [UPSEAT 2002]
(a) $5 I$
(b) $6 \underline{I}$
(c) $3 I$
(d) $4 I$

Hunza Gilgit

Numerade Educator

Problem 17

Four thin rods of same mass $M$ and same length $l$, form a square as shown in figure. Moment of inertia of this system about an axis through centre $O$ and perpendicular to its plane is
(a) $\frac{4}{3} M l^{2}$
(b) $\frac{M l^{2}}{3}$
(c) $\frac{M l^{2}}{6}$
(d) $\frac{2}{3} M l^{2}$

Hunza Gilgit

Numerade Educator

Problem 18

Three rings each of mass $M$ and radius $R$ are arranged as shown in the figure. The moment of inertia of the system about $Y Y$ ' will be
(a) $3 M R^{2}$
(b) $\frac{3}{2} M R^{2}$
(c) $5 M R^{2}$
(d) $\frac{7}{2} M R^{2}$

Hunza Gilgit

Numerade Educator

Problem 19

Let $l$ be the moment of inertia of an uniform square plate about an axis $A B$ that passes through its centre and is parallel to two of its sides. $C D$ is a line in the plane of the plate that passes through the centre of the plate and makes an angle $\theta$ with $A B .$ The moment of inertia of the plate about the axis $C D$ is then equal to [IIT-JEE 1998]
(a) $l$
(b) $l \sin ^{2} \theta$
(c) $l \cos ^{2} \theta$
(d) $l \cos ^{2} \frac{\theta}{2}$

Hunza Gilgit

Numerade Educator

Problem 20

Three rods each of length $L$ and mass $M$ are placed along $X, Y$ and $Z$-axes in such a way that one end of each of the rod is at the origin. The moment of inertia of this system about $Z$ axis is
(a) $\frac{2 M L^{2}}{3}$
(b) $\frac{4 M L^{2}}{3}$
(c) $\frac{5 M L^{2}}{3}$
(d) $\frac{M L^{2}}{3}$

Hunza Gilgit

Numerade Educator

Problem 21

Three point masses each of mass $m$ are placed at the corners of an equilateral triangle of side $a$. Then the moment of inertia of this system about an axis passing along one side of the triangle is [AIIMS 1995.
(a) $m a^{2}$
(b) $3 m a^{2}$
(c) $\frac{3}{4} m a^{2}$
(d) $\frac{2}{3} m a^{2}$

Hunza Gilgit

Numerade Educator

Problem 22

Two identical rods each of mass $M .$ and length $l$ are joined in crossed position as shown in figure. The moment of inertia of this system about a bisector would be
$\begin{array}{ll}\text { (a) } \frac{M l^{2}}{6} & \text { (b) } \frac{M l^{2}}{12}\end{array}$
(c) $\frac{M l^{2}}{3}$
(d) $\frac{M l^{2}}{4}$

Hunza Gilgit

Numerade Educator

Problem 23

The moment of inertia of a rod of length $l$ about an axis passing through its centre of mass and perpendicular to rod is $I$. The moment of inertia of hexagonal shape formed by six such rods, about an axis passing through its centre of mass and perpendicular to its plane will be
(a) $16 I$
(b) $40 I$
(c) $60 I$
(d) $80 I$

Hunza Gilgit

Numerade Educator

Problem 24

The moment of inertia of $H C l$ molecule about an axis passing through its centre of mass and perpendicular to the line joining the $H^{+}$and $C$ - ions will be, if the interatomic distance is $1 \AA$
(a) $0.61 \times 10^{-47} \mathrm{~kg} \cdot \mathrm{m}^{2}$
(b) $1.61 \times 10^{-47} \mathrm{~kg} \cdot \mathrm{m}^{2}$
(c) $0.061 \times 10^{-47} \mathrm{~kg} \cdot \mathrm{m}^{2}$
(d) 0

Hunza Gilgit

Numerade Educator

Problem 25

Four masses are joined to a light circular frame as shown in the figure. The radius of gyration of this system about an axis passing through the centre of the circular frame and perpendicular to its plane would be
$\begin{array}{lll}\text { (a) } a / \sqrt{2} & \text { (b) } & a / 2\end{array}$
$\begin{array}{lll}\text { (c) } a & \text { (d) } & 2 a\end{array}$

Hunza Gilgit

Numerade Educator

Problem 26

Four spheres, each of mass $M$ and radius $r$ are situated at the four corners of square of side $R$. The moment of inertia of the system about an axis perpendicular to the plane of square and passing through its centre will be
(a) $\frac{5}{2} M\left(4 r^{2}+5 R^{2}\right)$
(b) $\frac{2}{5} M\left(4 r^{2}+5 R^{2}\right)$
(c) $\frac{2}{5} M\left(4 r^{2}+5 r^{2}\right)$
(d) $\frac{5}{2} M\left(4 r^{2}+5 r^{2}\right)$

Hunza Gilgit

Numerade Educator

Problem 27

The moment of inertia of a solid sphere of density $\rho$ and radius $R$ about its diameter is
(a) $\frac{105}{176} R^{5} \rho$
(b) $\frac{105}{176} R^{2} \rho$
(c) $\frac{176}{105} R^{5} \rho$
(d) $\frac{176}{105} R^{2} \rho$

Hast Aggarwal

Numerade Educator

Problem 28

Two circular discs $A$ and $B$ are of equal masses and thickness but made of metals with densities $d_{A}$ and $d_{B}\left(d_{A}>d_{B}\right) .$ If their moments of inertia about an axis passing through centres and normal to the circular faces be $I_{A}$ and $I_{B}$, then(a) $I_{A}=I_{B}$
(b) $I_{A}>I_{B}$
(c) $I_{A}<I_{B}$
(d) $I_{A}>=\left\langle I_{B}\right.$

Hunza Gilgit

Numerade Educator

Problem 29

A force of $(2 \hat{i}-4 \hat{j}+2 \hat{k}) N$ acts at a point $(3 \hat{i}+2 \hat{j}-4 \hat{k})$ metre from the origin. The magnitude of torque is
(a) Zero
(b) $24.4 \mathrm{~N}-\mathrm{m}$
(c) $0.244 \mathrm{~N}-\mathrm{m}$
(d) $2.444 \mathrm{~N}-\mathrm{m}$

Hunza Gilgit

Numerade Educator

Problem 30

The resultant of the system in the figure is a force of $8 N$ parallel to the given force through $R$. The value of $P R$ equals to
(a) $1 / 4 R Q$
(b) $3 / 8 R Q$
(c) $3 / 5 R Q$
(d) $2 / 5 R Q$

Hunza Gilgit

Numerade Educator

Problem 31

A horizontal heavy uniform bar of weight $W$ is supported at its ends by two men. At the instant, one of the men lets go off his end of the rod, the other feels the force on his hand changed to
(a) $W$
(b) $\frac{W}{2}$
(c) $\frac{3 W}{4}$
(d) $\frac{W}{4}$Solution: (d) Let the mass of the rod is $M \quad \therefore$ Weight $(W)=M g$ Initially for the equilibrium $F+F=M g \Rightarrow F=M g / 2$
When one man withdraws, the torque on the rod
$\tau=I \alpha=M g \frac{l}{2}$
$\Rightarrow \frac{M l^{2}}{3} \alpha=M g \frac{l}{2} \quad\left[\right.$ As $\left.I=M l^{2} / 3\right]$
$\Rightarrow$ Angular acceleration $\alpha=\frac{3}{2} \frac{g}{l}$
and linear acceleration $a=\frac{l}{2} \alpha=\frac{3 g}{4}$
Now if the new normal force at $A$ is $F$ then $M g-F=M a$
$\Rightarrow F=M g-M a=M g-\frac{3 M g}{4}=\frac{M g}{4}=\frac{W}{4} .$

Hunza Gilgit

Numerade Educator

Problem 32

Consider a body, shown in figure, consisting of two identical balls, each of mass $M$ connected by a light rigid rod. If an impulse $J=M v$ is imparted to the body at one of its ends, what would be its angular velocity [IIT-JEE (Screening) 2003]
(a) $v / L$
(b) $2 v / L$
(c) $v / 3 L$
(d) $v / 4 L$

Hunza Gilgit

Numerade Educator

Problem 33

A thin circular ring of mass $M$ and radius $R$ is rotating about its axis with a constant angular velocity $\omega$. Four objects each of mass $m$, are kept gently to the opposite ends of two perpendicular diameters of the ring. The angular velocity of the ring will be
(a) $\frac{M \omega}{M+4 m}$
(b) $\frac{(M+4 m) \omega}{M}$
(c) $\frac{(M-4 m) \omega}{M+4 m}$
(d) $\frac{M \omega}{4 m}$

Hunza Gilgit

Numerade Educator

Problem 34

A circular platform is free to rotate in a horizontal plane about a vertical axis passing through its center. A tortoise is sitting at the edge of the platform. Now, the platform is given an angular velocity $\omega_{b}$. When the tortoise moves along a chord of the platform with a constant velocity (with respect to the platform), the angular velocity of the platform $\omega(t)$ will vary with time $t$ as
(a)
(c)
(d)

Hunza Gilgit

Numerade Educator

Problem 35

The position of a particle is given by : $\vec{r}=(\hat{i}+2 \hat{j}-\hat{k})$ and momentum $\vec{P}=(\hat{i}+4 \hat{j}-2 \hat{k})$. The angular momentum is perpendicular to
(a) $X$-axis
(b) $Y$-axis
(c) $Z$-axis
(d) Line at equal angles to all the three axes

Hunza Gilgit

Numerade Educator

Problem 36

Two discs of moment of inertia $I_{1}$ and $I_{2}$ and angular speeds $\omega_{1}$ and $\omega_{2}$ are rotating along collinear axes passing through their centre of mass and perpendicular to their plane. If the two are made to rotate together along the same axis the rotational $K E$ of system will be
(a) $\frac{I_{1} \omega_{1}+I_{2} \omega_{2}}{2\left(I_{1}+I_{2}\right)}$
(b) $\frac{\left(I_{1}+I_{2}\right)\left(\omega_{1}+\omega_{2}\right)^{2}}{2}$
(c) $\frac{\left(I_{1} \omega_{1}+I_{2} \omega_{2}\right)^{2}}{2\left(I_{1}+I_{2}\right)}$
(d) None of these

Hunza Gilgit

Numerade Educator

Problem 37

A smooth uniform rod of length $L$ and mass $M$ has two identical beads of negligible size, each of mass $m$, which can slide freely along the rod. Initially the two beads are at the centre of the rodvand the system is rotating with angular velocity $\omega_{0}$ about an axis perpendicular to the rod and passing through the mid point of the rod (see figure). There are no external forces. When the beads reach the ends of the rod, the angular velocity of the system is
(a) $\omega_{0}$
(b) $\frac{M \omega_{0}}{M+12 m}$
(c) $\frac{M \omega_{0}}{M+2 m}$
(d) $\frac{M \omega_{0}}{M+6 m}$

Hunza Gilgit

Numerade Educator

Problem 38

Moment of inertia of uniform rod of mass $M$ and length $L$ about an axis through its centre and perpendicular to its length is given by $\frac{M L^{2}}{12} .$ Now consider one such rod pivoted at its centre, free to rotate in a vertical plane. The rod is at rest in the vertical position. A bullet of mass $M$ moving horizontally at a speed $v$ strikes and embedded in one end of the rod. The angular velocity of the rod just after the collision will be
(a) $v / L$
(b) $2 v / L$
(c) $3 v / 2 L$
(d) $6 v / L$

Hunza Gilgit

Numerade Educator

Problem 39

A solid cylinder of mass $2 \mathrm{~kg}$ and radius $0.2 \mathrm{~m}$ is rotating about its own axis without friction with
angular velocity $3 \mathrm{rad} / \mathrm{s}$. A particle of mass $0.5 \mathrm{~kg}$ and moving with a velocity $5 \mathrm{~m} / \mathrm{s}$ strikes the cylinder and sticks to it as shown in figure. The angular momentum of the cylinder before collision will be
(a) $0.12 J-s$
(b)
(c) $1.2 J-S$
(d) $1.12 J-S$

Hunza Gilgit

Numerade Educator

Problem 40

In the above problem the angular velocity of the system after the particle sticks to it will be
(a) $0.3 \mathrm{rad} / \mathrm{s}$
(b) $5.3 \mathrm{rad} / \mathrm{s}$
(c) $10.3 \mathrm{rad} / \mathrm{s}$
(d) $89.3 \mathrm{rad} / \mathrm{s}$
Initial angular momentum of bullet + initial angular momentum of cylinder
= Final angular momentum of (bullet + cylinder) system
$\Rightarrow m v r+I_{1} \omega=\left(I_{1}+I_{2}\right) \omega^{\prime}$
$\Rightarrow m v r+I_{1} \omega=\left(\frac{1}{2} M r^{2}+m r^{2}\right) \omega^{\prime}$
$\Rightarrow 0.5 \times 5 \times 0.2+0.12=\left(\frac{1}{2} 2(0.2)^{2}+(0.5)(0.2)^{2}\right) \omega^{\prime}$
$\therefore \quad \omega^{\prime}=10.3 \mathrm{rad} / \mathrm{sec} .$

Hunza Gilgit

Numerade Educator

Problem 41

A ring of radrus $0.5 m$ and mass $10 \mathrm{~kg}$ is rotating about its diameter with an angular velocity of 20 $\mathrm{rad} / \mathrm{s}$. Its kinetic energy is
(a) $10 J$
(b) $100 J$
(c) $500 J$
(d) $250 \cdot J$

Hunza Gilgit

Numerade Educator

Problem 42

An automobile engine develops $100 \mathrm{~kW}$ when rotating at a speed of $1800 \mathrm{rev} / \mathrm{min}$. What torque does it deliver
(a) $350 \mathrm{~N}-\mathrm{m}$
(b) $440 \mathrm{~N}-\mathrm{m}$
(c) $531 \mathrm{~N}-\mathrm{m}$
(d) $628 \mathrm{~N}-\mathrm{m}$

Hunza Gilgit

Numerade Educator

Problem 43

An automobile engine develops $100 \mathrm{~kW}$ when rotating at a speed of $1800 \mathrm{rev} / \mathrm{min}$. What torque does it deliver
(a) $350 \mathrm{~N}-\mathrm{m}$
(b) $440 \mathrm{~N}-\mathrm{m}$
(c) $531 \mathrm{~N}-\mathrm{m}$
(d) $628 \mathrm{~N}-\mathrm{m}$

Hunza Gilgit

Numerade Educator

Problem 44

A disc and a ring of same mass are rolling and if their kinetic energies are equal, then the ratio of their velocities will be
(a) $\sqrt{4}: \sqrt{3}$
(b) $\sqrt{3}: \sqrt{4}$
(c) $\sqrt{3}: \sqrt{2}$
(d) $\sqrt{2}: \sqrt{3}$

Hunza Gilgit

Numerade Educator

Problem 45

A wheel is rotating with an angular speed of $20 \mathrm{rad} / \mathrm{sec} .$ It is stopped to rest by applying a constant torque in $4 s$. If the moment of inertia of the wheel about its axis is $0.20 \mathrm{~kg}-\mathrm{m}^{2}$, then the work done by the torque in two seconds will be
(a) $10 J$
(b) $20 J$
(c) $30^{\prime} J$
(d) $40 \cdot J$

Hunza Gilgit

Numerade Educator

Problem 46

If the angular momentum of a rotating body is increased by $200 \%$, then its kinetic energy of rotation will be increased by
(a) $400 \%$
(b) $800 \%$
(c) $200 \%$
(d) $100 \%$

Hunza Gilgit

Numerade Educator

Problem 47

A ring, a solid sphere and a thin disc of different masses rotate with the same kinetic energy. Equal torques are applied to stop them. Which will make the least number of rotations before coming to rest
(a) Disc
(b) Ring
(c) Solid sphere
(d) All will make same number of rotations

Hunza Gilgit

Numerade Educator

Problem 48

The angular velocity of a body is $\vec{\omega}=2 \hat{i}+3 \hat{j}+4 \hat{k}$ and a torque $\vec{\tau}=\hat{i}+2 \hat{j}+3 \hat{k}$ acts on it. The rotational power will be
(a) $20 \mathrm{~W}$
(b) $15 W$
(c) $\sqrt{17} W$
(d) $\sqrt{14} W$

Hunza Gilgit

Numerade Educator

Problem 49

A flywheel of moment of inertia $0.32 \mathrm{~kg}-\mathrm{m}^{2}$ is rotated steadily at $120 \mathrm{rad} / \mathrm{sec}$ by a $50 \mathrm{~W}$ electric motor. The kinetic energy of the flywheel is
(a) $4608 J$
(b) $1152 J$
(c) $2304 \mathrm{~J}$
(d) $6912 J$

Hunza Gilgit

Numerade Educator

Problem 50

A solid cylinder of mass $M$ and radius $R$ rolls without slipping down an inclined plane of length $L$ and height $h$. What is the speed of its centre of mass when the cylinder reaches its bottom
(a) $\sqrt{\frac{3}{4} g h}$
(b) $\sqrt{\frac{4}{3} g h}$
(c) $\sqrt{4 g h}$
(d) $\sqrt{2 g h}$

Hunza Gilgit

Numerade Educator

Problem 51

A sphere rolls down on an inclined plane of inclination $\theta$. What is the acceleration as the sphere reaches bottom
(a) $\frac{5}{7} g \sin \theta$
(b) $\frac{3}{5} g \sin \theta$
(c) $\frac{2}{7} g \sin \theta$
(d) $\frac{2}{5} g \sin \theta$

Hunza Gilgit

Numerade Educator

Problem 52

$R$
A ring solid sphere and a disc are rolling down from the top of the same height, then the sequence to reach on surface is
(a) Ring, disc, sphere
(b) Sphere, disc, ring
(c) Disc, ring, sphere
(d) Sphere, ring, disc

Hunza Gilgit

Numerade Educator

Problem 53

A thin uniform circular ring is rolling down an inclined plane of inclination $30^{\circ}$ without slipping. Its linear acceleration along the inclined plane will be
(a) $g / 2$
(b) $\mathrm{g} / 3$
(c) $g / 4$
(d) $2 \mathrm{~g} / 3$

Hunza Gilgit

Numerade Educator

Problem 54

A solid sphere and a disc of same mass and radius starts rolling down a rough inclined plane, from the same height the ratio of the time taken in the two cases is
(a)
(b) $\sqrt{15}+\sqrt{14}$
(c)
(d) $\sqrt{14} \cdot \sqrt{15}$

Hunza Gilgit

Numerade Educator

Problem 55

A solid sphere of mass $0.1 \mathrm{~kg}$ and radius $2 \mathrm{~cm}$ rolls down an inclined plane $1.4 \mathrm{~m}$ in length (slope in 10). Starting from rest its final velocity will be
(a) $1.4 \mathrm{~m} / \mathrm{sec}$
(b) $0.14 \mathrm{~m} / \mathrm{sec}$
(c) $14 \mathrm{~m} / \mathrm{sec}$
(d) $0.7 \mathrm{~m} / \mathrm{sec}$

Hunza Gilgit

Numerade Educator

Problem 56

A solid sphere rolls down an inclined plane and its velocity at the bottom is $v_{1}$. Then same sphere slides down the plane (without friction) and let its velocity at the bottom be $v_{2}$. Which of the following relation is correct
(a) $v_{1}=v_{2}$
(b) $v_{1}=\frac{5}{7} v_{2}$
(c) $v_{1}=\frac{7}{5} v_{2}$
(d) None of these

Hunza Gilgit

Numerade Educator

Problem 57

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 $m g$ 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
(a) $\sqrt{\frac{2 g h}{I+m r}}$
(b) $\sqrt{\frac{2 m g h}{I+m r^{2}}}$
(c) $\sqrt{\frac{2 m g h}{I+2 m r^{2}}}$
(d) $\sqrt{2 g h}$

Hunza Gilgit

Numerade Educator

Problem 58

In the following figure, a body of mass $m$ is tied at one end of a light string and this string is wrapped around the solid cylinder of mass $M$ and radius $R$. At the moment $t=0$ the system starts moving. If the friction is negligible, angular velocity at time $t$ would be
(a) $\frac{m g R t}{(M+m)}$
(b) $\frac{2 M g t}{(M+2 m)}$
(c) $\frac{2 m g t}{R(M-2 m)}$
(d) $\frac{2 m g t}{R(M+2 m)}$

Hunza Gilgit

Numerade Educator

Problem 59

{ A block of mass } 2} 2 \mathrm{~kg}$ hangs from the rim of a wheel of radius $0.5 \mathrm{~m} .$ On releasing from rest the block falls through $5 \mathrm{~m}$ height in $2 s .$ The moment of inertia of the wheel will be
(a) $1 \mathrm{~kg}-\mathrm{m}^{2}$
(b) $3.2 \mathrm{~kg}-\mathrm{m}^{2}$
(c) $2.5 \mathrm{~kg}-\mathrm{m}^{2}$
(d) $1.5 \mathrm{~kg}-\mathrm{m}^{2}$

Prem Bijarniya

Numerade Educator