System of Particles and Rotational Motion Archives

System of Particles and Rotational Motion

Figure shows two identical particles 1 and 2, each of mass m, moving in opposite directions with the same speed v along parallel lines. At a particular instant, $r_{1}$ and $r_{2}$ are their respective positions vectors drawn from point A which is in the plane of the parallel lines. Choose the correct options:
a) angular momentum $l_{1}$ of particle 1 is about A is $l_{1} =mvd_{1}$
b) angular momentum $l_{2}$ of particle 2 about A is $l_{2} = mvx_{2}$
c) total angular momentum of the system about A is $l = mv(r_{1}+r{2})$
d) total angular momentum of the system about A is $l = mv(d_{2}-d_{1})$

CBSE, Class 11, NCERT Exemplar, Physics, System of Particles and Rotational Motion

Solution: Correct answers is: d) total angular momentum of the system about A is $l = mv(d_{2}-d_{1})$ Angular momentum of particle 1 about A is given as, $\vec L_1=mvd_1$ Angular momentum of...

7.1. The centre of mass of which of the following is located outside the body? a) a pencil b) a shotput c) a dice d) a bangle

Class 11, Guides, NCERT, Physics, Physics, System of Particles and Rotational Motion

Answer: The correct option is d) a bangle. Explanation: The mass of a bangle is distributed along the circumference of the ring shape of the bangle as it is a hollow shape. The center of mass is...

Separation of Motion of a system of particles into motion of the centre of mass and motion about the centre of mass:
(i) Show $L=L^{\prime}+R \times M V$ where $L^{\prime}=\Sigma r^{\prime}{ }_{i} \times p_{i}^{\prime}$ is the angular momentum of the system about the centre of mass with velocities considered with respect to the centre of mass. Note $r_{i}=r_{i}-R$ , rest of the notation is the standard notation used in the lesson. Note L’ and MR $\times V$ can be said to be angular momenta, respectively, about and of the centre of mass of the system of particles.
(ii) Prove that : $\mathrm{dL}^{\prime} / \mathrm{dt}=\sum \mathrm{r}_{\mathrm{i}}^{\prime} \mathrm{x} \mathrm{dp}^{\prime} / \mathrm{dt}$ Further prove that: $\mathrm{dL}^{\prime} / \mathrm{dt}=\mathrm{T}^{\prime} \mathrm{ext}$ Where t’ext is the sum of all external torques acting on the system about the centre of mass. (Clue : A pply Newton’s Third Law and the definition of centre of mass. Consider that internal forces between any two particles act along the line connecting the particles.)

CBSE, Class 11, NCERT, Physics, System of Particles and Rotational Motion

Here $\vec{r}_{i}=\vec{r}_{i}+\vec{R}+R \ldots$ (1) also, $\vec{V}_{i}=\vec{V}_{i}+\vec{V} \ldots \ldots .$ (2) Where $\vec{r}_{i}^{\overrightarrow{3}}$ and $\vec{v}_{i}^{\overrightarrow{3}}$ denote...

Separation of Motion of a system of particles into motion of the centre of mass and motion about the centre of mass:
$(i)$ Show $p=p_{i}^{\prime}+m_{i} V$ Where $p_{i}$ is the momentum of the $i^{\text {th }}$ particle (of mass $\left.m_{i}\right)$ and $p_{i}=m_{i} v_{i}^{t} .$ Note $v_{i}^{\prime}$ is the velocity of the $\mathrm{i}^{\mathrm{ith}}$ particle with respect to the centre of mass.Also, verify using the definition of the centre of mass that $\Sigma p_{i}=0$
(ii) Prove that $\mathrm{K}=\mathrm{K}^{\prime}+1 / 2 \mathrm{MV}^{2}$ Where $K$ is the total kinetic energy of the system of particles, $K^{\prime}$ is the total kinetic energy of the system when the particle velocities are taken relative to the center of mass and $\mathrm{MV}^{2} / 2$ is the kinetic energy of the translation of the system as a whole.

CBSE, Class 11, NCERT, Physics, System of Particles and Rotational Motion

i)Here $\vec{r}_{i}=\vec{r}_{i}+\vec{R}+R \ldots$ also, $\vec{V}_{i}=\vec{V}_{i}+\vec{V} \ldots \ldots .$ Where $\vec{r}_{i}^{\overrightarrow{3}}$ and $\vec{v}_{i}^{\overrightarrow{3}}$ denote the...

Read the statement below carefully, and state, with reasons, if it is true or false; A wheel moving down a perfectly frictionless inclined plane will undergo slipping (not rolling) motion.

CBSE, Class 11, NCERT, Physics, System of Particles and Rotational Motion

True. Rolling happens only when there is a frictional force to provide the torque; otherwise, the wheel merely slips down the plane on its own weight.

Read each statement below carefully, and state, with reasons, if it is true or false;
(a) The instantaneous acceleration of the point of contact during rolling is zero.
(b) For perfect rolling motion, work done against friction is zero.

CBSE, Class 11, NCERT, Physics, System of Particles and Rotational Motion

(a) False. The instantaneous acceleration of a rolling object will have a value that is not zero. (b) True. Because the frictional force is zero during perfect rolling, no work is done against it.

Read each statement below carefully, and state, with reasons, if it is true or false;
(a) During rolling, the force of friction acts in the same direction as the direction of motion of the CM of the body.
(b) The instantaneous speed of the point of contact during rolling is zero.

CBSE, Class 11, NCERT, Physics, System of Particles and Rotational Motion

(a) False. The direction of frictional force is the polar opposite of the centre of mass's motion. Because the centre of mass of a rolling object moves backwards, the frictional force acts in the...

A cylinder of mass $10 \mathrm{~kg}$ and radius $15 \mathrm{~cm}$ is rolling perfectly on a plane of inclination $30^{\circ} .$ The coefficient of static friction $\mu_{\mathrm{s}}=0.25$ . If the inclination $\theta$ of the plane is increased, at what value of $\theta$ does the cylinder begin to skid, and not roll perfectly?

CBSE, Class 11, NCERT, Physics, System of Particles and Rotational Motion

The given situation can be depicted as: Mass is given as $m=10 \mathrm{~kg}$ Radius is given as $r=15 \mathrm{~cm}=0.15 \mathrm{~m}$ Co-efficient of kinetic friction is given as $\mu_{s}=0.25$ Angle...

A cylinder of mass $10 \mathrm{~kg}$ and radius $15 \mathrm{~cm}$ is rolling perfectly on a plane of inclination $30^{\circ} .$ The coefficient of static friction $\mu_{\mathrm{s}}=0.25$
(a) How much is the force of friction acting on the cylinder?
(b) What is the work done against friction during rolling?

CBSE, Class 11, NCERT, Physics, System of Particles and Rotational Motion

The above situation can be depicted as: Mass is given as $m=10 \mathrm{~kg}$ Radius is given as $r=15 \mathrm{~cm}=0.15 \mathrm{~m}$ Co-efficient of kinetic friction is given as $\mu_{s}=0.25$ Angle...

A solid disc and a ring, both of radius $10 \mathrm{~cm}$ are placed on a horizontal table simultaneously, with an initial angular speed equal to $10 \mathrm{~m} \mathrm{rad} \mathrm{s}^{-1}$ . Which of the two will start to roll earlier? The coefficient of kinetic friction is $\mu_{k}=0.2 .$

CBSE, Class 11, NCERT, Physics, System of Particles and Rotational Motion

The radius of the ring and the disc is given as r = 10 cm = 0.10 m Initial angular speed is given as ω0 =10 π rad s–1 The coefficient of kinetic friction is given as μk = 0.2 According to Newton’s...

A disc rotating about its axis with angular speed $\omega_{0}$ is placed lightly (without any translational push) on a perfectly frictionless table. The radius of the disc is R. What are the linear velocities of the points $A, B$ and $C$ on the disc shown in Figure. Will the disc roll in the direction indicated?

CBSE, Class 11, NCERT, Physics, System of Particles and Rotational Motion

Solution: The respective linear velocities are : For point $A, v_{A}=r \omega_{0}$ For point $B, v_{B}=r \omega_{0}$ both in the direction of arrow For point $C, v_{c}=(R / 2) \omega_{0}$ in the...

Prove the result that the velocity v of translation of a rolling body (like a ring, disc, cylinder or sphere) at the bottom of an inclined plane of a height $h$ is given by $\mathrm{v}^{2}=2 \mathrm{gh} /\left(1+\mathrm{k}^{2} / \mathrm{R}^{2}\right)$ using dynamical consideration (i.e. by consideration of forces and torques). Note $\mathrm{k}$ is the radius of gyration of the body about its symmetry axis, and $\mathbf{R}$ is the radius of the body. The body starts from rest at the top of the plane.

CBSE, Class 11, NCERT, Physics, System of Particles and Rotational Motion

The given question can be represented as: where, $R$ is the body's radius $g$ is the acceleration due to gravity $\mathrm{K}$ is the body's radius of gyration $v$ is the body's translational...

(a) Prove the theorem of perpendicular axes. (Hint: Square of the distance of a point $(x, y)$ in the $x-y$ plane from an axis through the origin and perpendicular to the plane is $\left.x^{2}+y^{2}\right)$ .
(b) Prove the theorem of parallel axes. (Hint: If the centre of mass of a system of n particles is chosen to be the origin $\left.\sum m_{i} r_{i}=0\right)$ .

CBSE, Class 11, NCERT, Physics, System of Particles and Rotational Motion

(a) The moment of inertia of a planar body (lamina) about an axis perpendicular to its plane is equal to the sum of the moments of inertia of the lamina about any two mutually perpendicular axes in...

Two discs of moments of inertia $I_{1}$ and $I_{2}$ about their respective axes (normal to the disc and passing through the centre), and rotating with angular speeds $\omega_{1}$ and $\omega_{2}$ are brought into contact face to face with their axes of rotation coincident.
(a) What is the angular speed of the two-disc system?
(b) Show that the kinetic energy of the combined system is less than the sum of the initial kinetic energies of the two discs. How do you account for this loss in energy? Take $\omega_{1} \neq \omega_{2}$

CBSE, Class 11, NCERT, Physics, System of Particles and Rotational Motion

(a) Let I1 and I2 be the moment of inertia of the two turntables respectively. Let ω1 and ω2 be the angular speed of the two turntables respectively. So, we can say, Angular momentum of turntable...

A bullet of mass $10 \mathrm{~g}$ and speed $\mathbf{5 0 0} \mathrm{m} / \mathrm{s}$ is fired into a door and gets embedded exactly at the centre of the door. The door is $1.0 \mathrm{~m}$ wide and weighs $12 \mathrm{~kg}$ . It is hinged at one end and rotates about a vertical axis practically without friction. Find the angular speed of the door just after the bullet embeds into it.

CBSE, Class 11, NCERT, Physics, System of Particles and Rotational Motion

Velocity is given as v = 500 m/s Mass of bullet is given as m = 10 g or 10 × 10–3 kg The width of the door is given as L = 1 m The radius of the door is given as r = 1 / 2 Mass of the door is given...

A man stands on a rotating platform, with his arms stretched horizontally holding a $5 \mathrm{~kg}$ weight in each hand. The angular speed of the platform is 30 revolutions per minute. The man then brings his arms close to his body with the distance of each weight from the axis changing from $90 \mathrm{~cm}$ to $20 \mathrm{~cm}$ . The moment of inertia of the man together with the platform may be taken to be constant and equal to $7.6 \mathrm{~kg} \mathbf{m}^{2}$ .
(a) What is his new angular speed? (Neglect friction.)
(b) Is kinetic energy conserved in the process? If not, from where does the change come about?

CBSE, Class 11, NCERT, Physics, System of Particles and Rotational Motion

Mass of each weight is given as 5 kg The moment of inertia of the man-platform system is given as 7.6 kg m2 So, the moment of inertia when his arms are fully stretched to 90 cm can be calculated as,...

As shown in Figure the two sides of a step ladder BA and CA are $1.6 \mathbf{m}$ long and hinged at A. A rope DE, $0.5$ m is tied halfway up. A weight $40 \mathrm{~kg}$ is suspended from a point F, $1.2 \mathrm{~m}$ from B along with the ladder BA. Assuming the floor to be frictionless and neglecting the weight of the ladder, find the tension in the rope and forces exerted by the floor on the ladder. (Take $g=9.8$ $\mathrm{m} / \mathbf{s}^{2}$ ) (Hint: Consider the equilibrium of each side of the ladder separately.)

CBSE, Class 11, NCERT, Physics, System of Particles and Rotational Motion

Solution: The above figure can be redrawn as, where, $N_{B}$ is the force being applied by floor point $B$ on the ladder $N_{c}$ is the force being applied by floor point $C$ on the ladder The...

A solid cylinder rolls up an inclined plane of the angle of inclination $30^{\circ} .$ At the bottom of the inclined plane, the centre of mass of the cylinder has a speed of $5 \mathrm{~m} / \mathrm{s}$ .
(a) How far will the cylinder go up the plane?
(b) How long will it take to return to the bottom?

CBSE, Class 11, NCERT, Physics, System of Particles and Rotational Motion

initial velocity of the solid cylinder is given $v=5 \mathrm{~m} / \mathrm{s}$ Angle of inclination is given as $\theta=30^{\circ}$ We assume that the cylinder goes up to a height of $h$, so we get:...

The oxygen molecule has a mass of $5.30 \times 10^{-26} \mathrm{~kg}$ and a moment of inertia of $1.94 \times 10^{-46} \mathrm{~kg}$ $m^{2}$ about an axis through its centre perpendicular to the lines joining the two atoms. Suppose the mean speed of such a molecule in a gas is $\mathbf{5 0 0} \mathbf{m} / \mathbf{s}$ and that its kinetic energy of rotation is two-thirds of its kinetic energy of translation. Find the average angular velocity of the molecule.

CBSE, Class 11, NCERT, Physics, System of Particles and Rotational Motion

Mass of one oxygen molecule is given as $m=5.30\times10^{-26}kg$ So, the mass of each oxygen atom will be $\frac{m}{2}$ Moment of inertia is given as $I=1.94\times10^{-46}kg m^{2}$ Velocity of the...

A hoop of radius $2 \mathrm{~m}$ weighs $100 \mathrm{~kg}$ . It rolls along a horizontal floor so that its centre of mass has a speed of $20 \mathrm{~cm} / \mathrm{s} .$ How much work has to be done to stop it?

CBSE, Class 11, NCERT, Physics, System of Particles and Rotational Motion

Radius of the ring is given as $r=2 \mathrm{~m}$ Mass of the ring s given as $m=100 \mathrm{~kg}$ Velocity of the hoop is given as $v=20 \mathrm{~cm} / \mathrm{s}=0.2 \mathrm{~m} / \mathrm{s}$ Total...

A solid sphere rolls down two different inclined planes of the same heights but different angles of inclination. (a) Will it reach the bottom with the same speed in each case? (b) Will it take longer to roll down one plane than the other? (c) If so, which one and why?

CBSE, Class 11, NCERT, Physics, System of Particles and Rotational Motion

(a) Let m be the mass of the ball let h be the height of the ball let v be the final velocity of the ball at the bottom of the plane The ball possesses Potential energy $mgh$ at the top of the...

A metre stick is balanced on a knife-edge at its centre. When two coins, each of mass $5 \mathrm{~g}$ are put one on top of the other at the $12.0 \mathrm{~cm}$ mark, the stick is found to be balanced at $45.0 \mathrm{~cm}$ . What is the mass of the metre stick?

CBSE, Class 11, NCERT, Physics, System of Particles and Rotational Motion

When two coins are inserted at the 12 cm point, the metre rule's centre of mass shifts to 45 cm from 50 cm. Mass of two coins is given as $m=10 \mathrm{~g}$ The distance at which the coins are...

From a uniform disc of radius $\mathbf{R}$ , a circular hole of radius $\mathrm{R} / 2$ is cut out. The centre of the hole is at $\mathbf{R} / \mathbf{2}$ from the centre of the original disc. Locate the centre of gravity of the resulting flat body

CBSE, Class 11, NCERT, Physics, System of Particles and Rotational Motion

Let the unit area of the original disc be $\sigma$ The radius of the original disc is given as $2r$ Mass of the original disc can be calculated as $m=\pi\left(2 r^{2}\right) \sigma=4 \pi r^{2}...

To maintain a rotor at a uniform angular speed of $200 \mathrm{rad} \mathrm{s}^{-1}$ , an engine needs to transmit a torque of $180 \mathrm{~N} \mathrm{~m} .$ What is the power required by the engine? (Note: uniform angular velocity in the absence of friction implies zero torque. In practice, applied torque is needed to counter frictional torque). Assume that the engine is $100 \%$ efficient.

CBSE, Class 11, NCERT, Physics, System of Particles and Rotational Motion

The angular speed of the rotor is given as $\omega=200 \mathrm{rad} / \mathrm{s}$ Torque is given as $T=180 \mathrm{Nm}$ So, power of the rotor (P) can be calculated as, $P=T \omega$ $=200 \times...

A rope of negligible mass is wound around a hollow cylinder of mass $3 \mathbf{k g}$ and radius 40 cm. What is the angular acceleration of the cylinder if the rope is pulled with a force of 30 N? What is the linear acceleration of the rope? Assume that there is no slipping.

CBSE, Class 11, NCERT, Physics, System of Particles and Rotational Motion

Mass of the hollow cylinder is given as $m=3 \mathrm{~kg}$ Radius of the hollow cylinder is given as $r=40 \mathrm{~cm}=0.4 \mathrm{~m}$ So, force applied will be $F=30 \mathrm{~N}$ Calculating the...

(a) A child stands at the centre of a turntable with his two arms outstretched. The turntable is set rotating with an angular speed of $40 \mathrm{rev} / \mathrm{min}$ . How much is the angular speed of the child if he folds his hands back and thereby reduces his moment of inertia to $2 / 5$ times the initial value? Assume that the turntable rotates without friction
(b) Show that the child’s new kinetic energy of rotation is more than the initial kinetic energy of rotation. How do you account for this increase in kinetic energy?

CBSE, Class 11, NCERT, Physics, System of Particles and Rotational Motion

(a) Initial angular velocity, is given as $\omega_{1}=40 \mathrm{rev} / \mathrm{min}$ Suppose the final angular velocity be $\omega_{2}$ Let the moment of inertia of the boy with stretched hands be...

A solid cylinder of mass $20 \mathrm{~kg}$ rotates about its axis with angular speed $100 \mathrm{rad} \mathrm{s}^{-1}$ . The radius of the cylinder is $0.25 \mathrm{~m}$ . What is the kinetic energy associated with the rotation of the cylinder? What is the magnitude of angular momentum of the cylinder about its axis?

CBSE, Class 11, NCERT, Physics, System of Particles and Rotational Motion

Mass of the cylinder is given as $m=20 \mathrm{~kg}$ Angular speed is given as $\omega=100 \mathrm{rad} \mathrm{s}^{-1}$ Radius of the cylinder is given as $r=0.25 \mathrm{~m}$ So, the moment of...

Torques of equal magnitude are applied to a hollow cylinder and a solid sphere, both having the same mass and radius. The cylinder is free to rotate about its standard axis of symmetry, and the sphere is free to rotate about an axis passing through its centre. Which of the two will acquire a greater angular speed after a given time?

CBSE, Class 11, NCERT, Physics, System of Particles and Rotational Motion

Let the mass radius of the solid sphere and also the hollow cylinder be m and r. The moment of inertia of the hollow cylinder about its standard axis is given as ${{I}_{1}}=M{{R}^{2}}$ Moment of...

(a) Find the moment of inertia of a sphere about a tangent to the sphere, given the moment of inertia of the sphere about any of its diameters to be $2 \mathrm{MR}^{2} / 5$ , where $M$ is the mass of the sphere and $\mathbf{R}$ is the radius of the sphere.
(b) Given the moment of inertia of a disc of mass M and radius $\mathbf{R}$ about any of its diameters to be $\mathrm{MR}^{2} / 4$ , find its moment of inertia about an axis normal to the disc and passing through a point on its edge.

CBSE, Class 11, NCERT, Physics, System of Particles and Rotational Motion

The moment of inertia of a sphere about its diameter is $=2 \mathrm{MR}^{2} / 5$ and is also shown in the figure, As the the theorem of parallel axes says, M.I of a sphere about a tangent to the...

A car weighs $1800 \mathrm{~kg}$ . The distance between its front and back axles is $1.8 \mathrm{~m}$ . Its centre of gravity is $1.05 \mathrm{~m}$ behind the front axle. Determine the force exerted by the level ground on each front wheel and each back wheel.

CBSE, Class 11, NCERT, Physics, System of Particles and Rotational Motion

Mass of the car is given as $m=1800 \mathrm{~kg}$ Distance between the two axles is given as $d=1.8 \mathrm{~m}$ Distance of the centre of gravity from the front axle is given as $=1.05 \mathrm{~m}$...

A $2 \mathrm{~m}$ irregular plank weighing $\mathrm{W} \mathrm{kg}$ is suspended in the manner shown below, by strings of negligible weight. If the strings make an angle of $35^{0}$ and $55^{\circ}$ respectively with the vertical, find the location of center of gravity of the plank from the left end.

CBSE, Class 11, NCERT, Physics, System of Particles and Rotational Motion

Following is the FBD(Free Body Diagram) for the above figure: Length of the plank is given as $\mid=2 \mathrm{~m}$ $\theta_{1}=35^{\circ}$ and $\theta_{2}=55^{\circ}$ Let the tensions produced in...

Two particles, each of mass $m$ and speed v, travel in opposite directions along parallel lines separated by a distance $d$ . Show that the angular momentum vector of the two-particle system is the same whatever be the point about which the angular momentum is taken

CBSE, Class 11, NCERT, Physics, System of Particles and Rotational Motion

Considering three points $Z, C$ and $X$ : Angular momentum at Z will be given as, $\mathrm{Lz}=\mathrm{mv} \times 0+\mathrm{mv} \times \mathrm{d}$ $=\mathrm{mvd}-(1)$ Angular momentum about $x$ will...

Find the components along the $x, y, z$ axes of the angular momentum I of a particle, whose position vector is $r$ with components $x, y, z$ and momentum is $p$ with components $p_{x}, p_{y}$ and $p_{z}$ . Show that if the particle moves only in the $x-y$ plane the angular momentum has only a zcomponent.

CBSE, Class 11, NCERT, Physics, System of Particles and Rotational Motion

Linear momentum is given by $\vec{p}=p_{x} \hat{i}+p_{y} \hat{j}+p_{z} \hat{k}$ Positional vector of the body is given by $\vec{r}=x \hat{i}+y \hat{j}+z \hat{k}$ Angular momentum is given by...

Show that $a \cdot(b \times c)$ is equal in magnitude to the volume of the parallelepiped formed on the three vectors, $a, b$ and $c$ .

CBSE, Class 11, NCERT, Physics, System of Particles and Rotational Motion

Let the parallelepiped formed be: where, $\overrightarrow{O J}=\vec{a}, \overrightarrow{O L}=\vec{b}$ and $\overrightarrow{O K}=\vec{c}$ $\hat{n}$ is a unit vector along $\mathrm{OJ}$ and is...

Show that the area of the triangle contained between the vectors a and $b$ is one half of the magnitude of $\mathbf{a} \times \mathbf{b}$

CBSE, Class 11, NCERT, Physics, System of Particles and Rotational Motion

The side $A D$ is equal to $\vec{a}$ and $A B$ is equal to $\vec{b}$ For $\triangle \mathrm{ADN}$ : $\sin \theta=\mathrm{DN} / \mathrm{AD}=\mathrm{DN} / \vec{a}$ $\mathrm{DN}=\vec{a} \sin \theta$...

A child sits stationary at one end of a long trolley moving uniformly with a speed $\mathbf{V}$ on a smooth horizontal floor. If the child gets up and runs about on the trolley in any manner, what is the speed of the CM of the (trolley + child) system?

CBSE, Class 11, NCERT, Physics, System of Particles and Rotational Motion

The child and the trolley are one system, and the youngster's movement within the cart is entirely internal. The velocity of the system's centre of mass will not change because there is no external...

In the HCI molecule, the separation between the nuclei of the two atoms is about $1.27 \AA(1 \AA$ $=10^{-10} \mathrm{~m}$ ). Find the approximate location of the CM of the molecule, given that a chlorine atom is about $35.5$ times as massive as a hydrogen atom and nearly all the mass of an atom is concentrated in its nucleus.

CBSE, Class 11, NCERT, Physics, System of Particles and Rotational Motion

Mass of hydrogen atom is known as $1$ unit Mass of chlorine atom is known as $35.5$ unit Let the center of mass to be $x$ metre from the chlorine atom So, the distance of center of mass from the...

7.1. The centre of mass of which of the following is located outside the body? a) a pencil b) a shotput c) a dice d) a bangle

Read the statement below carefully, and state, with reasons, if it is true or false; A wheel moving down a perfectly frictionless inclined plane will undergo slipping (not rolling) motion.

Read each statement below carefully, and state, with reasons, if it is true or false;
(a) The instantaneous acceleration of the point of contact during rolling is zero.
(b) For perfect rolling motion, work done against friction is zero.

Read each statement below carefully, and state, with reasons, if it is true or false;
(a) During rolling, the force of friction acts in the same direction as the direction of motion of the CM of the body.
(b) The instantaneous speed of the point of contact during rolling is zero.

A solid disc and a ring, both of radius $10 \mathrm{~cm}$ are placed on a horizontal table simultaneously, with an initial angular speed equal to $10 \mathrm{~m} \mathrm{rad} \mathrm{s}^{-1}$ . Which of the two will start to roll earlier? The coefficient of kinetic friction is $\mu_{k}=0.2 .$

A hoop of radius $2 \mathrm{~m}$ weighs $100 \mathrm{~kg}$ . It rolls along a horizontal floor so that its centre of mass has a speed of $20 \mathrm{~cm} / \mathrm{s} .$ How much work has to be done to stop it?

A solid sphere rolls down two different inclined planes of the same heights but different angles of inclination. (a) Will it reach the bottom with the same speed in each case? (b) Will it take longer to roll down one plane than the other? (c) If so, which one and why?

A metre stick is balanced on a knife-edge at its centre. When two coins, each of mass $5 \mathrm{~g}$ are put one on top of the other at the $12.0 \mathrm{~cm}$ mark, the stick is found to be balanced at $45.0 \mathrm{~cm}$ . What is the mass of the metre stick?

From a uniform disc of radius $\mathbf{R}$ , a circular hole of radius $\mathrm{R} / 2$ is cut out. The centre of the hole is at $\mathbf{R} / \mathbf{2}$ from the centre of the original disc. Locate the centre of gravity of the resulting flat body

A rope of negligible mass is wound around a hollow cylinder of mass $3 \mathbf{k g}$ and radius 40 cm. What is the angular acceleration of the cylinder if the rope is pulled with a force of 30 N? What is the linear acceleration of the rope? Assume that there is no slipping.

A car weighs $1800 \mathrm{~kg}$ . The distance between its front and back axles is $1.8 \mathrm{~m}$ . Its centre of gravity is $1.05 \mathrm{~m}$ behind the front axle. Determine the force exerted by the level ground on each front wheel and each back wheel.

Two particles, each of mass $m$ and speed v, travel in opposite directions along parallel lines separated by a distance $d$ . Show that the angular momentum vector of the two-particle system is the same whatever be the point about which the angular momentum is taken

Show that $a \cdot(b \times c)$ is equal in magnitude to the volume of the parallelepiped formed on the three vectors, $a, b$ and $c$ .

Show that the area of the triangle contained between the vectors a and $b$ is one half of the magnitude of $\mathbf{a} \times \mathbf{b}$

A child sits stationary at one end of a long trolley moving uniformly with a speed $\mathbf{V}$ on a smooth horizontal floor. If the child gets up and runs about on the trolley in any manner, what is the speed of the CM of the (trolley + child) system?

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