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, and 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 of particle 1 is about A is b) angular momentum of particle 2 about A is c) total angular momentum of the system about A is d) total angular momentum of the system about A is

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

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 where is the angular momentum of the system about the centre of mass with velocities considered with respect to the centre of mass. Note , rest of the notation is the standard notation used in the lesson. Note L’ and MR can be said to be angular momenta, respectively, about and of the centre of mass of the system of particles.(ii) Prove that : Further prove that: 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.)

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: Show Where is the momentum of the particle (of mass and Note is the velocity of the particle with respect to the centre of mass.Also, verify using the definition of the centre of mass that (ii) Prove that Where is the total kinetic energy of the system of particles, is the total kinetic energy of the system when the particle velocities are taken relative to the center of mass and is the kinetic energy of the translation of the system as a whole.

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.

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.

(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.

(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 and radius is rolling perfectly on a plane of inclination The coefficient of static friction . If the inclination of the plane is increased, at what value of does the cylinder begin to skid, and not roll perfectly?

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 and radius is rolling perfectly on a plane of inclination The coefficient of static friction (a) How much is the force of friction acting on the cylinder?(b) What is the work done against friction during rolling?

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 are placed on a horizontal table simultaneously, with an initial angular speed equal to . Which of the two will start to roll earlier? The coefficient of kinetic friction is

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 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 and on the disc shown in Figure. Will the disc roll in the direction indicated?

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 is given by using dynamical consideration (i.e. by consideration of forces and torques). Note is the radius of gyration of the body about its symmetry axis, and is the radius of the body. The body starts from rest at the top of the plane.

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 in the plane from an axis through the origin and perpendicular to the plane is . (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 .

(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 and about their respective axes (normal to the disc and passing through the centre), and rotating with angular speeds and 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

(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 and speed is fired into a door and gets embedded exactly at the centre of the door. The door is wide and weighs . 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.

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 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 to . The moment of inertia of the man together with the platform may be taken to be constant and equal to .(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?

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 long and hinged at A. A rope DE, m is tied halfway up. A weight is suspended from a point F, 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 ) (Hint: Consider the equilibrium of each side of the ladder separately.)

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 At the bottom of the inclined plane, the centre of mass of the cylinder has a speed of .(a) How far will the cylinder go up the plane?(b) How long will it take to return to the bottom?

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 and a moment of inertia of 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 and that its kinetic energy of rotation is two-thirds of its kinetic energy of translation. Find the average angular velocity of the molecule.

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 weighs . It rolls along a horizontal floor so that its centre of mass has a speed of How much work has to be done to stop it?

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?

(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 are put one on top of the other at the mark, the stick is found to be balanced at . What is the mass of the metre stick?

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...

### A rope of negligible mass is wound around a hollow cylinder of mass 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.

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 . How much is the angular speed of the child if he folds his hands back and thereby reduces his moment of inertia to 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?

(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 rotates about its axis with angular speed . The radius of the cylinder is . 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?

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?

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 , where is the mass of the sphere and is the radius of the sphere.(b) Given the moment of inertia of a disc of mass M and radius about any of its diameters to be , find its moment of inertia about an axis normal to the disc and passing through a point on its edge.

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 . The distance between its front and back axles is . Its centre of gravity is behind the front axle. Determine the force exerted by the level ground on each front wheel and each back wheel.

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 irregular plank weighing is suspended in the manner shown below, by strings of negligible weight. If the strings make an angle of and respectively with the vertical, find the location of center of gravity of the plank from the left end.

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 and speed v, travel in opposite directions along parallel lines separated by a distance . 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

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 axes of the angular momentum I of a particle, whose position vector is with components and momentum is with components and . Show that if the particle moves only in the plane the angular momentum has only a zcomponent.

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 is equal in magnitude to the volume of the parallelepiped formed on the three vectors, and .

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 is one half of the magnitude of

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

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...