Chapter 7 - SYSTEM OF PARTICLES AND ROTATIONAL MOTION

A disc of radius R is rotating with an angular speed o
ω about a
horizontal axis. It is placed on a horizontal table. The coefficient of
kinetic friction is μ k.
(a) What was the velocity of its centre of mass before being brought
in contact with the table?
(b) What happens to the linear velocity of a point on its rim when
placed in contact with the table?
(c) What happens to the linear speed of the centre of mass when
disc is placed in contact with the table?
(d) Which force is responsible for the effects in (b) and (c).
(e) What condition should be satisfied for rolling to begin?
(f ) Calculate the time taken for the rolling to begin.

Two cylindrical hollow drums of radii R and 2R, and of a common
height h, are rotating with angular velocities ω (anti-clockwise) and
ω (clockwise), respectively. Their axes, fixed are parallel and in a
horizontal plane separated by (3R +δ ) . They are now brought in
contact (δ → 0) .
(a) Show the frictional forces just after contact.
(b) Identify forces and torques external to the system just after
contact.
(c) What would be the ratio of final angular velocities when friction
ceases?

A disc of radius R is rotating with an angular speed o
ω about a
horizontal axis. It is placed on a horizontal table. The coefficient of
kinetic friction is μ k.
(a) What was the velocity of its centre of mass before being brought
in contact with the table?
(b) What happens to the linear velocity of a point on its rim when
placed in contact with the table?
(c) What happens to the linear speed of the centre of mass when
disc is placed in contact with the table?
(d) Which force is responsible for the effects in (b) and (c).
(e) What condition should be satisfied for rolling to begin?
(f ) Calculate the time taken for the rolling to begin.

Two discs of moments of inertia l1 and l2 andI about their respective
axes (normal to the disc and passing through the centre), and
rotating with angular speed ω1 andω2 are brought into contact
face to face with their axes of rotation coincident.
(a) Does the law of conservation of angular momentum apply to
the situation? why?
(b) Find the angular speed of the two-disc system.
(c) Calculate the loss in kinetic energy of the system in the process.
(d) Account for this loss.

Find the centre of mass of a uniform (a) half-disc, (b) quarter-disc.

(n-1) equal point masses each of mass m are placed at the vertices
of a regular n-polygon. The vacant vertex has a position vector a
with respect to the centre of the polygon. Find the position vector of
centre of mass.

A wheel in uniform motion about an axis passing through its centre
and perpendicular to its plane is considered to be in mechanical
(translational plus rotational) equilibrium because no net external
force or torque is required to sustain its motion. However, the
particles that constitute the wheel do experience a centripetal
acceleration directed towards the centre. How do you reconcile this
fact with the wheel being in equilibrium?
How would you set a half-wheel into uniform motion about an
axis passing through the centre of mass of the wheel and
perpendicular to its plane? Will you require external forces to
sustain the motion?

The vector sum of a system of non-collinear forces acting on a rigid
body is given to be non-zero. If the vector sum of all the torques due
to the system of forces about a certain point is found to be zero,
does this mean that it is necessarily zero about any arbitrary point?

Why does a solid sphere have smaller moment of inertia than a
hollow cylinder of same mass and radius, about an axis passing
through their axes of symmetry?

The centre of gravity of a body on the earth coincides with its centre
of mass for a ‘small’ object whereas for an ‘extended’ object it may
not. What is the qualitative meaning of ‘small’ and ‘extended’ in
this regard?
For which of the following the two coincides? A building, a pond, a
lake, a mountain?

(a) Why is moment of inertia called rotational inertia?
(b) Calculate M.I of a uniform circular disc of mass 500gm and radius 10cm about
(i) Diameter (ii) axis tangent to the disc and parallel to diameter
(c) Axis passing trough centre and perpendicular to its plane?

Four particles of mass 1kg, 2kg, 3kg and 4kg are placed at the four vertices A, B, C
and D of square of side 1m. Find the position of centre of mass of the particle.

The moment of inertia of a solid sphere about a tangent is 7/5 MR² . Find the

moment of inertia about a diameter?

Obtain the equationw = w_o+ at ?

A planet revolves around on massive star in a highly elliptical orbit is its angular
momentum constant over the entire orbit. Give reason?

How is angular momentum related to linear momentum?

State the condition for translational equilibrium of a body?

A wheel 0.5m in radius is moving with a speed of 12m/s. find its angular
speed?

The angle θ covered by a body in rotational motion is give by the equation θ = 6t +

5t² + 2t³. Determine the value of instantaneous angular velocity and angular

acceleration at time t = 2S.

Show that for an isolated system the centre of mass moves with uniform velocity
along a straight line path?

What is the position of centre of mass of a rectangular lamina?

Which component of a force does not contribute towards torque?

Two particles in an isolated system under go head on collision. What is the
acceleration of the centre of mass of the system?

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.
(c) The instantaneous acceleration of the point of contact during rolling is zero.
(d) For perfect rolling motion, work done against friction is zero.
(e) A wheel moving down a perfectly frictionless inclined plane will undergo slipping (not rolling) motion

A cylinder of mass 10 kg and radius 15 cm is rolling perfectly on a plane of inclination 30°. The coefficient of static friction µ_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?
(c) If the inclination θ of the plane is increased, at what value of θ does the cylinder begin to skid, and not roll perfectly?

A solid disc and a ring, both of radius 10 cm are placed on a horizontal table simultaneously, with initial angular speed equal to 10 π rad s^-1. Which of the two will start to roll earlier? The co-efficient of kinetic friction is μ_k = 0.2.

Explain why friction is necessary to make the disc in Fig. 7.41 roll in the direction indicated. (a) Give the direction of frictional force at B, and the sense of frictional torque, before perfect rolling begins.
(b) What is the force of friction after perfect rolling begins?

A disc rotating about its axis with angular speed ω_ois 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 Fig. 7.41? Will the disc roll in the direction indicated?

 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 v² = 2gh / [1 + (k²/R²) ]
Using dynamical consideration (i.e. by consideration of forces and torques). Note k is the radius of gyration of the body about its symmetry axis, and R is the radius of the body. The body starts from rest at the top of the plane. 

(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 perpendicular to the plane is x² + y²).
(b) Prove the theorem of parallel axes.
(Hint: If the centre of mass is chosen to be the origin ∑ m_ir_i = 0).

Two discs of moments of inertia I₁ and I₂ about their respective axes (normal to the disc and passing through the centre), and rotating with angular speeds ω₁ and ω₂ 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 ω₁ ≠ ω₂.

A bullet of mass 10 g and speed 500 m/s is fired into a door and gets embedded exactly at the centre of the door. The door is 1.0 m wide and weighs 12 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.
(Hint: The moment of inertia of the door about the vertical axis at one end is ML²/3.)

A man stands on a rotating platform, with his arms stretched horizontally holding a 5 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 90cm to 20cm. The moment of inertia of the man together with the platform may be taken to be constant and equal to 7.6 kg m². 
(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?

(Hint: Consider the equilibrium of each side of the ladder separately.)

A solid cylinder rolls up an inclined plane of angle of inclination 30°. At the bottom of the inclined plane the centre of mass of the cylinder has a speed of 5 m/s.

The oxygen molecule has a mass of 5.30 × 10^–26 kg and a moment of inertia of 1.94×10^–46 kg m² 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 500 m/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.

A hoop of radius 2 m weighs 100 kg. It rolls along a horizontal floor so that its centre of mass has a speed of 20 cm/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 g are put one on top of the other at the 12.0 cm mark, the stick is found to be balanced at 45.0 cm. What is the mass of the metre stick?

 From a uniform disk of radius R, a circular hole of radius R/2 is cut out. The centre of the hole is at R/2 from the centre of the original disc. Locate the centre of gravity of the resulting flat body.

o maintain a rotor at a uniform angular speed of 200 rad s^–1, an engine needs to transmit a torque of 180 Nm. 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. 

A rope of negligible mass is wound round a hollow cylinder of mass 3 kg 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) 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 rev/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.

A solid cylinder of mass 20 kg rotates about its axis with angular speed 100 rad s^–1. The radius of the cylinder is 0.25 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?

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?

 (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 2MR²/5, where M is the mass of the sphere and R is the radius of the sphere.
(b) Given the moment of inertia of a disc of mass M and radius R about any of its diameters to be MR²/4, find its moment of inertia about an axis normal to the disc and passing through a point on its edge.

A non-uniform bar of weight W is suspended at rest by two strings of negligible weight as shown in Fig.7.39. The angles made by the strings with the vertical are 36.9° and 53.1° respectively. The bar is 2 m long. Calculate the distance d of the centre of gravity of the bar from its left end.

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

Find the components along the x, y, z axes of the angular momentum l of a particle, whose position vector is r with components x, y, z and momentum is p with components px, py and pz. Show that if the particle moves only in the x-yplane the angular momentum has only a z-component.

Show that a. (b × 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 a × b.

A child sits stationary at one end of a long trolley moving uniformly with a speed 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?

. In the HCl molecule, the separation between the nuclei of the two atoms is about 1.27 Å (1 Å = 10^–10 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.

Give the location of the centre of mass of a (i) sphere, (ii) cylinder, (iii) ring, and (iv) cube, each of uniform mass density. Does the centre of mass of a body necessarily lie inside the body?