SYSTEMS OF PARTICLES AND RIGID BODY MOTION

VERY SHORT ANSWER TYPE

1. A particle moves in a circular path with decreasing speed. What happens to its angular

momentum?

A. Decreases in magnitude and remains the same in direction (along the axis)

2. Give the geometrical meaning of angular momentum.

A. Angular momentum is geometrically the product of twice the mass and aerial

velocity.

3. Is it possible to open a pen cap with one finger? Why?

A. No, since torque cannot be applied.

4. Name the quantity which can bring rolling without slipping.

A. Friction with the surface.

5. When a sphere of radius r rolls with its COM having a linear velocity v, what is the

velocity of (i) the lower most point in contact with the surface and (ii) the top most

point?

A. (i) at lower most point , v = 0, (ii) velocity will be 2v at top most point.

6. Write two factors on which centre of mass of a body do not depend.

A. Two factors on which centre of mass of a body does not depend-

(i) Choice of coordinate system

(ii) Rotatory motion of a body

7. A rifle barrel has a special groove which imparts spin to the bullet. Why?

A. Angular momentum gained by the bullet provides better accuracy.

8. Why is it easier to balance a bicycle in motion?

A. The wheel does not fall as long as the angular momentum remains constant. Due to

friction, angular momentum changes and when the wheel stops, it comes to unstable

equilibrium.

9. What is the torque provided by force acting through the centre of mass of a sphere?

A. Zero. Since τ = r1F and r1 = 0 for all points on the axis.

10. What is the angular velocity of the hour hand of clock?

A. Angular velocity of hour hand

ω = 2∏/12 hour = 2∏/12 hour x 3600 radians/sec.

11. For uniform circular motion, does the direction of the centripetal force depend on the

sense of rotation (i.e.) clockwise or anticlockwise?

A. No.

12. A child sits stationary at the 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 (trolley + child) system?

A. Since there is no external force, the centre of mass will continue to have same

velocity.

13. If a string of a rotating stone breaks, in which direction will the stone move?

A. The stone will move along the tangent at the point of breaking.

14. A projectile fired into the air suddenly explodes into several fragments. What can you

say about the motion of the fragments after the collision?

A. No external force acts. The centre of mass will follow its original path with every

particle scattered.

15. What is the position vector of centre of mass of two particles of equal masses?

A. For two equal masses, the centre of mass lies at the mid-point of the line joining them

→ → →

rcm = (r1 + r2)/2

16. Why does a pilot not fall down when his aeroplane takes a vertical loop?

A. Weight provides necessary centripetal force at the highest point.

17. If one of the particles is heavier than the other, to which side will their centre of mass

shift?

A. Closer to the heavier body and on the line joining the two masses.

18. In a flywheel, most of the mass is concentrated at the rim. Explain why?

A. Concentration of mass at the rim increases the moment of inertia and thereby brings

uniform motion.

19. Two satellites of equal masses, which can be considered as particles are orbiting the

earth at different heights. Will their moments of inertia be same or different?

A. Different. The satellite at larger height has more moment of inertia since

I = mass x (distance)2

20. A planet revolves around a massive star in a highly elliptical orbit. Is its angular

momentum conserved over the entire orbit?

A. Yes. Since no external torque acts on the planets.

21. Explain why the speed of a whirl wind in a tornado is alarmingly high.

A. In a whirl wind, air from nearby regions gets concentrated in a small space. Moment

of inertia I decrease on account of decrease in distance. As L = Iω, = constant,

therefore angular speed ω increases to alarming high values.

22. If the ice on the polar caps of the earth melts, how will it affect the duration of the

day? Explain.

A. Earth rotates about its polar axis. When ice of polar caps of earth melts, mass

concentrated near the axis of rotation spreads out. Therefore, moment of inertia I

increase. As no external torque acts

L = Iω = I(2∏/T) = constant. With increase of I, T will increase, i.e. length of the

day increase.

23. A disc of metal is melted and recast in the form of a solid sphere. What will happen to

the moment of inertia about a vertical axis passing through the centre?

A. Moment of inertia will decrease because Id = ½ mr2 and Is = 2/5 mr

2. The radius of

sphere formed on recasting the disc will also decrease.

24. A ballet dancer stretches her hand out for slowing down. Name the conservation

obeyed.

A. This is based on the principle of conservation of angular momentum.

25. What are the factors on which moment of inertia of a body depends?

A. Moment of inertia of a body depends on position and orientation of axis of rotation. It

also depends on shape, size of the body and the distribution of mass of the body about

the given axis.

26. Two solid spheres of the same mass are made of metals of different densities. Which

of them has a larger moment of inertia about a diameter?

A. The sphere of metal with smaller density shall be bigger in size and hence it will have

larger moment of inertia.

27. A labourer standing near the top of an old wooden step ladder feels unstable. Why?

A. The point of contact of the ladder with the ground is the point about which the ladder

can rotate. When the labourer is at the top of the ladder the lever arm of force is large

and so the turning effect.

28. A ladder is at rest with its upper end against a wall and the lower end on the ground.

Is the ladder more likely to slip, when a man stands on it at the bottom or at the top?

Give reasons?

A. It is more likely to slip when a man stands at the top of the ladder. This is due to the

fact that the man’s weight will provide an extra torque for the slipping of the ladder.

29. If earth were to shrink suddenly, what would happen to the length of the day?

A. Length of the day reduces as earth shrinks

30. Why do we prefer to use a wrench with a long arm?

A. To have more torque with less force.

2 marks

Q1- What is centre of mass of a system of particles? How is it useful?

Q2-Does centre of mass of a body necessarily falls inside the body?

Q3-If two particles of masses M1 and M2 are moving towards each other with speed v1 and

v2, what is speed of their centre of mass?

Q4-What happens to moment of force about a point if line of action of force shifts towards

the point of rotation?

Q5- A body is in rotational motion. Is it necessary that a torque be acting on it?

Q6- Why do we prefer to use wrench of longer arm?

Q7- Why it is easier to open a tap with two fingers than with one finger?

Q8- Can the mass of a body be taken to be concentrated at centre of mass for the purpose of

calculating its rotational inertia?

Q9- A disc is recast into a thin walled cylinder of same radius. Which will have large moment

of inertia?

Q10- A faulty beam balance with unequal arms has its beam horizontal. Are the weights of

the two pans equal? Explain.

Q11- Will two spheres of equal masses and size( one hollow metallic and other wooden) will

have equal moments of inertia? Give reason.

Q12- How does a ballet dancer take advantage of the principle of conservation of angular

momentum?

Q13-When a rigid body said to be in equilibrium? State the necessary condition for a body to

be in equilibrium.

Q14- If earth shrinks suddenly, how the length o0f day will be affected?

Q15- State parallel axes theorem of moment of inertia with help of a labeled diagram.

Q16- Handle to open the door, is always provided near the free edge of the door. Why?

Q17-Draw a uniform triangular lamina and mark its centre of mass.

Q18- A person is standing on a rotating table with metal spheres in his stretched out hands. If

he withdraws his hands to his chest, what will be the effect on his angular velocity?

Q19-The angular speed of the earth around the sun increases, when it comes closer to the sun.

Why?

Q20- Prove that the centre of mass of two particles divides the line joining the particles in the

inverse ratio of their masses.

Q21- Torque and work are both equal to force times distance. Then how they differ?

(i)Work is a scalar quantity while torque is a vector quantity.

(ii)Work= force x distance moved in the direction of force

Torque = force x perpendicular distance from the axis of rotation

Q22- What do you mean by moment of inertia?

The moment of inertia of a rigid body about an axis is the sum of the products of the

masses of its various particles and squares of their perpendicular distance from the axis

of rotation. I = Σ miri2

Q23- A planet revolves around the sun under the gravitational force exerted by sun. Why is

the torque on the planet due to gravitation of sun is zero/

Because the gravitational force acts along the line joining the planet to the sun. Vector

r and F are always parallel

Therefore Torque = r Fsinθ = r F sin00 = 0

Q24- What is torque? Give its S.I. unit.

The turning or rotating effect of a force about the axis of rotation is called moment of

force or torque.

S.I. unit N.m

Q25-Name the factors on which moment of inertia of a body depends

(i) Mass of the body

(ii) Shape and size

(iii) Distribution of mass about axis of rotation

(iv) Position and orientation of the axis of rotation w.r.t. the body

(3M QUESTIONS)

1. What is a rigid body? Give examples.

2. Explain that torque is only due to transverse component of force .Radial

component has nothing to do with torque.

3. Show that centre of mass of an isolated system moves with a uniform velocity

along a straight line path.

4. Locate the centre of mass of uniform triangular lamina and a uniform cone.

5. Explain the concepts of torque and angular momentum.

6. Define the radius of gyration. What is its physical significance?

7. Define a rigid body. Name two kinds of motion which a rigid body can execute.

8. Define a couple.

9. Define centre of mass of a system. How does it differ from the centre of gravity?

10. Find the condition for the cylinder to roll down without slipping.

11. What is rolling motion?

12. State the law of conservation of angular momentum.

13. Define the term angular momentum.

14. On what factors does the turning effect of a force depend? What is the turning

effect of force called?

15. Define the term torque or moment of force. Give its units and dimensions.

5 MARK QUESTIONS

1. What do you mean by centre of mass of a system ? Prove that the centre of mass of a

system moves as if all the mass of the system is concentrated at the at the centre of mass and

all the external forces acting on the system are applied directly at this point .

2. Define the centre of mass of a system of n particles . Prove that in the absence of any

external force , the centre of mass of a system moves with constant velocity .

3. Define the term torque . Show that the torque at a point is the product of :

a) The magnitude of force and the moment arm of the force (lever arm) .

b) The magnitude of position vector of the point and the transverse component of the force .

4. i) Define angular momentum . Prove that angular momentum of a particle is equal to the

product of its linear momentum and perpendicular distance from the axis of rotation .

ii) Prove that the time rate of change of the angular momentum of a particle is equal to the

torque acting on it .

5. What is centre of mass? Show that the centre of mass of a two particle system of equal

masses lies at the centre of the line joining them . Obtain an expression for the position

vector of the centre of mass of a system consisting of n particles .

6. Explain the vector product of two vectors . Mention its any two properties . Show that the

vectors

A = 2i - 3j - k and B = - 6i + 9j + 3k are parallel.

7. State and explain the principle of conservation of angular momentum . Apply this

principle to explain the following observations :

i) An ice skater can increase her angular momentum by folding her arms and bringing her

stretched leg close to the other leg .

ii) The speed of the inner layers of the whirlwind about its axis in a tornado is alarmingly

high .

8. Define moment of inertia . What is its physical significance ? Establish the relation

between moment of inertia and torque on a rigid body.

9. Derive an expression for the rotational kinetic energy of a rigid body rotating with an

angular velocity w and hence define moment of inertia .

A disc of mass 5 kg and radius 0.5 m rolls on the ground at the rate of 10 m/s . Calculate the

kinetic energy of the disc .

10 . a) State the theorems of parallel and perpendicular axes of moment of inertia .

b) What is the moment of inertia of a rod of mass M and length l about an axis perpendicular

to it through one end ?

HOTS

1. 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?

Hint: When the vertical height of the object is very small as compared to earth’s radius,

we call the object

small, otherwise it is extended.

(a) Building and pond are small objects.

(b) A deep lake and a mountain are examples of extended

objects.

2. 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?

Hint: All the mass in a cylinder lies at distance R from the axis of symmetry but most of

the mass of a solid

sphere lies at a smaller distance than R.

3. The variation of angular position θ , of a point on a rotating rigid body,

with time t is shown in Fig. 7.7.

Is the body rotating clock-wise or anti-clockwise?

Hint: Positive slope indicates anticlockwise rotation which taken as positive.

4. A uniform cube of mass m and side a is placed on a frictionless horizontal surface. A

vertical force F is applied to the edge as shown in Fig. 7.8. Match the following (most

appropriate choice):

(a) mg/4 <F < mg /2 (i) Cube will move up.

(b) F > mg/2 (ii) Cube will not exhibit motion.

(c) F > mg (iii) Cube will begin to rotate and

slip at A.

(d) F = mg/4 (iv) Normal reaction effectively at

a/3 from A, no motion.

Hint: (a) ii, (b) iii, (c) i (d) iv

5. A uniform sphere of mass m and radius R is placed on a rough horizontal surface (Fig.

7.9). The sphere is struck horizontally at a height h from the floor. Match the following.

(a) h = R/2 (i) Sphere rolls without slipping with a

constant velocity and no loss of energy.

(b) h = R (ii) Sphere spins clockwise, loses energy by

friction.

(c) h = 3R/2 (iii) Sphere spins anti-clockwise, loses energy

by friction.

(d) h = 7R/5 (iv) Sphere has only a translational motion, looses

energy by friction.

Hint: (a) iii, (b) iv, (c) ii, (d) i

6. 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?

Hint: No. Given ∑ Fi ≠ 0

i

The sum of torques about a certain point ‘0

∑ ri X Fi = 0

i

The sum of torques about any other point O’,

∑ (ri – a) X Fi = ∑ ri X Fi – a X ∑ Fi

i i i

Here, the second term need not vanish.

7. 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?

Hint: The centripetal acceleration in a wheel arise due to the internal elastic forces which

in pairs cancel

each other; being part of a symmetrical system.

In a half wheel the distribution of mass about its centre of mass

(axis of rotation) is not symmetrical. Therefore, the direction

of angular momentum does not coincide with the direction of

angular velocity and hence an external torque is required to

maintain rotation.

8. A door is hinged at one end and is free to rotate about a vertical axis (Fig. 7.10).

Does its weight cause any torque about this axis? Give reason for your answer.

Hint: No. A force can produce torque only along a direction normal to

itself as τ = r × f . So, when the door is in the xy-plane, the

torque produced by gravity can only be along ± Z direction,

never about an axis passing through y direction.

9. 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 speed ω1 and ω2 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.

Hint:

(a) Yes, because there is no net external torque on the system. External forces,

gravitation and normal reaction, act through the axis of rotation, hence produce no

torque.

(b) By angular momentum conservation

I I11 I 22

ω=

(c) =

=

-

(d) The loss in kinetic energy is due to the work against the friction between the two

discs.

10. A disc of radius R is rotating with an angular speed ωo about a

horizontal axis. It is placed on a horizontal table. The coefficient

of kinetic friction is μk.

(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.

Hint: (a) Zero (b) Decreases (c) Increases (d) Friction (e) vcm = Rω.

(f) Acceleration produced in centre of mass due to friction:

acm F/m = k mg/m = k g

Angular acceleration produced by the torque due to friction,

k mgR

I I

vcm ucm acmt ⇒ vcm k gt

and o t ⇒ o k mgRt/I

For rolling without slipping

Vcm/R = ω0 - k mgRt/I

k gt/R = ω0 - k mgRt/I

11. A uniform square plate S (side c) and a uniform rectangular plate R (sides b, a)

Have identical areas and masses (Fig. 7.11). Show that

(i) IxR /I xS < 1; (ii) IyR /IyS > 1; (iii) I zR / IzS > 1.

Hint: Area of square = area of rectangle c2 = ab

(i). and (ii)

and

(iii)

and a2+b

2-2ab > 0

( ) > 0

12. A uniform disc of radius R, is resting on a table on its rim.The coefficient of friction

between

disc and table is μ (Fig 7.12). Now the disc is pulled with a force F as shown in the

figure.

What is the maximum value of F for which the disc rolls without slipping ?

Hint: Let the acceleration of the centre of mass of disc be ‘a’a, then

Ma = F – f

The angular acceleration of the disc is α = a/R. (if there is no sliding)

Then

(½ MR2) α = Rf

Ma = 2f

Thus f = F/3. Since there is no sliding,

f ≤ µ mg

F ≤ 3µ Mg.

Value based questions

(SYSTEMS OF PARTICLES AND RIGID BODY MOTION)

1. Gopal and Radha lived with their parents who worked very hard to make their ends

meet. They both decided to buy a washing machine from the savings of their pocket

money as they felt the utility value of a washing machine and their mother’s

sufferings. Within a year they presented a washing machine bought from their savings

to their mother.

(i) What qualities of children do you appreciate?

(ii) The spin dryer of washing machine revolves at 15 rps and after sometime it slows

down to 5 rps while making 50 revolutions. Find (a) angular acceleration (b) time

taken.

(iii) For rotation about fixed axis, which part of angular velocity ω change?

A. (i) Both the children are good natured and love their parents.

(ii) n1 = 15 rps, n2 = 5 rps. Angle traced in 50 revolutions

θ = 50 x 2∏ rad.

(ω22 – ω1

2) = 2αθ

α = -4∏rad/s2

ω2 = ω1 αt

t = (ω2 –ω1)/α

t = 5s.

(iii) The direction of ω does not change but magnitude may change.

2. Ravi was sleeping on the roof top when he heard some commotion downstairs. He

saw two thieves holding his dad at gun point. He couldn’t find anything nearby except

a big drum and in a split of second he rolled the drum from the stairs which hit hard

one thief holding a gun. The thief fell down with pain and on seeing this other thief

ran away. Finally the thief was caught and handed over to police.

(i) What trait of Ravi saved his dad?

(ii) What do you mean by rolling without slipping?

(iii) A thin hollow cylinder opened at both the ends and weighing 5 kg rolls with a

speed of 10m/s without slipping. Find the kinetic energy.

A. (i) Bravery and presence of mind

(ii) When an objects rolls on a surface where there is no relative motion between

them then the motion is called rolling without slipping.

(iii) m = 5 kg, v = 10m/s

when cylinder rolls without slipping it has both translational and rotational

K.E.

E = ½ mv2 ½ Iω

2

I = mr2 and rω = v

E = ½ mv2 + ½ (mr

2)ω

2

½ mv2 + ½ mv

2 = mv

2

E = 5 x 102

= 500 J

3. Knowing the effects of rise in temperature, pollution and global warming, Balu

decided to contribute through his small effort which could lead to pollution free

environment. So instead of going to school by his car alone, he preferred going by bus

along with his classmates.

(i) What is mainly responsible for global warming?

(ii) What value did balu show?

(iii) If the ice on polar caps melts due to pollution, how will it affect the duration of the

day.

A. (i) pollution

(ii) Balu is very considerate towards environment and wanted to lead in protecting

the environment.

(iii) Earth rotates about its polar axis. When ice of polar caps melts mass

concentrated near axis of rotation spreads out increasing moment of inertia I.

As no external torque acts

L = Iω = I(2∏/T) = constant. With increase of I, T will increase,

i.e. length of the day increase.

4. During Diwali this year, Shyam realized the effects of pollution and smoke created

due to bursting of crackers and his mother had to be hospitalized last year owing to

the ill effects of pollution. So he decided to put an end to this menace by awakening

the people about the harmful effects of crackers and how they can celebrate happy and

safe diwali.

(i) Is bursting of crackers environmental friendly?

(ii) Which force leads to explosion of bomb?

(iii) A cracker of mass 30 kg at rest explodes into two pieces of masses 18 kg and 12

kg. the velocity of 18 kg mass is 6 m/s. what is the K.E. of other mass.

A. (i). No, it causes noise and pollution which is harmful.

(ii) Internal forces within the crackers lead to its explosion while

external forces like gravity remains same.

(iii). m1 = 18 kg, m2 = 12 kg.

v1 = 6m/s, v2 = ?

according to principle of conservation of linear momentum

m1v1 + m2v2 = 0

v2 = -m1v1/m2 = - 9m/s

E2 = ½ m2v22

= 486 J

5. Kapil had been seeing his mother grinding the flour manually since his childhood and

the agony and pain his mother went through during the ordeal of grinding. He thought

of a plan and connected a small electric motor to the wheels of the grindstone. Now

the grinding was effortless and his mother was happy and felt proud of her son.

(i) What does this show about Kapil?

(ii) After connecting motor to grindstone, what energy transformation takes place?

(iii) A grinding stone of radius 2 m revolving at 120 rpm accelerates to 660 rpm in 9

seconds. Find the angular acceleration and linear acceleration.

A. (i) Kapil is caring, thoughtful and applies his practical knowledge.

(ii) Electrical to Mechanical energy.

(iii). r = 2m, n1 = 120 rpm = 2 rps, n2 = 660 rpm = 11 rps.

t= 9 sec

angular acc. α = ? linear acc. a = ?

α = (ω2 – ω1)/t = 2∏(n2-n1)/t = 2∏ x 9 = 2∏ rad/s2

a = r x d = 2 x 2∏ = 4∏m/s2