MCQs

1. [2082] What is the difference between moment of inertia of a rod about an axis passing through its one end and through its centre perpendicular to its length?
   a. [latex]\frac{ML^2}{3}[/latex]
   b. [latex]\frac{ML^2}{4}[/latex]
   c. [latex]\frac{ML^2}{6}[/latex]
   d. [latex]\frac{ML^2}{12}[/latex]
2. [2081 GIE ‘A’] A merry-go-round is rotating with an angular speed [latex]\omega[/latex]. If a child sits in it, which of the following physical quantity is conserved?
   a. kinetic energy
   b. potential energy
   c. linear momentum
   d. angular momentum
3. [2081 GIE ‘B’] The torque due to the gravitational force on a body about its centre of mass is
   a. finite
   b. infinite
   c. zero
   d. not measurable
4. [2081 ‘B/C’] If the rotational kinetic energy of a body is E and moment of inertia is [latex]I[/latex] then angular momentum will be:
   a. [latex]EI[/latex]
   b. [latex]\sqrt{2EI}[/latex]
   c. [latex]\frac{E}{I}[/latex]
   d. [latex]\sqrt{IE}[/latex]
5. [2081 ‘D’] A disc of moment of inertia [latex]I[/latex] is rotating about an axis passing through its centre and perpendicular to its plane. If a small wax of mass m is dropped at distance r from the axis of rotation, then what will be the new moment of inertia of the disc?
   a. [latex]I[/latex]
   b. [latex]I – mr^2[/latex]
   c. [latex]I + mr^2[/latex]
   d. [latex]\frac{I}{mr^2}[/latex]
6. [2080 GIE ‘A’] The moment of inertia of a body of mass M about a given axis is [latex]I[/latex]. What is the radius of gyration?
   a. [latex]\frac{I}{M}[/latex]
   b. [latex]IM[/latex]
   c. [latex]\sqrt{\frac{I}{M}}[/latex]
   d. [latex]\sqrt{IM}[/latex]
7. [2080 GIE ‘B’] A body applied with constant torque changes the angular momentum [latex]L_o[/latex] to final angular momentum [latex]4L_o[/latex] in 3 sec., then the torque is
   a. [latex]3L_o[/latex]
   b. [latex]L_o[/latex]
   c. [latex]4L_o[/latex]
   d. [latex]2L_o[/latex]
8. [2080 ‘P’] Which quantity in rotational motion is analogous to force in linear motion?
   a. Torque
   b. Moment of inertia
   c. Angular velocity
   d. Angular momentum
9. [2080 ‘R’] When the torque acting upon a system is zero, which of the following will be constant?
   a. Force
   b. Linear momentum
   c. Angular momentum
   d. Angular acceleration
10. [2079 GIE ‘A’] When torque acting upon a system is zero, which of the following will be constant?
    a. Force
    b. Linear momentum
    c. Angular momentum
    d. Impulse
11. [2079 GIE ‘B’] The spokes are used in bicycle wheel to
    a. Increase frictional force
    b. to decrease frictional force
    c. decrease moment of inertia
    d. increase moment of inertia
12. [2079 ‘O’] In rotational motion, the physical quantity that imparts angular acceleration is,
    a. Force
    b. Torque
    c. Moment of inertia
    d. Angular momentum
13. [2079 ‘V’] If L represents momentum [latex]I[/latex], represents moment of inertia, then [latex]\frac{L^2}{2I}[/latex] represents,
    a. Rotational kinetic energy
    b. Torque
    c. Power
    d. Potential energy

ANSWERS:

1.b	2.d	3.c	4.b	5.c	6.c	7.b	8.a	9.c	10.c
11.d	12.b	13.a

THEORETICAL QUESTIONS

New Course

1. [2082]
   1. A ballet dancer sometimes stretched and sometimes folds her arms during her performance, why? Justify. [2]
   2. Establish a relation between torque and moment of inertia for a rigid body. [2]
   3. Why do we prefer a wrench of longer arm over a wrench of shorter arm? [1]

2. [2081 GIE ‘A’]
   The moment of inertia is analogous to the mass in linear motion.
   1. Define the moment of inertia and prove that the kinetic energy of a rotational body is [latex]\frac{1}{2}I\omega^2[/latex] where symbols have their usual meanings. [1+2]
   2. The ratio of moment of inertia of two spheres having same K.E. is 1:4, calculate the ratio of their angular velocities. [2] Ans: 0.845

3. [2081 GIE ‘B’] Define angular momentum and write its SI unit. [2]
4. [2081 ‘B/C’]
   1. Define the moment of inertia. [1]
   2. A uniform rod of mass m and length L is rotating about an axis AB passing through its one end and perpendicular to the length. Calculate the moment of inertia of the rod about the axis AB. [2]

5. [2081 GIE ‘A’]
   1. Define moment of inertia. [1]
   2. State principle of conservation of angular momentum with one example. [1+1]

6. [2080 GIE ‘B’] State and explain the principle of conservation of angular momentum. [2]
7. [2080 ‘P’] Define angular momentum. Write its SI unit. [1+1]
8. [2080 ‘R’]
   1. Define radius of gyration. [1]
   2. Calculate the moment of inertia of a thin uniform rod about an axis passing through its centre and perpendicular to its length. [2]

9. [2079 GIE ‘A’] State the principle of conservation of angular momentum. [2]
10. [2079 GIE ‘B’]
    1. Define moment of inertia. [1]
    2. Define the relationship between angular momentum and moment of inertia. [2]

11. [2079 ‘O’] The angular speed is inversely proportional to the moment of inertia, that is given by the principle of conservation of energy.
    1. In a flywheel, most of the mass is concentrated at the rim? Explain why? [1]
    2. The angular velocity of the earth around the sun increases, when it comes closer to the sun. why? [2]
    3. If the earth were to shrink suddenly, what would happen to the length of the day? [2]

12. [2079 ‘V’]
    1. What do you mean by moment of inertia? [1]
    2. State principle of conservation of angular momentum. [1]

Old Course
SHORT ANSWER QUESTION

13. [2076 ‘B’] If the ice on the polar caps of the earth melts, how will it affect the duration of the day? Explain. [2]
14. [2076 ‘C’] Can you distinguish a raw egg and a hardboiled egg by spinning each one on the table? Explain. [2]
15. [2075 ‘A’] Experienced cooks can tell whether an egg is raw or hard boiled by rolling it down a slope. How is this possible? What are they looking for? [2]
16. [2075 ‘B’] Both “the work done by a force” and “the torque produced by a force” are the product of force and the position vector. How can one make the difference between the two? Explain.[2]
17. [2074 ‘S’] If earth shrinks, how will be the duration of a day affected? [2]
18. [2074 ‘A’] Does the angular momentum of a body, moving in a circular path change? Give explanation to your answer. [2]
19. [2072 ‘S’] The cap of a bottle can be easily opened with the help of two fingers than with one finger. Why? [2]

LONG ANSWER QUESTION

20. [2076 GIE ‘A’] Define moment of inertia. Show that K.E. of a rotating body is [latex]\frac{1}{2}I\omega^2[/latex].
21. [2076 GIE ‘B’] What is moment of inertia? Show that in rotational motion, power is the product of torque and angular velocity.
22. [2076 ‘B’] Define moment of inertia. Derive an expression for the moment of inertia of thin uniform rod about an axis through its centre and perpendicular to its length.
23. [2075 ‘A’] Show that in rotational motion, power is the product of torque and angular velocity.
24. [2075 ‘B’] What is the physical meaning of moment of inertia of a rigid body? Also derive its expression in the case of a thin and uniform rod about an axis passing through one end and perpendicular to its length.              
25. [2074 ‘S’] What is meant by moment of inertia? How is it related with the rotational kinetic energy of a body?    
26. [2074 ‘A’] Explain the concept of torque and angular acceleration in the case of a rigid body. Derive a relation between them.
27. [2074 ‘B’] Define moment of inertia. How is this related with angular momentum of a body rotating about an axis of rotation?
28. [2073 ‘S’] Define torque and couple in rotational motion. Also derive an expression for the work done by a couple.      
29. [2073 ‘C’] Define moment of inertia. Obtain an expression for the moment of inertia of a thin and uniform rod about an axis passing through the centre and perpendicular to its length.
30. [2072 ‘S’] What is radius of gyration? Show that the acceleration of a body rolling down an inclined plane is [latex]\alpha = \frac{MgSin\theta}{(M+\frac{I}{R^2})}[/latex], where [latex]\theta[/latex] is the angle of inclination of the plane through which a body rolls down. M is the mass of the body.  is the moment of inertia and R is the radius.           
31. [2072 ‘D’] Define moment of inertia and radius of gyration. Derive an expression for the kinetic energy of rotation of a rigid body.   
32. [2072 ‘E’] State and explain the principle of conservation of angular momentum with example.

NUMERICAL PROBLEMS

New Course

1. [2081 GIE ‘B’] A disc of moment of inertia [latex]5\times 10^{-4}kgm^2[/latex] is rotating freely about the axis through its centre at 60 rpm. Calculate the new revolution per minute if some wax of mass 0.01 kg dropped gently on the disc 0.06 m from the axis.  [3] Ans: 56 rpm
2. [2081 ‘B/C’] A flywheel of moment of inertia 0.32 kgm² is rotated steadily at 120 rad/sec by a 50 watt electric motor. Calculate:
   1. the kinetic energy of the flywheel.
   2. the frictional couple opposing the rotation. [2] Ans: (i) 2304 J (ii) 0.42 Nm
3. [2081 ‘D’] A wheel starts from rest and accelerates with constant angular acceleration to an angular velocity of 8 revolutions per second in 5 seconds. Calculate:
   1. angular acceleration and
   2. angle which the wheel has rotated at the end of 3 sec. [2] Ans: (i) 10.05 rad/sec² (ii) 45.24 radian
4. [2080 GIE ‘B’] A playground merry-go-round of radius 2.0 m has a moment of inertia 250 kgm² and is rotating at 10 rev/min. A 25 kg child jumps onto the edge of the merry-go-round. What is the new angular speed of the merry-go-round? [3] Ans: 7.14 rpm
5. [2080 ‘P’] The speed of a motor engine decreases from 900 rev/min. to 600 rev/min. in 10 seconds. Calculate:
   1. The angular acceleration
   2. Number of revolutions made by the motor during this interval
   3. How many additional seconds are required for motor to come to rest in the same rate. [3] Ans: (i) – 3.14 rad/sec², (ii) 125 rev, (iii) 20 sec.
6. [2080 ‘R’] A physics teacher stands on a freely rotating platform. He holds a dumbbell in each hand of his outstretched arms while a student gives him a push until his angular velocity reaches 1.5 rad/s. When the freely spinning teacher pulls his hands in close to his body, his angular velocity increases to 5.0 rad/s. What is the ratio of his final kinetic energy to initial kinetic energy? [2] Ans: 3.33
7. [2079 GIE ‘A’/ 2075 GIE] A ballet dancer spins with 2.4 rev/s with her arms outstretched when the moment of inertia about the axis of rotation is [latex]I[/latex]. With her arms folded; the moment of inertia about the same axis becomes [latex]0.6I[/latex]. Calculate the new rate of spin. [3] Ans: 4 rev/sec
8. [2079 GIE ‘B’] A constant torque of 500 Nm turns a wheel which has a moment of inertia 20 kgm² about its centre. Find the angular velocity gained in two seconds. [3] Ans: 50 rad/sec
9. [2079 ‘V’] A disc of moment of inertia [latex]5\times 10^{-4}[/latex] kg m² is rotating freely about the axis through its centre at 40 rpm. Calculate the new revolution per minute if some wax of mass 0.02 kg dropped gently on to the disc 0.08 m from the axis. [3] Ans: 32 rpm

Old Course

10. [2076 ‘C’] A wheel starts from rest and accelerates with constant angular acceleration to an angular velocity of 15 revolutions per second in 10 seconds. Calculate the angular acceleration and angle which the wheel has rotated at the end of 2 second. Ans: 9.42 rad/s², 18.84 rad
11. [2073 ‘D’] An electric fan is turned off, and its angular velocity decreases uniformly from 500 rev/min to 200 rev/min in 4 seconds. Find the angular acceleration and the number of revolutions made by the motor in the 4 sec interval. Ans: 7.85 rad/s², 23.3 revolutions
12. [2072 ‘C’] A constant torque of 500 Nm turns wheel which has a moment of inertia 20 kgm² about its center. Find the angular velocity gained in 2 second and the kinetic energy gained. Ans: 50 rads^-1, 25000 J
13. [2072 ‘E’] A disc of radius 1 m and mass 5 kg is rolling along a horizontal plane, its moment of inertia about its centre is 2.5 kgm². If its velocity along the plane is 2ms^-1, find its angular velocity and the total energy. Ans: 2 rads^-1, 15 J
14. A constant torque of 500 Nm turns a wheel about its centre. The moment of inertia about this axis is 100 kgm². Find the angular velocity gained in 4 seconds and kinetic energy gained after 20 revolutions. Ans: 20 rad/s, 62800 J
15. A computer disc drive is turned on starting from the rest and has constant angular acceleration, (a) how long did it take to make the first complete rotation, and (b) what is its angular acceleration? Given that the disk took 0.750 sec for the drive to make its second complete revolution. Ans: 1.81 sec, 3.83 rad/s
16. A constant torque of 200 Nm turns a wheel about its centre. The moment of inertia about the axis is 100 kgm². Find the angular velocity gained in 4 seconds and the kinetic energy gained after 10 revolutions. Ans: 8 rad/s, 12566.4 J
17. A ballet dancer spins with 2.4 rev/s with her arms outstretched when the moment of inertia about the axis of rotation is [latex]I[/latex]. With her arms folded, the moment of inertia about the same axis becomes [latex]0.6I[/latex]. Calculate the new rate of spin. Ans: 4
18. A constant torque of 200 N turns a wheel about its center. The moment of inertia of it about the axis is 100 kgm². Find the K.E. gained after 20 revolutions when it starts from rest. Ans: 25132.7 J
19. A ballet dancer spins about a vertical axis at 1 revolution per second with her arms stretched. With her arms folded, her moment of inertia about the axis decreases by 40%, calculate the new rate of revolution. Ans: 1.67 rps
20. An electric fan is turned off, and its angular velocity decreases uniformly from 500 rev/min to 200 rev/min in 4.00 s. (a) Find the angular acceleration and the number of revolutions made by the motor in 4.00 s interval. (b) How many more seconds are required for the fan to come to rest if the angular acceleration remains constant? Ans: [latex]-7.85\ rads^{-1}[/latex], 23.3, 2.68 s
21. Speed of a body spinning about an axis increase from rest to 100 [latex]rev.min^{-1}[/latex] in 5 sec., if a constant torque of 20 Nm is applied. The external torque is then removed and the body comes to rest in 100 sec. due to friction. Calculate the frictional torque. Ans: 1 Nm