System of Particles and Rotational Motion

Question 1

Centre of Mass

Answer 1

Point which moves in the same way in which a single particle having the total mass of the system and acted upon by the same external force would move

Question 2

Centre of mass of a two-particle system

Answer 2

R = (m1r1 + m2r2) / (m1 + m2)

Question 3

Prove centre of mass of two particle system from ab-initio

Answer 3

Proved

Question 4

Centre of mass coordinates of n-particle system

Answer 4

xCM = (m1x1 + m2x2 + ….. + mnxn) / (m1+m2+m3+m4+…mn)
Similarly, y and z

Question 5

Binary Systems example

Answer 5

• Binary stars
• Diatomic molecule
• Earth-moon system

Question 6

Rigid Body

Answer 6

Does not undergo size and shape, however large the external force may be acting on it

Question 7

Axis of rotation

Answer 7

• Body possessing rotational motion if all particles move along circles in parallel planes
• Centres of these cricles lie on a fixed line perpendicular to parallel planes

Question 8

Equation of rotational motion

Answer 8

1. w = w0 + (alpha)(t)
2. theta = w0 + 1/2(alpha)(t)^2
3. w^2 - w0^2 = 2(alpha)(theta)

Question 9

Torque

Answer 9

• Depends on magnitude of force
• Depends on Perpendicular distance
• Measurement of product of magnitude of force and perpendicular distance between line of action of force and axis of rotation
• r * F

Question 10

Principal of moments

Answer 10

When a body is in rotational equilibrium, sum of clockwise moments about any point is equal to the sum of anticlockwise moments about that point or algebraic sum of moments about any point is zero

Question 11

Relation between Power and torque

Answer 11

P = torque * omega

Question 12

Angular momentum

Answer 12

Moment of linear momentum of particle about that axis
L = r * p

Question 13

Relation between torque and angular momentum + derivation

Answer 13

d(L) / dt = torque
d (r * p) / dt = dr/dt * p +r * dp/dt
= 0 + r * F (0 as v * p = 0)
= torque

Question 14

2 kinds of motion of a rigid body

Answer 14

• Translation - Particles move with same velocity
• Rotational motion - About an axis

Question 15

Rotational Equilibrium

Answer 15

Resultant of torques due to all forces acting on body about any point must be zero

Question 16

Translational equilibrium

Answer 16

Resultant of all external forces acting on the body must be zero

Question 17

Moment of inertia

Answer 17

Moment of inertia of a rigid body about a fixed axis is defined as sum of products of masses of the particles constituting the body and the squares of their respective distances from axis of rotation

Question 18

Moment of Inertia formula

Answer 18

I = sigma (mr^2)

Question 19

Factors on which I depends upon

Answer 19

• Mass of body
• Size and shape of body
• Distribution of mass about axis of rotation
• Position and orientation of axis of rotation

Question 20

Relation between rotational KE and I

Answer 20

KE = 1/2 * I * (omega)^2

Question 21

Total KE of a rolling body

Answer 21

Translational KE + Rotational KE
= 1/2 mv^2 + 1/2 * I (omega)^2

Question 22

Relation between MI and angular momentum

Answer 22

L = (I) * (omega)

Question 23

Relation between MI and Torque

Answer 23

torque = I * alpha

Question 24

Radius of gyration

Answer 24

I = MK^2 or K = root (I / M)

Question 25

Moment of inertia for:
1. Thin rod of length L (passing through centre)
2. Thin rod of length L (through one end and perpendicular to length)
3. Circular ring (passing through centre and perpendicular to plane)
4. Circular ring (passing through diameter)
5. Circular ring (tangent in its plane)
6. Circular disc (passing through centre and perpendicular to plane)
7. Circular disc (passing through diameter)
8. Circular disc (tangent in its plane)
9. Right circular solid
10. Hollow cylinder
11. Solid sphere (through diameter)
12. Hollow sphere (symmetry axis)

Answer 25

1. I = 1/12 * ML^2
2. I = 1/3 * ML^2
3. I = MR^2
4. I = MR^2 / 2
5. I = 3/2 * MR^2
6. I = 1/2 * MR^2
7. I = 1/4 * MR^2
8. I = 5/4 * MR^2
9. I = 1/2 * MR^2
10. I = MR^2
11. I = 2/5 * MR^2
12. I = 2/3 * MR^2