Rotational Motion

Question 1

Condition for rigid body dynamics

Answer 1

Relative distance between two points should not change

Question 2

Moment of inertia of discrete particle system

Answer 2

Σmiri^2, where mi is the total mass of the body and ri is the perpendicular distance of the body from the axis of rotation

Question 3

Moment of inertia of a continuous body

Answer 3

∫miri^2, because we are adding small-small particles forming the body

Question 4

M.O.I of rod when the axis is the passing from one end

Answer 4

ML^2/3, where m= mass of the body and L = length of the body from the axis

Question 5

M.O.I of rod when the axis is the passing through the centre of mass

Answer 5

ML^2/12, where m= mass of the body and L = length of the body from the axis

Question 6

M.O.I of ring when the axis is the passing through the centre of mass, perpendicular to the plane of the ring

Answer 6

MR^2, where m = mass of the body and r = radius of the body

Question 7

M.O.I of disc when the axis is the passing through the centre of mass, perpendicular to the plane of the ring

Answer 7

MR^2/2, where m = mass of the body and r = radius of the body

Question 8

M.O.I of hollow cylinder, when the axis is the passing through the centre of mass, perpendicular to the plane of the ring

Answer 8

MR^2, where m = mass of the body and r = radius of the body

Question 9

M.O.I of solid cylinder, when the axis is the passing through the centre of mass, perpendicular to the plane of the ring

Answer 9

MR^2/2, where m = mass of the body and r = radius of the body

Question 10

M.O.I of solid sphere, when the axis is the passing through the centre of mass, perpendicular to the plane of the ring

Answer 10

2/5MR^2, where m = mass of the body and r = radius of the body

Question 11

M.O.I of hollow sphere, when the axis is the passing through the centre of mass, perpendicular to the plane of the ring

Answer 11

2/3MR^2, where m = mass of the body and r = radius of the body

Question 12

Parallel Axis Theorem

Answer 12

I = Ic +md^2, where Ic = moment of inertia from the centre of the body and d = perpendicular distance.
This theorem should only be applied on a axis parallel to Ic

Question 13

Calculate the MOI of a solid sphere with axis on the tangent of the sphere.

Answer 13

MOI = 2/5Mr^2 + M(R)^2

Question 14

Perpendicular Axis Theorem

Answer 14

Iz = Ix + Iy, where all the three are perpendicular to each
Iz is perpendicular to the plane of the body, whereas Ix and Iy are parallel to the body

Question 15

Calculate MOI of ring when the axis is perpendicular to the surface.

Answer 15

Iz = MR^2+Mr^2 = 2MR^2

Question 16

Radius of gyration

Answer 16

i = Mk^2,
where k = radius of gyration, distance a single particle has to be placed for it to have the same MOI

Question 17

Torque(τ) and its three definitions

Answer 17

Force because of the rotational effect of a body
τ=rFsinθ
τ = rfperpendicular
τ = rperpendicular F

Question 18

What does equilibrium mean in a rotating and translating body?

Answer 18

Translational Equilibrium = Fnet = 0
Rotational Equilibrium = τnet = 0(about any point)

Question 19

Which point should we take the torque about in a fixed axis rotation?

Answer 19

Hinge Point

Question 20

Angular Momentum

Answer 20

For a body: I(about axis)*ω, ω= angular velocity of the body
For a particle: Mvr, where v = velocity vector and r = radius vector

Question 21

Rotational kinetic energy

Answer 21

Iω^2/2

Question 22

Work Energy Theorem

Answer 22

Work done by all forces = Change in kinetic Energy

Question 23

Acceleration of a pulley when two masses(m1 and m2) are attached by a string with a pulley having mass

Answer 23

Acceleration = (m2g-m1g)/m1+m2+I/R^2

Question 24

What is the conditions for conservation of angular momemtum?

Answer 24

Torque net = 0
Initial Momentum = Final Momentum

Question 25

Angular Impulse

Answer 25

l*J, where j = impulse and l = length from which it is applied
Change in angular momentum

Question 26

What is the conditions for pure rolling?

Answer 26

The velocity of the lowest point of the body should be at rest w.r.t ground.
v = Rω
a = Rα

Question 27

Acceleration of a body rolling from an inclined plane

Answer 27

a= gsinθ/1+I/mR^2

Question 28

What can you say about the kinetic energy of various body rolling down from an inclined plane?

Answer 28

The work done by gravity is same, hence the kinetic energy is same even though everything is different.

Question 29

Angular Momentum of a body rolling

Answer 29

Icω + Mrvc, Ic = MOI about COM and vc = velocity of centre of mass

Question 30

Angle of repose and Angle of toppling

Answer 30

Angle of repose = μs(Coefficient of static friction)
Angle of topping = L/h