Energy Formulation

Torque and Angular Acceleration of a Rigid Body

For a rigid body, net torque acting

τ = lα

Where, α = angular acceleration of rigid body

I = moment of inertia about axis of rotation

• Kinetic energy of a rigid body rotating about fixed axis

• Angular moment of a particle about same point

L = r × P

L = m(r × v)

Where L = angular displacement

• Angular moment of a rigid body rotating about a fixed axis.

• Angular moment of a rigid body in combined rotation and transiation

L = L_CM + M(r₀×ν₀)

• Conservation of angular momentum

• Kinetic energy of rigid body in combined translational and rotational motion.

1. Kinetic energy associated with the motion of the centre of mass CM of the body as if the total mass were concentrated at that point.
2. Kinetic energy associated with the rotation of the body about an appropriate axis through CM.

Sum of (i) and (ii) is stated as kinetic energy of rigid body.

Uniform Pure Rolling

Pure rolling means no relative motion or no slipping at point of contact between two bodies.

If V_p = V_Q ⇒ no slipping

ν = Rω

If V_p > V_Q ⇒ forward slipping

ν > Rω

If V_p < V_Q ⇒ backward slipping

ν < Rω

No slippig s = 2πR

Forward slipping s > 2πR

Backward slipping s < 2πR

Accelerated Pure Rolling

A pure rolling is equivalent to pure translation and pure rotation. It follows a uniform rolling and accelerated pure rolling can be defined as

F + f = Ma

(F – f).R = lα

F = force them acting on a body

f = friction on that body

Angular Impulse

The angular impulse of a torque in a given time interval is defined as

where L₂ and L₁ are the angular momentum at time t₂ and t₁ respectively.

Collision: A Collision is an isolated event in which two or more moving bodies exert forces on each other for a relatively short time. Collision between two bodies may be classified in two ways: Head-on collision, and Oblique collision.

Heat-on Collision: Let the two balls of masses m₁ and m₂ collide directly with each other with velocities v₁ and v₂ in direction as shown in figure. After collision the velocity become  and  along the same line.

Where, m₁ = mass of body 1

m₂ = mass of body 2

v₁ = velocity of body 1

v₂ = velocity of body 2

 = velocity of body 1 after collision

 = velocity of body 2 after collision

Where e = coefficient restitution

• In case of head-on elastic collision

e = 1

• In case of head-on inelastic collision

0 < e < 1

• In case of head-on perfectly inelastic collision

e = 0

If e is coefficient of restitution between ball and ground, then after nth collision with the floor, the speed of ball will remain eⁿv₀ and it will go upto a height e²ⁿ h.

Oblique Collision: In case of oblique collision linear momentum of individual particle do change along the common normal direction. No component of impulse act along common tangent direction. So, linear momentum or linear velocity remains unchanged along tangential direction. Net momentum of both the particle remains conserved before and after collision in any direction.