ROTATIONAL MOTION IIITorque, Angular Acceleration and Rotational Energy

NEWTON’S 2ND LAW & ROTATIONAL MOTION

The net force on a particle is proportional to its TANGENTIAL acceleration.

The net torque on a particle is proportional to its ANGULAR acceleration.

ROTATIONAL INERTIA

I = ∑mr2

Rot. Inertia = ∑ masses of particles x radius2

Smaller radius Lower Rotational Inertia

Larger radius Higher Rotational Inertia

EXAMPLE The motor of an electric saw brings the circular blade

up to the rated angular speed of 80 rev/s in 240 rev. If the rotational inertia of the blade is 1.41 x 10 -3 kg m2, what net torque must the motor apply to the blade?

ROTATIONAL WORK

The rotational work WR done by a constant torque τ in turning an object through an angle θ isWR = τθ Θ must be in radians; Unit – Joule (J)

The rotational work done by a net external torque equals the change in rotational kinetic energy

Therefor the formulas for rotational work and energy are analogous to the translational formulas

ROTATIONAL WORK AND KINETIC ENERGY

Rotational Translational

Work: W = W = Fd = Fx

Kin. Energy: K = ½ I 2 K = ½ mv2

Power: P = P = Fv

Work-Energy Theorem:W = K = ½ I 2 - ½ I o

2 W = ½ mv2 - ½ mvo2

ROLLING BODIES The kinetic energy of a rolling body (without

slipping) relative to an axis through the contact point is the sum of the rotational kinetic energy about an axis through the center of mass and the translational kinetic energy of the center of mass.

K = ½ ICM 2 + ½ mvCM2

total = rotational + translationalKE KE + KE

If the rolling body experiences a change in height then potential energy (mgh) must also be included

CONCEPT CHECK

EXAMPLE A 1 kg cylinder with a rotational inertia of 0.0625 kg

m2 rolls without slipping down a one meter high incline. At the bottom of the incline the cylinder’s translational speed is 3.13 m/s. What is the cylinder’s angular velocity?