Rotational Work and Kinetic Energy

1. Rotational Work and Kinetic Energy Dual Credit Physics Montwood High School R. Casao

2. Rotational Work • Work W is the net work if more than one force or torque acts on an object. • For rotational motion, the rotational work, W = F·s, done by a single force F acting tangentially along an arc length s is: W = F·s = F·(r·) = · •  has to be in radians. • For a single torque acting through and angle , W = ·

3. When the torque and angular displacement are in opposite directions the torque does negative work and slows the rotation of the body. • When the torque and angular displacement are in the same direction, the torque does positive work and increases the rotation of the body.

4. Rotational Power • Instantaneous rotational power: •  must be in rad/s. • Unit: Watts

5. Work-Energy Theorem and Kinetic Energy • The relationship between the net rotational work done on a rigid body and the change in rotational kinetic energy of the body can be derived from the equation for rotational work: Wnet = · = I·· • Assuming the torques are due to constant force,  is constant. • From

6. Rotational kinetic energy, K: K = 0.5·I·2 • The net rotational work done on an object is equal to the change in rotational kinetic energy of the object (with 0 J linear kinetic energy).

7. To change the rotational kinetic energy of an object, a net torque must be applied. • Rotating bodies often have two types of kinetic energy: the kinetic energy due to the linear motion of the object and the kinetic energy due to the rotational motion of the object. • Linear kinetic energy: Klinear = 0.5·m·v2 • Rotational kinetic energy: Krotational = 0.5·I·2 • Total kinetic energy: Ktotal = Klinear +Krotational

8. Example: Rolling Ball

9. For an object rolling down an incline without slipping, vcm is independent of mass and radius. • The masses and the radii cancel out, so all objects of a particular shape (with the same equation for the moment of inertia) roll with the same speed regardless of their size or density. • The rolling speed does vary with the moment of inertia, which varies with the shape. • Rigid bodies with different shapes roll with different speeds.

10. If you release a cylindrical hoop (I = m·r2), a solid cylinder (I = 0.5·m·r2), and a uniform sphere (I = 0.4·m·r2) at the same time from the top of an incline, the sphere reaches the bottom first, followed by the cylinder and the hoop. • The smaller the rotational inertia, the greater the angular acceleration.

11. Conservation of Energy for Rotational Motion • Energy before = Energy after • Ugi + Ki linear + Ki rotational = Ugf + Kf linear + Kf rotational