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

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 =  · 

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.

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

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: W net =  ·  = I·  ·  Assuming the torques are due to constant force,  is constant. From

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).

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 : K linear = 0.5·m·v 2 – Rotational kinetic energy: K rotational = 0.5·I·  2 – Total kinetic energy: K total = K linear +K rotational

Example: Rolling Ball

For an object rolling down an incline without slipping, v cm 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.

If you release a cylindrical hoop (I = m·r 2 ), a solid cylinder (I = 0.5·m·r 2 ), and a uniform sphere (I = 0.4·m·r 2 ) 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.

Conservation of Energy for Rotational Motion Energy before = Energy after Ug i + K i linear + K i rotational = Ug f + K f linear + K f rotational