ROTATIONAL AND TRANSLATIONAL QUANTITIES

Position

Velocity = d /dt = d/dt

Acceleration = d /dt = d/dt

Force = m =

Kinetic energy k = mv/2 k = I/2

Work W = W =

Power P = P =

Potential Energy U = -.d U = -.d

Momentum = m = or = I

Mass m Moment of inertia I = mr

Impulse = dt = = dt =

Thrust (dm/dt) (dI/dt)

For a constant acceleration only: For constant angular acceleration only:

r = vt + 0.5at = t + 0.5t

v = v + at = + t

v= v + 2ar = + 2

r = ( + v) (t/2) = (+ )(t/2)

For a system of particles with a CONSTANT total mass, M:

= M and = d/dt

= (r)/M and = M(d/dt).

For systems where the mass may or may not vary:

= d/dt = m(d/dt) + (dm/dt)

The second term is the linear thrust which acts like a force and produces an acceleration. A linear thrust exists if mass enters or leaves the system with a non- zero relative velocity. For a rocket, dm/dt is negative. The thrust of a rocket is in the negative or opposite direction of the exhaust velocity. A rocket will thrust up if mass is ejected down.

A force that acts for a finite time is called a linear impulse () which produces a finite change in the linear momentum. A torque that acts for a finite time is called an angular impulse () which produces a finite change in the angular momentum.

A torque is a force multiplied by a non-zero lever arm. The magnitude of a torque is:

Fl = Frsin

where r is a vector that starts at the pivot point and extends to where the force hits the body, is the angle between r and the force, and l = rsin is the lever arm or perpendicular distance of the force with respect to the pivot. For a rigid body, a body with a constant moment of inertia with respect to the pivot point,

= I

The general form for the angular momentum is:

=

For a constrained system or an object with a pivot axis through the center of mass which is a principle axis (axis of symmetry),

= I

Newton’s law for momentum applied to rotation is:

= d/dt

For = I,

= I(d/dt) + (dI/dt)

The second term is the angular thrust. An angular thrust exists if the moment of inertia (distribution of mass) is changing with a non-zero relative angular velocity about the pivot. The angular thrust acts like a torque and produces an angular acceleration.