ROTATIONAL AND TRANSLATIONAL QUANTITIES
Position
Velocity
Acceleration
Force
Kinetic energy
Work
Power
Potential Energy
Momentum
Mass
Impulse
Thrust

r


v = d r /dt


a = d v /dt


 F = ma
k = mv 2 /2
 
W =  F .ds
 
P = F .V


U = -  Fcons. .d s


p = mv
m



I =  F dt = p

v rel (dm/dt)
For a constant acceleration only:
 r = v o t + 0.5at 2
v = v 0 + at




 = d  /dt
 = d  /dt


 = r xF
k = I  2 /2
 
W =   .d

P =  .


U = -   cons. .d s



 
L = r xp or L = I 
Moment of inertia I =  m i r i



H =   dt = L

 rel (dI p /dt)
2
For constant angular acceleration only:
 =  0 t + 0.5  t 2
 = 0 +  t
 2 =  0 2 + 2 
 = (  0 +  )(t/2)
v 2 = v 0 2 + 2a  r
 r = ( v0 + v) (t/2)
For a system of particles with a CONSTANT total mass, M:

Fext



= M acm and Fext = d Pcm /dt



rcm = (  mi r i )/M and Pcm = M(d rcm /dt).
For systems where the mass may or may not vary:




Fext = d p /dt = m(d v /dt) + v
rel
(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 ( I ) which produces a finite
change in the linear momentum. A torque that acts for a finite time is called an angular

impulse ( H ) 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,


 ext = I 
The general form for the angular momentum is:
  
L = r xp
For a constrained system or an object with a pivot axis through the center of mass which
is a principle axis (axis of symmetry),


L = I
Newton’s law for momentum applied to rotation is:


 ext = d L /dt


For L = I  ,



 ext = I(d  /dt) + 
rel
(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.