Energy and Angular Momentum
In solving motion problems we can sometimes use Conservation of Energy and
Conservation of Angular Momentum to make the problems easier to solve than the
straightforward way using Newton's Second Law alone.
1. Conservation of Energy
If Fdr is independent of path (and this will be true if F = 0), then we can define a
potential energy: V(r) = - rsr Fdr (where rs is some standard position). If all the forces
in the problem can be put in terms of potential energies, then we can use the
Conservation of Energy to relate the velocity to the position. (For this to hold, we have
no outside work or energies added, and we lose no energy or do no work on the outside.)
This is a first order differential equation (since v and r are related) involving scalars,
instead of the second order differential equation involving vectors from Newton's Second
law (where a and r are related):
½mvf2 + Vi(rf) = ½mvo2 + Vi(ro) = E = constant.
2. Conservation of Angular Momentum
We have defined angular momentum, L, to be:
L = r  p where p = mv .
dL/dt = (dr/dt  p) + (r  dp/dt) .
We note that dr/dt = v, and vv=0. We also note that dp/dt = F. Therefore, we have:
dL/dt = r  F =  ,
where  is the torque. If there is no torque, then dL/dt = 0, which means that L is a
constant.
In a two-dimensional situation, we can use polar form with the z component added
(cylindrical coordinates). In this case we have:
Lz = r  p = r p = r m v = r m r ' = mr2' .
dLz/dt = d(mr2')/dt = 2mrr'' + mr2''
Using Newton's Second Law, and using the acceleration in polar form, we have:
Fr = mar = m(r'' - r'2)
F = ma = m(r'' + 2r'')
z = r F = r(m[r'' + 2r'']) = mr2'' + 2mrr'' = dLz/dt .
In this case, if F = 0, then Lz = mr2' = constant. This can be useful in eliminating ' in
terms of r in either the Fr equation of Newton's Second Law [Fr = mar = m(r''-r'2)] or in
the Conservation of Energy equation (v = 'r) !
Homework Problem #18: A particle of mass, m, moves according to the
equations:
x(t) = xo + at2 (where a is a constant)
y(t) = bt3 (where b is a constant)
z(t) = ct (where c is a constant).
a) Find an expression for the angular momentum, L(t).
b) Find the force, F(t).
c) Find the torque, (t).
d) Verify that (t) = dL(t)/dt .
Homework Problem #19: Determine if the force is conservative, and determine
the potential energy function if it is:
a) Fx = 18abyz3 - 20bx3y2 ; Fy = 18abxz3 – 10bx4y, Fz = 6abxyz2 (where a and
b are constants).
b) F = Fx(x)x + Fy(y)y + Fz(z)z.
HINT: if F is conservative, then the line integral of the force is Independent Of
Path