PHY5200 F06

Chapter 4: Energy

Reading

Taylor 4.2-4.4 (today) and 4.3-4.5 (Wednesday).

Recall

T_tot = T_CM + T_rot = ½Mv_CM² + ½I_CMω²

Example: Kinetic Energy of the Swinging Rod

Calculate the kinetic energies of CM motion and relative motion for a swinging rod.

T_CM = ½mv_CM² = (1/8)m²lω²

T_rel = ∑_i ½m_iv'_i² = ½ω² ∑_i m_ir'_i² --> ½ω² ∫ r²dm = 2½ω² ∫₀^l/2 r²dm = (1/24) mω²l²

T_tot = ½ω² ∫₀^l r²dm = (1/6)mω²l²

A quick check confirms that T_tot = T_CM + T_rel.

Potential Energy and Conservative Forces

You probably recall that potential energy is a useful concept, but that not every force has an associated potential energy. Forces which can be associated with a potential energy are called conservative forces. You probably recall that gravity (constant or 1/r²) and the electrostatic force are conservative, while friction isn't. We'd like a simple way of differentiating conservative forces from non-conservative forces.

First consider a force in one dimension, F = F xhat where F is some scalar function. The idea of a potential energy is to associate a scalar function, U(x) with the force F such that F = -dU(x)/dx. Then, the integral for the work simplifies as

W = ∫₁² Fdx = -∫₁² (dU/dx)dx = U(1) - U(2)

That is, the work done in moving from position 1 to position 2 only depends on the value of a function at points 1 and 2, and nothing in between. Furthermore, for the potential energy to be a useful concept, the function U can't depend on time (we don't want to get a different value for the work if we move the particle now, or a litle be later) or velocity (we don't want a different value for the work if we move quickly from point 1 to 2 or more slowly). What we want is a function that only depends on position, and determines the total work required to move from point 1 to point 2 independent of the path followed, time of motion, or velocity of the motion. We can easily generalize this to three dimensions as follows:

Conditions for a Force to be Conservative (Taylor)

A force F acting on a particle (object) is conservative if and only if it satisfies two conditions:

F only depends on the particle's position r (and not on the velocity v, or the time t, or any other variable); that is F = F(r).
For any two points 1 and 2, the work W(1-->2) done by F is the same for all paths between 1 and 2.

A conservative force has an associated potential energy function U(r). We can then define a particle's mechanical energy (or simply energy) as the sum of the particle's kinetic energy and its potential energy:

E = KE + PE = T + U(r)

and as long as there are no non-conservative forces acting on the particle, the total energy is conserved. This is a direct consequence of the work -- kinetic energy theorem:

T(2) - T(1) = W = U(1) - U(2)

or after rearranging:

T(2) + U(2) = T(1) + U(1)

and since 1 and 2 can be any points, this shows that T+U=E is constant.

Determining U(r)

We can use the definition of work to determine the form of U(r) for a given force. Choose a reference point to be point 1, where we define the potential to be zero.

U(r) - U(1) = U(r) = -W(1-->r) = -∫₁^rF(r')⋅dr'.

Some Example Potential Energy Functions

Near the earth's surface the force of gravity is nearly constant, F=-mg khat. The potential energy is U(r) = ∫₁^rmgkhat⋅dr' = mg(z - z₁) = mgh, where z and z₁ are the vertical coordinates of the limits, and, if we pick the reference point to be the surface of the earth, h is the height above the surface.

At greater distances from the earth (or any other celestial object), the force of gravity is given by Newton's law of universal gravitation, F = GmM/r² rhat, where m and M are the masses of the test object and the gravitating body, r is the distance between their centers, and G is the constant of gravitation. The potential energy is U(r) = -GmM ∫₁^r(1/r'²)r'hat⋅dr' = GmM(1/r - 1/r₁).

The spring force (in the x direction) is given by F=-kx. The potential energy is U(x) = k ∫₁^x x'dx' = ½k(x² - x₁²).

If we choose r=r1, then the potential energy must be 0, independent of the path taken to go from and return to the starting point:

∫_loop F(r')⋅dr' = 0

This integral is known as a loop integral, and Stokes Theorem relates it to an integral over any surface bounded by the loop

∫_loop F(r')⋅dr' = ∫surface (∇×F)⋅dnhat = 0

Since the loop is arbitrary, so is the surface, resulting in the conclusion that ∇×F = 0 everywhere for the force to be conservative (to have a potential energy function).