*Originally posted by Jirakon*

**A mass (***m*) is hanging on a massless, unstretchable rod of length *l*, and it swings like a pendulum. There is a frictinal force proportional to the velocity of the mass.

a) What is the equation of motion for the mass?

b) In the approximation of small ampiltudes (sin x = x), show that the natural frequency is (g/*l*)^(1/2), where g is the ...[text shortened]... ecting friction, but considering large amplitudes ( sin x =/= x), determine the period *T*.

Missed this one! Now let's see...

**(a) What is the equation of motion for the mass?**
The arc length "s" swept out through an angle "theta" by a pendulum of length "L" is given by:

s = L * theta

To find the velocity, we differentiate with respect to time:

v = ds/dt = L * dtheta/dt

The friction force is proportional to the velocity, so we have:

Ff = kL * dtheta/dt

Differentiating "v" with respect to time we get:

a = dv/dt = L * d^2theta/dt^2

Multiplying this by the mass gives us the net force:

F = ma = mL * d^2theta/dt^2

Also, a force balance on a free-swing mass attached to the pendulum that is constrained to swing on a circular arc reveals that the resultant force is:

F = -mg * sin(theta)

This value is negative, because a positive displacement results in a force back along the arc. Summing this force and the friction force gives us:

F(total) = kL * dtheta/dt - mg * sin(theta)

Equating this expression with our other force expression, we get:

mL * d^2theta/dt^2 = kL * dtheta/dt - mg * sin(theta)

Simplifying, we have:

d^2theta/dt^2 - (k/m) * dtheta/dt + (g/L) * sin(theta) = 0

This is the expression for the motion of the pendulum when the friction force is proportional to the velocity. I think it can be solved using Laplace transforms, but I haven't done them in a while so I'll leave them for another day. ðŸ˜‰

**(b) In the approximation of small ampiltudes (sin x = x), show that the natural frequency is (g/l)^(1/2), where g is the gravitational field strength.**
In approximating small amplitudes, I'll make the assumption that the velocity is small as well and so we can neglect friction. Therefore, our expression above reduces to:

d^2theta/dt^2 + (g/L) * theta = 0

Using the D-operator method to solve this, we get:

(D^2 + (g/L))(theta) = 0

D = +/- i*(g/L)^0.5

We could plug this value into the characteristic expression, but all we're really interested in is the magnitude of the constant, which denotes the frequency (g/L)^0.5 as required.

**(c) Neglecting friction, but considering large amplitudes ( sin x =/= x), determine the period T.**
For large amplitudes with no friction, our expression reduces to:

d^2theta/dt^2 + (g/L) * sin(theta) = 0

According to Wikipedia, after solving this expression for dtheta/dt and then inverting, the result can't be integrated so you can only approximate the answer using a power series.

http://en.wikipedia.org/wiki/Pendulum_(mathematics)#Arbitrary-amplitude_period