- #1

JD_PM

- 1,131

- 158

- Homework Statement
- None

- Relevant Equations
- $$\ddot \theta + \frac{g}{l} \theta = 0$$

The equation of motion of a simple pendulum is:

$$\ddot \theta + \frac{g}{l} \theta = 0$$

Our Physics professor told us: 'If you want to become a good Physicist you have to be able to analytically check your answers to see whether they make sense'.

In class he took the limits of constant quantities of the problem, like the mass, to see what would physically happen.

However, here we are dealing with SHM, which doesn't depend on the mass of the oscillating object.

Thus, I only see two quantities we can play with: the length ##l## and gravity.

$$lim_{l -> \infty} = \frac{g}{l} = 0$$Then:

$$\ddot \theta = 0$$OK, so if the length of the rope is infinitely large, we expect no SHM at all.

What I think is that, as the rope approaches infinity (and the angle is very small), we can think of the rope as being in the vertical position. That's the equilibrium position,in which the restoring force (i.e. the ##\frac{g}{l}## term) is zero.

$$lim_{g -> \infty} = \frac{g}{l} = \infty$$

In this case we have an infinite restoring force, which means that in absence of any kind of friction the mass will swing infinitely many times.

Do these analytical checks make sense to you?

$$\ddot \theta + \frac{g}{l} \theta = 0$$

Our Physics professor told us: 'If you want to become a good Physicist you have to be able to analytically check your answers to see whether they make sense'.

In class he took the limits of constant quantities of the problem, like the mass, to see what would physically happen.

However, here we are dealing with SHM, which doesn't depend on the mass of the oscillating object.

Thus, I only see two quantities we can play with: the length ##l## and gravity.

__Analytical checking of l____ength ##l##__$$lim_{l -> \infty} = \frac{g}{l} = 0$$Then:

$$\ddot \theta = 0$$OK, so if the length of the rope is infinitely large, we expect no SHM at all.

**But how to justify it?**What I think is that, as the rope approaches infinity (and the angle is very small), we can think of the rope as being in the vertical position. That's the equilibrium position,in which the restoring force (i.e. the ##\frac{g}{l}## term) is zero.

__Analytical checking of gravity__$$lim_{g -> \infty} = \frac{g}{l} = \infty$$

In this case we have an infinite restoring force, which means that in absence of any kind of friction the mass will swing infinitely many times.

Do these analytical checks make sense to you?