nabaa said:
to be honest I don't understand what's wrong with them? isn't the double derivative of 3g/2Lcost = −3g/2Lcost
Yes, but this is not equal to -3g/2L times the original function. If the original function is (3g/2L)cos(t), then -3g/2L*(original function) would be -(9g
2/4L
2)cos(t), which is NOT what you get when you differentiate twice.
So, your proposed function does NOT satisfy your differential equation.
Think about this for a second. What do we know? SOME SORT of sinusoidal function will satisfy the differential equation, but we don't know its exact form. So, in order to figure it out, we need to take into account everything that we DO know:
- we know that at t = 0, angle(t) = A, which means that the function has to be of the type that is non-zero when you give it a zero argument (this helps you decide between sine and cosine)
- we know that A is the maximum displacement, which means that the amplitude of the oscillation is A, which means that the rod goes back and forth between an angular position of +A and -A. This tells you that A has to be multiplying the sinusoidal function in front (and nothing else).
- we know that the frequency has to be in there somewhere and should be set by the physical properties of the rod and its environment. We also know that it should be consistent with the differential equation, which contains that constant C that appears out front AFTER twice differentiating (but maybe now I am giving away too much...)