Period of a Pendulum
Consider a block of mass released at the top of a frictionless quarter-circle incline of radius . How much time does it take the block to travel to the bottom? Assume the dimensions of the box are much smaller than and that the block stays in contact with the surface of the incline at all times. The acceleration due to gravity is given by .
You may make use of the integral
and use in your solution.
Look at the general case described below. Here we will use:
From the diagram,
The motion of a block down a circular incline is identical to that of a simple pendulum. In this case, the period of a simple pendulum of varying amplitude can be derived by changing the height $h$ from which the block is released.
Putting in terms of the amplitude , it can be seen that . Therefore, .
Period of a Pendulum of Amplitude
Using the previous diagram,
4 General Case
Consider a block of mass released at a height on a frictionless, curved incline described by the function . How much time does it take the block to travel to the bottom? Assume the dimensions of the box are much smaller than and that the block stays in contact with the surface of the incline at all times. The acceleration due to gravity is given by .
When the block is located at a height , its velocity can be derived from conservation of energy.
The time it takes the block to travel down the incline is related to the distance given by:
From the diagram:
From above, is given by . is given by . Ignoring first order differentials, .
Restrictions on :
We will require in our interval so that the block’s velocity vector never points above the surface and hence the block will stay on the incline.
gives the exact expression for the time taken, however no elementary antiderivative exists for most functions. We will make a practical approximation that will make it easy to evaulate the integral. If , we can rewrite as:
We can make the u-substitution and .
If we have , then
which is the time it takes for the block to fall a height when released from rest. We can see that is sufficient in most cases.