A pendulum consisting of a mass m and a string of length L = 22 cm is hung from the edge of a bicycle wheel of radius r = 26 cm whose axis of rotation has been fixed vertically. The wheel is spun around its axis faster and faster until the pendulum is seen to make an angle of approximately q = 45o from the vertical, as shown in the diagram below. At this point, what is the period of rotation for the wheel assembly?
The first thing to note about this problem is that it involves a mass moving in circular motion. The most common mistake that people make in this problem is a basic one: they get the radius of the circular motion wrong. Don’t just grab for a symbol that looks like the one you need — make sure that it’s really the quantity that you’re looking for! In particular, the radius of the circular motion is not r = 0.26 m! To make this perfectly clear (if it’s not already), let’s add a little something to the figure given with the problem statement. Let’s draw in the horizontal circle that the mass m is moving in:
From the figure it is clearly seen that the radius R of the mass’ circular motion is given by
R = r + L sinq = 0.42 m .
We are now ready to tackle the problem at hand. The reasoning goes as follows: We want to find the period of rotation, T. There is only one place that this comes into play when dealing with forces, and that is in the centripetal acceleration:
Substituting the expression for v into that for ac gives us that
To apply Newton’s 2nd law in component form, we must first draw a FBD for the mass m:
We then get that:
Using this result for the string tension along with the result above for ac in the c-direction equation then gives us that
Were you able to follow all of that? Work carefully through the steps, showing your own work on your notebook paper. This is a good problem showing the fundamentals of how circular motion problems tend to work. If you can fully understand this one, you are well on your way to being able to work with circular motion!