Part of bike wheel is cut out, find period of oscillations

[alalalash_kachok]

Homework Statement

Part of bike wheel with angle ##\alpha## is cut out. All the mass of wheel is concetrated in wheel rim.

Relevant Equations

Find period of wheel's oscillations, if attach wheel's center to rod.

Assume the part of wheel with angle ##\phi## not is cut out, but is located next to cut out part. Turn this from right to left. Then wheel's center of mass will be higher by $$R\phi \sin(\alpha/2)$$. It allows to express potential energy. But I don't know how I can express kinetic energy in this case. Help me. please

 

[jbriggs444]

Homework Statement: Part of bike wheel with angle ##\alpha## is cut out. All the mass of wheel is concetrated in wheel rim.
Relevant Equations: Find period of wheel's oscillations, if attach wheel's center to rod.

Assume the part of wheel with angle ##\phi## not is cut out, but is located next to cut out part. Turn this from right to left. Then wheel's center of mass will be higher by $$R\phi \sin(\alpha/2)$$.

Let us see your calculation for this center of mass increase. It looks wrong to me.

Also, if we are cutting out a hole at a positive (first or second quadrant) angle ##\phi_0##, does that mean that the new center of mass is going to be higher or lower than the original center?

Possibly I am misunderstanding your interpretation of the problem statement. We have cut out a piece of the wheel but then pasted it back in next to where we cut it out? And then flipped the wheel in a mirror image fashion from right to left? That seems unnecessarily baroque.

It allows to express potential energy. But I don't know how I can express kinetic energy in this case. Help me. please

Once we have potential energy as a function of ##\phi##, we can look at kinetic energy in terms of rotation rate ##\omega##.

You should be able to write that formula now.