Frequencies of Small Oscillations for a Hoop Rolling in a Pipe

• CAF123
In summary: No, the hoop does not rotate about CM here, it rotates about the centre of pipe so ##v=(R-a)\dot{\theta}##.
CAF123
Gold Member

Homework Statement

A hoop of mass ##m## and radius ##a## rolls without slipping inside a pipe of radius ##R## (the motion is 2D). Write down the kinetic and potential energy. Hence find the frequency of small oscillations about equilibrium.

Homework Equations

Moment of inertia of a hoop: I=Mr2
Rotational kinetic energy

The Attempt at a Solution

I envisaged the problem as a hoop rolling backwards and forwards in a pipe shaped like an arc of radius R.
Kinetic energy is sum of rotational and kinetic, so $$T = \frac{1}{2}mv^2 + \frac{1}{2} (ma^2) \left(\frac{v^2}{a^2}\right) = mv^2.$$

The path of the C.O.M of the hoop is arc shaped. Let ##\theta## be the angle between a vertical passing through the centre of the hoop and the C.O.M. Draw a horizontal at the base of the pipe. After a little time, the C.O.M is at angle ##\theta##. For small oscillations the increase in height of the C.O.M from its initial position (at a height a above the base of the pipe) is a + a(1-cosθ). So V = mg(a+a(1-cosθ)).

Is this correct? I am wondering if this can be solved more easily using the centre of momentum frame.

V doesn't look correct. The CM of hoop is at a constant distance (R-a) from the centre of pipe. When you displace the hoop by angle ##\theta##, what is the distance of CM from the base?

Hi Pranav-Arora,
Pranav-Arora said:
V doesn't look correct. The CM of hoop is at a constant distance (R-a) from the centre of pipe. When you displace the hoop by angle ##\theta##, what is the distance of CM from the base?

I believe that would be a + (R-a)(1-cosθ)

CAF123 said:
I believe that would be a + (R-a)(1-cosθ)

Yes, looks right to me.

Pranav-Arora said:
Yes, looks right to me.

Ok, so ##E = mv^2 + mg(a+(R-a)(1-\cos \theta))##. I think to find the freq of small oscillations, I should find a form ##\ddot{\theta}+ \omega^2 \theta = g##, where g is not a function of θ. Differentiating E would give me $$\ddot{\theta} + \left(\frac{g(R-a) - ga}{2a^2}\right)\theta = 0,$$from which I can extract ##\omega## and hence f.

CAF123 said:
Ok, so ##E = mv^2 + mg(a+(R-a)(1-\cos \theta))##. I think to find the freq of small oscillations, I should find a form ##\ddot{\theta}+ \omega^2 \theta = g##, where g is not a function of θ. Differentiating E would give me $$\ddot{\theta} + \left(\frac{g(R-a) - ga}{2a^2}\right)\theta = 0,$$from which I can extract ##\omega## and hence f.

That doesn't look right.

Rewrite the potential energy as mg(R-(R-a)cosθ). Substitute in the energy equation and differentiate again.

What did you substitute for v?

Pranav-Arora said:
That doesn't look right.
Could you please explain why? The equation is certainly dimensionally consistent.

What did you substitute for v?
Because of no slip, I said ##v_{COM} = a \dot{\theta}##.

CAF123 said:
Because of no slip, I said ##v_{COM} = a \dot{\theta}##.

No, the hoop does not rotate about CM here, it rotates about the centre of pipe so ##v=(R-a)\dot{\theta}##.

What is hoop rolling in a pipe?

Hoop rolling in a pipe is a physical phenomenon in which a hoop (a circular object) is rolled inside a pipe (a cylindrical object).

What causes a hoop to roll in a pipe?

The hoop rolling in a pipe is caused by the transfer of kinetic energy from the hoop to the pipe. This is due to the friction between the hoop and the pipe, as well as the shape of the pipe which creates a channel for the hoop to roll in.

What are the real-life applications of hoop rolling in a pipe?

Hoop rolling in a pipe has several practical applications, such as in the manufacturing of pipes and tubes, where it can be used to test the strength and durability of the pipes. It is also used in amusement parks for rides such as the roller coaster.

What factors affect the speed of a hoop rolling in a pipe?

The speed of a hoop rolling in a pipe is affected by various factors including the size and weight of the hoop, the diameter and length of the pipe, the surface texture of the pipe, and the force applied to the hoop.

Are there any challenges in studying hoop rolling in a pipe?

Yes, there are challenges in studying hoop rolling in a pipe as it involves complex mathematical calculations and requires precise measurements. Additionally, external factors such as air resistance and surface imperfections can affect the results.

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