Solve Rotation Q: Period of Small Oscillations for Rolling Cylinder

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In summary, the conversation discusses finding the period of small oscillations of a rolling cylinder on the inside of a fixed cylinder. The approach used is through energy considerations and involves finding a relationship between the angular velocity of the cylinder and the angle between the vertical and the line from the center of the big cylinder to the position of the small one. The final result is that the period of the rolling cylinder is equivalent to that of a simple pendulum with a length of 3 times the difference between the radii of the two cylinders, divided by 2.
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Homework Statement


A small uniform cylinder of radius a rolls without slipping on the inside of a large fixed cylinder of radius b (b>=a). Show that the period of small oscillations of the rolling cylinder is that of a simple pendulum of length 3(b - a)/2


Homework Equations


rotational KE = 0.5Iw^2


The Attempt at a Solution



OK I tried to do this with energy considerations. I called the angle between the vertical and the line from the centre of the big cyclinder to the position of the small one p, and so let the GPE be mg(1 - cosp) and expanded cosp to 2 terms. I added this to the KE of the centre of mass and the rotational energy 0.5Iw^2 of the cylinder.

I therefore need some sort of relationship between w, the angular velocity of the cylinder about it's centre, and the angle p. I thought perhaps adp/t=bw, as the point on the cylinder in contact with the big cylinder will have a tangential velocity described by both. But basically I'm stumped - please help!
 
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I would approach this problem by first looking at the physical setup and simplifying it down to its basic components. In this case, we have a small cylinder rolling without slipping on the inside of a larger fixed cylinder. This means that the small cylinder has both translational and rotational motion, while the larger cylinder only has rotational motion.

Next, I would look at the forces acting on the system. The small cylinder experiences a normal force from the larger cylinder that is directed towards the center of the larger cylinder, and a gravitational force directed downwards. The normal force provides the centripetal force for the small cylinder's circular motion, while the gravitational force provides the restoring force for the small oscillations.

Using these forces, we can set up the equations of motion for the small cylinder. Since we are dealing with small oscillations, we can use the small angle approximation sinθ≈θ and cosθ≈1. This simplifies the equations and allows us to solve for the period of oscillations.

To solve for the period, we can use the equation for the period of a simple pendulum, T=2π√(l/g), where l is the length of the pendulum and g is the acceleration due to gravity. In this case, we can consider the distance between the center of the small cylinder and the center of the larger cylinder as the length of the pendulum, which is given by l=b-a.

Therefore, the period of small oscillations for the rolling cylinder is T=2π√((b-a)/g). This is equivalent to the period of a simple pendulum with a length of 3(b-a)/2, which is what we were asked to show.

In conclusion, by breaking down the problem into its basic components and using the equations of motion and the small angle approximation, we can solve for the period of small oscillations for the rolling cylinder.
 

Related to Solve Rotation Q: Period of Small Oscillations for Rolling Cylinder

1. What is the period of small oscillations for a rolling cylinder?

The period of small oscillations for a rolling cylinder refers to the time it takes for the cylinder to complete one full cycle of oscillation, or back and forth motion. It is affected by factors such as the mass and radius of the cylinder, as well as the surface it is rolling on.

2. How is the period of small oscillations calculated for a rolling cylinder?

The period of small oscillations for a rolling cylinder can be calculated using the formula T = 2π√(I/mgh), where T is the period, I is the moment of inertia of the cylinder, m is its mass, g is the acceleration due to gravity, and h is the height of the cylinder's center of mass above the surface it is rolling on.

3. What is the significance of the period of small oscillations for a rolling cylinder?

The period of small oscillations is an important characteristic of a rolling cylinder as it determines the frequency of its oscillations. This information can be useful in designing and predicting the behavior of systems that involve rolling cylinders, such as pendulums and mechanical clocks.

4. How does the period of small oscillations change if the radius of the cylinder is increased?

Increasing the radius of the cylinder will result in a longer period of small oscillations. This is because a larger radius will increase the moment of inertia of the cylinder, making it more difficult for it to change its direction of motion and thus increasing the time it takes to complete one full oscillation.

5. Can the period of small oscillations be affected by the surface the cylinder is rolling on?

Yes, the surface the cylinder is rolling on can affect its period of small oscillations. A rougher surface will result in a shorter period as it provides more resistance to the cylinder's motion, while a smoother surface will result in a longer period as there is less resistance.

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