How Does Mass and Height Affect the Angular Speed of a Rotating Bicycle Wheel?

[PascalPanther]

"Consider the bicycle wheel is not turning initially. A block of mass m is attached (with a massless string) to a wheel. The block is allowed to fall a distance of h. Assume that the wheel has a moment of inertia I about its rotation axis.

Find the angular speed of the wheel after the block has fallen a distance of h in terms of m,g,h, r(of the wheel) and I"

This is what I did:

U + K + W(other) = U + K
mgh + 0 + 0 = 0 + 1/2mv^2 + 1/2 I*omega^2

omega = Sqrt[(2mgh - mv^2)/I]

This is wrong... what am I doing wrong? I know I can use omega = v^2/r, but not sure where that would go...

 

[OlderDan]

"Consider the bicycle wheel is not turning initially. A block of mass m is attached (with a massless string) to a wheel. The block is allowed to fall a distance of h. Assume that the wheel has a moment of inertia I about its rotation axis.

Find the angular speed of the wheel after the block has fallen a distance of h in terms of m,g,h, r(of the wheel) and I"

This is what I did:

U + K + W(other) = U + K
mgh + 0 + 0 = 0 + 1/2mv^2 + 1/2 I*omega^2

omega = Sqrt[(2mgh - mv^2)/I]

This is wrong... what am I doing wrong? I know I can use omega = v^2/r, but not sure where that would go...

ω = v^2/r is not correct, but it is sort of close. When you get it right, you can replace either v or ω using the correct relationship. Since you want to solve for ω, replace the v and solve away.

 

[kyle8921]

Your approach is on the right track, but there are a few errors in your calculation. Let's break down the steps and correct them:

1. Identify the initial and final states of the system: In this case, the initial state is when the wheel is not turning and the block is at a height h above the ground. The final state is when the block has fallen to the ground and the wheel is rotating.

2. Write the conservation of energy equation: As you correctly did, the total energy of the system (U + K + W) in the initial state must equal the total energy in the final state.

3. Calculate the initial and final potential energies: In the initial state, the potential energy is mgh, as you correctly identified. In the final state, the potential energy is 0, since the block is now at the ground.

4. Calculate the initial and final kinetic energies: In the initial state, the kinetic energy is 0 since the wheel is not rotating. In the final state, the kinetic energy is 1/2mv^2 for the block and 1/2Iω^2 for the wheel, as you correctly identified.

5. Set the initial and final energies equal to each other: This is where your calculation went wrong. You should set mgh = 1/2mv^2 + 1/2Iω^2, since the total energy must be conserved.

6. Use the relationship between linear and rotational speed: As you correctly mentioned, the linear speed of the block is related to the angular speed of the wheel by v = ωr, where r is the radius of the wheel. Substituting this into the equation from step 5, we get mgh = 1/2m(ωr)^2 + 1/2Iω^2.

7. Solve for ω: Rearranging the equation from step 6, we get ω = √[2mgh/(mr^2 + I)]. This is the correct equation for the angular speed of the wheel after the block has fallen a distance of h.

In summary, your approach was correct, but there were a few errors in your calculation. By carefully identifying the initial and final states, writing the conservation of energy equation, and correctly substituting the relationship between linear and rotational speed, we arrive at the correct equation for the angular speed of the wheel.