Question

Consider a bicycle wheel that initially is not rotating. A block of mass m is attached to the wheel via a string

and is allowed to fall a distance h. Assume that the wheel has a moment of inertia I about its rotation axis.
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Part A
Consider the case that the string tied to the block is wrapped around the outside of the wheel, which has a radius r as shown in (Figure 1). Find w, the angular speed of the wheel after the block has fallen a distance h, for this case.
Express w in terms of m, g, h, rA, and I.
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Part B
WB=
Now consider the case that the string tied to the block is wrapped around a smaller inside axle of the wheel of radius rp as shown in (Figure 2). Find wp, the angular speed of the wheel after the block has fallen
Express wp in terms of m, g, h, rB, and I.
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Part C
17| ΑΣΦ
VE ΑΣΦ
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?
O WA > WB
O WB > WA
O WA = WB
Which of the following describes the relationship between wĄ and wp?
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?
distance h, for this case.

Answer 1

Using conservation of energy, we calculated the angular speed of the bicycle wheel when a block falls, given as w = sqrt((2mgh)/I) for the larger radius and wp = sqrt((2mgh)/(I * rB^2)) for the smaller radius. Due to the smaller radius, wp will be greater than w.

Step-by-step explanation:

To solve for the angular speed of the bicycle wheel after the block has fallen a distance h, we will use energy conservation principles. The potential energy of the mass m as it falls a distance h is converted into the rotational kinetic energy of the wheel.

Part A

For the case where the string is wrapped around the outside of the wheel with radius rA:

Initial potential energy (PE) of the block = mgh

Rotational kinetic energy (RE) of the wheel = (1/2)Iw2

By energy conservation, PE = RE:

mgh = (1/2)Iw2

Rearrange to solve for w:

w = sqrt((2mgh)/I)

Part B

For the case where the string is wrapped around the inner axle with radius rB :

wp will be higher than w since the radius rB is smaller, leading to a higher angular velocity for the same potential energy drop.

The angular speed wp is given by:

wp = sqrt((2mgh)/(I * rB2))

Part C

Since rB < rA, it follows that the angular velocity wp will be greater than w after the block has fallen a distance h.