A wheel of mass 0.47 kg and radius 49 cm is spinning with an angular velocity of 24 rad/s. You then push your hand against the edge of the wheel, exerting a force F on the wheel as shown in the figure below. If the wheel comes to a stop after traveling 1/4 of a turn, what is F?
Torque*Angle = (Kinetic energy lost)
F*R*(pi/2) = (1/2)*I*wo^2
= (1/4)*M*R^2*24^2
Solve for F.
wo = 24 rad/s
R = 0.49 m
M = 0.47 kg
To find the force F exerted on the wheel, we can use the principle of conservation of angular momentum. This principle states that the total angular momentum of a system remains constant if no external torque acts on it.
The angular momentum of the wheel can be calculated using the formula:
L = I * ω,
where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity.
The moment of inertia of a solid disk can be calculated using the formula:
I = 1/2 * m * r^2,
where m is the mass of the wheel and r is the radius.
Given:
Mass of the wheel, m = 0.47 kg,
Radius of the wheel, r = 49 cm = 0.49 m,
Angular velocity, ω = 24 rad/s.
First, we need to find the initial angular momentum of the wheel. Substituting the values into the formula:
L_initial = I * ω
= (1/2 * m * r^2) * ω.
Next, we need to find the final angular momentum when the wheel comes to a stop after traveling 1/4 of a turn. Since the wheel is at rest, the final angular velocity is zero. Hence, the final angular momentum is:
L_final = I * ω_final
= I * 0.
Since the total angular momentum is conserved, we can set the initial angular momentum equal to the final angular momentum:
L_initial = L_final.
Substituting the values:
(1/2 * m * r^2) * ω = 0.
Now, rearranging the equation to solve for F:
F = 2 * (L_final / r) = 2 * (L_initial / r) = 2 * I * ω / r.
Substituting the values of I, ω, and r, we have:
F = 2 * (0.5 * m * r^2 * ω / r)
= m * r * ω.
Substituting the given values:
F = 0.47 kg * 0.49 m * 24 rad/s
= 5.6576 N.
Therefore, the force F exerted on the wheel is approximately 5.6576 N.