Final answer:
To find the average torque on a game show wheel, the angular deceleration is first determined using angular displacement and initial angular speed. The moment of inertia for a disk is then calculated, and the torque is found by multiplying it with the angular deceleration.
Step-by-step explanation:
To find the average torque exerted on a game show wheel that comes to rest after 0.75 of a turn from an initial angular speed of 1.42 rad/s, we need to follow a series of steps. First, let's determine the final angular velocity (ω_f), which is 0 since the wheel comes to rest. The initial angular velocity (ω_i) is given as 1.42 rad/s.
Since the wheel comes to a stop after 0.75 of a turn, we need to convert this into radians: 0.75 turns * 2π rad/turn = 1.5π rad. This is the angular displacement (θ).
To find the angular deceleration (α), we use the formula θ = ω_i * t + (1/2) * α * t^2. Since ω_f = ω_i + α * t, and ω_f = 0, we can rearrange to α = - (ω_i^2) / (2 * θ). Plugging in the values we get α = - (1.42^2) / (2 * 1.5π) rad/s^2.
The moment of inertia (I) for a disk is (1/2) * mass * radius^2. Here, I = (1/2) * 6.4 kg * (0.71 m)^2. Torque (τ) can be found using τ = I * α. Applying these calculations gives us the average torque.