In the figure below, wheel A of radius rA = 11 cm is coupled by belt B to wheel C of radius rC = 23 cm. The angular speed of wheel A is increased from rest at a constant rate of 1.6 rad/s2. Find the time needed for wheel C to reach an angular speed of 110 rev/min, assuming the belt does not slip.
To find the time needed for wheel C to reach an angular speed of 110 rev/min, we need to follow a step-by-step process:
Step 1: Convert the angular speed of wheel C from rev/min to rad/s.
1 revolution = 2π radians
110 rev/min * (2π radians / 1 revolution) = 220π radians/min
Step 2: Convert the angular speed from radians/min to radians/s.
220π radians/min * (1 min / 60 s) = 11π/3 radians/s
Step 3: Use kinematic equations to find the time needed for wheel C to reach the desired angular speed.
We have the initial angular speed of wheel A, ωA = 0 rad/s.
The final angular speed of wheel A, ωA = 1.6 rad/s^2.
The final angular speed of wheel C, ωC = 11π/3 rad/s.
The equation relating angular speed, angular acceleration, and time is:
ωC = ωA + α * t (1)
Given that ωA = 0 rad/s and α = 1.6 rad/s^2, equation (1) simplifies to:
11π/3 rad/s = 1.6 rad/s^2 * t
Solving for t, we get:
t = (11π/3 rad/s) / (1.6 rad/s^2)
t ≈ 6.854 s
Therefore, the time needed for wheel C to reach an angular speed of 110 rev/min is approximately 6.854 seconds.