In the figure, wheel A of radius rA is coupled by belt B to wheel C of radius rC. The angular speed of wheel A is increased from rest at a constant rate α. Find the time needed for wheel C to reach angular speed ω assuming the belt does not slip. (Hint: If the belt does not slip, the linear speeds at the two rims must be equal.) State your answer in terms of the given variables.
To find the time needed for wheel C to reach angular speed ω, we can use the relationship between angular speed and linear speed. Since the belt does not slip, the linear speeds at the rims of wheel A and wheel C must be equal.
First, let's find the linear speed of wheel A when it reaches angular speed ω. The linear speed vA of wheel A is given by vA = rA * ω, where rA is the radius of wheel A and ω is its angular speed.
Next, let's find the linear speed of wheel C when it reaches angular speed ω. The linear speed vC of wheel C is given by vC = rC * ω, where rC is the radius of wheel C and ω is its angular speed.
Since the linear speeds at the rims of the two wheels must be equal, we can equate them:
vA = vC
rA * ω = rC * ω
Now, we can solve for the time needed for wheel C to reach angular speed ω by solving this equation for ω:
rA * ω = rC * ω
Dividing both sides by ω, we get:
rA = rC
Therefore, the time needed for wheel C to reach angular speed ω is given by the equation:
t = ω / α
where t is the time, ω is the angular speed, and α is the constant rate at which the angular speed of wheel A is increased.