In the figure below, wheel A of radius rA = 10 cm is coupled by belt B to wheel C of radius rC = 25 cm. The angular speed of wheel A is increased from rest at a constant rate of 1.6 rad/s2. Find the time in seconds needed for wheel C to reach an angular speed of 130 rev/min, assuming the belt does not slip.
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There is no diagram but I'll take an educated guess. You have to take care to get units matching.
2.5 rotations of A produces 1 rotation of C.
(If this isn't obvious, note the circumference of A = 2π x 10cm and the circumference of C is 2π x 25cm.
After 1 rotation of A the belt moves 2π x 10cm.
I rotation of C requires the belt to move 2π x 25cm.
So for 1 rotation of C, the number of rotations of A = (2π x 25cm)/(2π x 10)cm = 2.5.)
When the angular speed of C=130 rev/min, the angular speed of A = 130 x 2.5 = 325rev/min.
This is 325/60 = 5.417 revs per second. (as there are 60 seconds in 1 minute).
This is 5.417 x 2π = 34.0 radians/second (as there are 2π radians in 1 revolution).
If starting from rest, angular speed = angular acceleration x time:
ω = αt (this is the rotational equivalent of v = at)
t = ω/α = 34.0/1.6 = 21s
2.5 rotations of A produces 1 rotation of C.
(If this isn't obvious, note the circumference of A = 2π x 10cm and the circumference of C is 2π x 25cm.
After 1 rotation of A the belt moves 2π x 10cm.
I rotation of C requires the belt to move 2π x 25cm.
So for 1 rotation of C, the number of rotations of A = (2π x 25cm)/(2π x 10)cm = 2.5.)
When the angular speed of C=130 rev/min, the angular speed of A = 130 x 2.5 = 325rev/min.
This is 325/60 = 5.417 revs per second. (as there are 60 seconds in 1 minute).
This is 5.417 x 2π = 34.0 radians/second (as there are 2π radians in 1 revolution).
If starting from rest, angular speed = angular acceleration x time:
ω = αt (this is the rotational equivalent of v = at)
t = ω/α = 34.0/1.6 = 21s