Answer:
1080 rpm
Explanation:
The drum will follow a constant acceleration for a given time and then maintain its speed.
The constant acceleration will follow:
w(t) = w0 + γ * t
Where
w0 = initial angular speed (zero in this case)
γ = angular acceleration
The angular acceleration will depend on the torque applied and the inertia moment of the drum
T = γ * J
The block will apply a torque of:
T = P * r
T = 250 * 0.5 = 125 lb*ft
The drum weights 50lb, so it has a mass of
m = P/g = 50/32.2 = 1.55 lb*s^2/ft
The drum has a moment of inertia of
J = 1/2 * m * r^2
J = 1/2 * 1.55 * 0.5^2 = 0.194 lb*s^2*ft
(remember that lb is a unit of force not mass, but lb*s^2/ft is an indirect unit of mass)
So, the angular acceleration is
γ = T/J
γ = 125/0.194 = 644 rad/s^2
Now we need to know how long will it take the block to fall 5 ft.
Since wee know the drum has radius r = 0.5 f, we need to find how much it needs to spin to release 5 ft of rope.
The circumference is 2π*r = 3.14 feet, it will take it 5/3.14 = 1.59 turns to release 5 feet of rope. 1.59 turns is 2π*1.59 = 10 radians.
The equation for the rotation is:
a(t) = a0 + w0 * t + 1/2 * γ * t^2
a0 and w0 are zero
10 = 1/2 * 644 * t^2
t^2 = 0.031
t = 0.176 s
Then, the angular speed is:
w(0.176) = 644 * 0.176 = 113.3 rad/s
Which is 113.3/2π = 18 turns per second or
18 * 60 = 1080 rpm
