5-111. A box having a weight of 8 lb is moving around in a circle of radius rA = 2 ft with a speed of (vA) 1 = 5 ft>s while connected to the end of a rope. If the rope is pulled inward with a constant speed of vr = 4 ft>s, determine the speed of the box at the instant rB = 1 ft. How much work is done after pulling in the rope from A to B? Neglect friction and the size of the box.

a) vB = 10.77 ft/s

b) W = 11.30 lb*ft

Explanation:

a) W = 8 lb ⇒ m = W/g = 8 lb/32.2 ft/s² = 0.2484 slug

vA lin = 5 ft/s

rA = 2 ft

v rad = 4 ft/s

vB = ?

rB = 1 ft

W = ?

We can apply The law of conservation of angular momentum

Lin = Lfin

m*vA*rA = m*vB*rB ⇒ vB = vA*rA / rB

⇒ vB = (5 ft/s) * (2 ft) / (1 ft) = 10 ft/s (tangential speed)

then we get

vB = √ (vB tang² + vB rad²) ⇒ vB = √ ((10 ft/s) ² + (4 ft/s) ²)

⇒ vB = 10.77 ft/s

b) W = ΔK = KB - KA = 0.5*m*vB² - 0.5*m*vA²

⇒ W = 0.5*m * (vB² - vA²) = 0.5*0.2484 slug * ((10.77 ft/s) ² - (5 ft/s) ²)

⇒ W = 11.30 lb*ft