A heavy pulley is used to descend a height without landing a

A heavy pulley is used to descend a height without landing at a great speed. Let the pulley be a uniform disk of mass “M” and radius “R”. The descent has a height “h” and you want the landing speed to be “v”.

a. What is the amount of mass that must be added in order to produce a certain landing speed?

b.What is the value for the added mass if the person is 80 kg, the pulley is 100 kg, the height to be descended is 5 meters, and the landing speed is to be 2 m/s?

Solution

A) on mass m,

Using Fnet = ma in vertical direction,

mg - T = ma .......(i)

on pulley , torque = I x alpha

RT = ((M + m\' )R^2 / 2) (a/R)

T = ( M +m\') a/2

putting in (i)

mg - ( M +m\') a/2 = ma

a = 2mg / (2m + M +m\')

using v^2 - u^2 = 2ad

v^2 - 0 = 2(2mg / (2m + M +m\') ) (h)

v = 2 sqrt(mgh / (2m + M + m\') )

b) plug in values,

2 = 2 sqrt ( 80 x 9.8 x 5 / (160 + 100 + m\' ) )

1 = 80 x 9.8 x 5 / (260 + m\' )

m\' + 260 = 3920

m\' = 3660 kg