Question

A 1.53kg bucket hangs on a rope wrapped around a pulley of mass 7.07kg and radius 66cm. This pulley is frictionless in its axle, and has the shape of a solid uniform disk.

A. Explain conceptually why the moment of inertia of this pulley is less than the moment of inertia of a hoop around its center with the same mass and circumference as the pulley.

B. What is the angular acceleration of the pulley?

C. What is the acceleration of the bucket?

Answer 1

here.

mass of bucket , m1 = 1.53 kg

mass of pulley , m2 = 7.07 kg

radius , r = 66 cm = 0.66 m

a)

moment of inertia = M * r^2

r-is the distance from the axis of rotation

for same mass, the moment of inertia is proportional to r

for hoop

the all the mass is concentrated r distance from the center

and for solid disk(pulley)

the mass is evenly distributed from the center to the edge

so, for some masses ,r is lesser than the radius

so, the moment of inertia of pulley is less than hoop

b)

the accelration of system , a = net force /effective mass

a = ( m1 * g)/(m1 + I/r^2)

a = ( m1 * g)/(m1 + 0.5 * m2 * r^2/r^2)

a = ( 1.53 * 9.81)/(1.53 + 0.5 * 7.07)

a = 2.96 m/s^2

the angular accelration , alpha = a/r

alpha = 2.96/0.66 = 4.49 rad/s^2

c)

the accelration of the bucket is 2.96 m/s^2