A device consists of eight balls each of mass 0.5 kg attached to the ends of low

Question

A device consists of eight balls each of mass 0.5 kg attached to the ends of low-mass spokes of length 1.2 m, so the radius of rotation of the balls is 0.6 m. The device is mounted in the vertical plane. The axle is held up by supports that are not shown, and the wheel is free to rotate on the nearly frictionless axle. A lump of clay with mass 0.25 kg falls and sticks to one of the balls at the location shown, when the spoke attached to that ball is at 45 degrees to the horizontal. Just before the impact the clay has a speed 8 m/s, and the wheel is rotating counterclockwise with angular speed 0.52 radians/s. Which of the following statements are true about the device and the clay, for angular momentum relative to the axle of the device? Just before the impact, what is the angular momentum of the combined system of device plus clay about the center C? (As usual, x is to the right, y is up, and z is out of the screen, toward you.) Just after the impact, what is the angular momentum of the combined system of device plus clay about the center C? Just after the impact, what is the angular velocity of the device?

Explanation / Answer

a)

option 5

this is because of conservation of angular momentum

b)

for the clay :

mc = mass of clay = 0.25 kg

rc = 0.6 Cos45 = 0.42 m

v = 8 m/s

Lclay = mc v rc = 0.25 x 8 x 0.42 = 0.84 kgm/s

for the wheel :

Iwheel = (8) M r2 = (8) (0.5) (0.6)2 = (8) (0.5) (0.6)2 = 1.44 kgm2

w = 0.52 rad/s

Lwheel = Iwheel w = 1.44 x 0.52 = 0.75 kgm/s

Total angular momentum is given as

L = Lclay + Lwheel = 0.84 + 0.75 = 1.59 kgm/s

c)

using conservation of angular momentum

L' = momentum after the clay sticks to mass at end = initial angular momentum = 1.59 kgm/s

d)

I' = final moment of inertia = (8 M + m) r2 = (8 x 0.5 + 0.25) (0.6)2 = 1.53 kgm2

final angular momentum is given as

L' = I' w'

1.59 = 1.53 w'

w' = 1.04 rad/s

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