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 1.6 m, so the radius of rotation of the balls is 0.8 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 7 m/s, and the wheel is rotating counterclockwise with angular speed 0.21 radians/s.

(a) Which of the following statements are true about the device and the clay, for angular momentum relative to the axle of the device?

The angular momentum of the device + clay just after the collision is equal to the angular momentum of the device + clay just before the collision.The angular momentum of the falling clay is zero because the clay is moving in a straight line.Just before the collision the angular momentum of the wheel is 0.The angular momentum of the device is the sum of the angular momenta of all eight balls.The angular momentum of the device is the same before and after the collision.

radians/s
(e) Qualitatively, what happens to the total linear momentum of the combined system? Why?

Some of the linear momentum is changed into angular momentum.The downward linear momentum decreases because the axle exerts an upward force.    Some of the linear momentum is changed into energy.There is no change because linear momentum is always conserved.

(f) Qualitatively, what happens to the total kinetic energy of the combined system? Why?

There is no change because kinetic energy is always conserved.The total kinetic energy decreases because there is an increase of thermal energy in this inelastic collision.    Some of the kinetic energy is changed into linear momentum.Some of the kinetic energy is changed into angular momentum.

A device consists of eight balls each of mass 0.5 kg attached to the ends of low-mass spokes of length 1.6 m, so the radius of rotation of the balls is 0.8 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 7 m/s, and the wheel is rotating counterclockwise with angular speed 0.21 radians/s. (a) Which of the following statements are true about the device and the clay, for angular momentum relative to the axle of the device? The angular momentum of the device + clay just after the collision is equal to the angular momentum of the device + clay just before the collision. The angular momentum of the falling clay is zero because the clay is moving in a straight line. Just before the collision the angular momentum of the wheel is 0. The angular momentum of the device is the sum of the angular momenta of all eight balls. The angular momentum of the device is the same before and after the collision. (b) 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.) L with arrowC,i = kg · m2/s (c) Just after the impact, what is the angular momentum of the combined system of device plus clay about the center C? L with arrowC,f = kg · m2/s (d) Just after the impact, what is the angular velocity of the device? vector omegaf = radians/s (e) Qualitatively, what happens to the total linear momentum of the combined system? Why? Some of the linear momentum is changed into angular momentum. The downward linear momentum decreases because the axle exerts an upward force. Some of the linear momentum is changed into energy. There is no change because linear momentum is always conserved. (f) Qualitatively, what happens to the total kinetic energy of the combined system? Why? There is no change because kinetic energy is always conserved. The total kinetic energy decreases because there is an increase of thermal energy in this inelastic collision. Some of the kinetic energy is changed into linear momentum. Some of the kinetic energy is changed into angular momentum.

Explanation / Answer

given 8 balls, m = 0.5 kg each
r = 0.8 m
M = 0.25 kg
theta = 45 deg
u = 7 m/s
w = 0.21 rad/s

a. The angular momentum of the device + clay just after the collision is equal to the angular momentum of the device + clay just before the collision.

this is the statement of conservation of angular momentu
and this is true

The angular momentum of the device is the sum of the angular momenta of all eight balls. This is true for before the collision

b. Just before collision
angular momentum of the clay ball = -M*u*rcos(theta) = -0.9899 kg m^2/s
angular momentum of the appratus = 8*m*r^2*w = 0.5376 kg m^2/s
hecne total angular momentum of the device before collision = -0.4523 kg m^2/s
-ve sign indicates the angular momentum is into the page

c. after collision, angular momentum of the system = -0.4523 kg m^2/s ( from conservation of angular moemtnum)
d. let angular speed of the device after collision be w'
then
-0.4523 = (8*0.5 + 0.25)0.8^2*w'
w' = -0.16628 rad/s
-ve sign indicates angular speed is now clockwise

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