?A device consists of eight balls each of mass 0.61kg attached to the ends of lo

Question

?A device consists of eight balls each of mass 0.61kg attached to the ends of low-mass spokes of length 2.4m, so that the radius of rotation of the balls is 1.2m. 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 0.42kg falls and sticks to one of the balls at the location shown, when the spoke attached to that ball is at 45° to the horizontal. Just before the impact the clay has a speed 3m/s, and the wheel is rotating counterclockwise with angular speed 0.21rad/s.

Just before the impact, what is the angular momentum of the combined system of device plus clay about the center? (As usual, [^(x)] is to the right, [^(y)] is up, and [^(z)] is out of the page.)

[(L)ec]i=?   ,  ,   ?

Just after the impact, what is the angular momentum of the combined system of device plus clay about the center?

[(L)ec]f=?   ,  ,   ?

Just after impact, what is the angular velocity of this device?

[?ec]f=?   ,  ,   ?

Explanation / Answer

Given: radus of ball's rotation (r) = 2.1 m ; mass of each ball (M) = 0.61 Kg ; mass of lump (m) = 0.42 Kg

speed of clay (v) = 3 m/s angluar speed () = 0.21 rad/s

Now moment of inertia of the wheel before impact (I) = 8*M*r^2 = 8*0.61*2.1*2.1 = 21.52 Kg-m^2

so angular momentum of wheel (L1) = I* = 21.52*0.21 = 4.52 Kg-m^2/s (counter-clockwise)

also the angular momentum of the lump about axis of rotation (L2) = m*v*r*sin(45) = 1.87 Kg-m^2/s (clockwise)

Hence total angular momentum before impact (L) = 4.52-1.87 = 2.65 Kg-m^2/s (counter-clockwise)

Also since there are no external torque acting on the system so total angular momentum of the system remains conserved.

Hence total angular momentum after the impact is also (L) = 4.52-1.87 = 2.65 Kg-m^2/s (counter-clockwise)

Now after the impact total moment of inertia of the system (Inew) = 8*M*r^2 + m*r^2 = 21.52 + 1.85 = 23.37 Kg-m^2

hence new angular velocity of the device = L/Inew  = 2.65/23.37 = 0.113 rad/s

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