A plate of mass M = 1.0 kg, radius R= 20.0 cm and angular speed w= 1.0 s rotates

Question

A plate of mass M = 1.0 kg, radius R= 20.0 cm and angular speed w= 1.0 s rotates without friction about a vertical axis through its center of mass. Sand is trickled straight down onto the plate at a distance 10 cm from the center. What is the angular speed of the plate and sand after 0.5 kg of sand has been dropped? (It doesn't really make any difference to the final angular speed, if a 10 cm radius ring of sand is dropped all at once.) Please explain in detail. I have no idea how to approach this question. Thank you!

Explanation / Answer

As it has been said, it doesn't really make any difference to the final angular speed, if a 10 cm radius ring of sand is dropped all at once....so lets approach that way....by assuming the entire sand has been dropped at once... Intial moment of inertia, of disk , I1= 0.5 m*r^2 = 0.5 * 1 * 0.2^2 = 0.02 kg m^2... w1 = 1 rad /s Now, Final Moment of inertia, I2 = I1 + I sand = 0.02 + 0.5* 0.1^2 = 0.025 kg m^2 Now, I1 * w1 = I2 * w2 w2 = 0.02 * 1 / 0.025 = 0.8 rad/s

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