Your cat wants to leave. You are sitting in your comfy chair and don\'t want to

Question

Your cat wants to leave. You are sitting in your comfy chair and don't want to gel up, but the door is closed. The door is freely pivoted on well-oiled hinges and whilst it is closed, it is not latched (i.e. it is free to open if struck). The door has a mass of 20kg, a height of 2m and a width of 1m. You throw a ball of mass 200g at 20m/s at the door, striking it perpendicularly exactly in the centre and the ball rebounds elastically. As a consequence, the door begins to open (i.e. it has an angular velocity). How long does it take the door to rotate through 30 degrees? (Which is enough so that the cat can escape)? [The final angular velocity of the door is omega =mvL / (I +m(L/2)^2), so time is pi/(6 omega) seconds. Since m=200g, L=1m, v=20m/s, l=(1/3)*20*(1)^2, so t=0.879s

Explanation / Answer

here,

mass of ball , m = 200 g = 0.2 kg

l = 1 m

speed of ball , v = 20 m/s

mass of door , M = 20 kg
let the final angular speed be w

using conservation of angular momentum about the hinge

w*( I + m*(L/2)^2 ) = m*v*L

w*( (1/3)*M*L^2 + m*(L/2)^2 ) = m*v*L

w*( (1/3) * 20*1^2 + 0.2*(0.5)^2) = 0.2*20*1

w = 0.6 rad/s

angle covered , theta = 30 degree = 0.523 rad

time taken , t = theta/w

t = 0.523/0.6

t = 0.87 s

the time taken for the door to rotate 30 degree is 0.87 s

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