It\'s mainly part e) that I\'m not sure how to do. A child of mass m runs toward

Question

It's mainly part e) that I'm not sure how to do.

A child of mass m runs towards a stationary merry-go-round, travelling tangentially to the merry-go-round with velocity v. The merry-go-round can be modelled as a horizontal disc of mass M and radius R mounted on a vertical frictionless axis. The child jumps on to the merry-go-round and holds on to a handle at the perimeter of the disc. The moment of inertia of a disc about the axis perpendicular to the disc and passing through its centre is MR2. b) Find the final angular velocity vector of the merry-go-round. 5 marks] c) Determine the fraction of the initial kinetic energy that remains in the final system [5 marks] d) The child gets off the merry-go-round and stops it. He then runs once around the outside applying a tangential force of magnitude F on the merry-go-round in his direction of motion. Find his final velocity. [6 marks] e) Two children are sitting opposite each other on the merry-go-round, which is now rotating with an angular speed w. One throws a ball directly at the other at a speed v Rw. Find an expression in terms of v,w and R for the distance by which it misses the second child 6 marks

Explanation / Answer

e) when the ball is thrown, it travels in a straight line towards the seconds child, meanwhile the second child is diplaced from his original position by some distance because of the motion of the merry go round.

let the ball take time 't' to reach the other side, then this time will be,

t= 2R/v;

meanwhile let the child is displaced by a distance 'd',

then, its angular displacement,a =d/R {since, angle= arc/radius}

time taken by the second child to be displaced by a distance 'd' is the same time as taken by the ball,

t=a/w; t= d/Rw

now equating both the values of 't', we get

d/Rw=2R/v

d=2R2w/v

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