Question

Suppose you’re eating in yet another restaurant where the dishes are shared at the table and all placed uniformly on a rotating disk-like surface. Model this surface as a thin disk of radius 45.3 cm. Someone else has spun the surface, such that it is initially at an angular speed of 0.4 rev/s. The surface and food has a combined mass of 3.3 kg. The waiter, to show off, throws a new dish of dumplings (mass 0.8 kg) onto the surface at a speed of 0.5 m/s, such that the dish lands on and sticks to the very edge of the surface moving in the same direction as the rotating food. While this is happening, you quickly calculate the final angular speed of the food so that you can predict its location at any time before others have a chance to eat the dumplings. What is this speed, in rad/s?

Answer 1

Since there are no external torques for the system of disc and the new dish, angular momentum is conserved.

Angular momentum=I where I is moment of inertia and is angular velocity.

Also, angular momentum=mvr, where m is mass, v is velocity and r is position of the object relative to the point where angular momentum is calculated.

Moment of inertia of disc is given by: mr²/2 where m is mass and r is radius,

Moment of inertia of point mass is given by: mr² where m is mass and r is the distance of the point mass from axis of rotation.

So, moment of inertia of disc=3.3*0.453*0.453/2 ( here, m=3.3 kg and radius =45.3 cm=0.453 m)

= 0.339 kg-m²

Initial angular velocity=0.4 rev/s=0.4*2 rad/sec=2.51 rad/s

So, initial angular momentum of disc=0.339*2.51=0.851 kgm²/s

Initial angular momentum of the new dish=0.8*0.5*0.453 ( here m=0.8 kg,v=0.5 m/s,r=45.3 cm=0.453 m)

=0.181 kgm²/s

So,total initial angular momentum=0.851+0.181=1.032 kgm²/s

Final moment of inertia=moment of inertia of disc+moment of inertia of new dish=0.339+mr²=0.339+0.8*0.453*0.453

=0.503 kgm²

Let the final angular velocity be . So, final angular momentum=0.503

Initial angular momentum= final angular momentum => 0.503=1.032 => =1.032/0.503=2.05 rad/s