QUESTION: A circular platform of radius Rp = 4 m and mass Mp = 400 kg rotates on
ID: 1968114 • Letter: Q
Question
QUESTION:
A circular platform of radius Rp = 4 m and mass Mp = 400 kg rotates on frictionless air bearings about its vertical axis at 6 rpm. An 80 kg man standing at the very center of the platform starts walking (at t=0) radially outward at a speed of 0.5 m/s with respect to the platform. Approximating the man by a vertical cylinder of radius Rm = 0.2 m, determine an equation (specific expression) for angular velocity of the platform as a function of time. What is the angular velocity when the man reaches the edge of the platform?
Book Provides, a table, Momement of Inertia and Value of Constant c
I= 1/2 M R^2 (Solid cylinder or disk) c=1/2
Please provide solution and steps to solve this problem.
Explanation / Answer
Def Leppard rocks...
Anyway... this is a conservation of angular momentum problem. This means that
initial moment of inertia x angular speed = final moment of inertia x angular speed
You are given initial angular speed, and you want to solve for final angular speed. So all you need is expressions for the initial and final moment of inertia for the system.
Moment for the disk: ½ Mp Rp2 = ½ * 400 * 42 = 3200
Moment for the cylinder (man) : ½ Mm Rm2 = ½ * 80 * 0.22 = 1.6
Total moment of inertia, initially: 3200 + 1.6 = 3201.6
Now... as the man moves outward, we simply have to use the parallel axis theorem, which states that the moment of inertia for the man is the original value (of 1.6) PLUS his mass times his distance from the center squared. In this case this means that
Total moment of inertia, final: 3201.6 + mass of man x distance from center squared
= 3201.6 + 80 * (speed * time)2 = 3201.6 + 80 * (0.5 t )2
= 3201.6 + 20 t2
Now... we can drop these into the first equation above (conservation of angular momentum):
3201.6 x initial angular speed = [ 3201.6 + 20 t2 ] x final angular speed
Rearrange and you get
final angular speed = initial angular speed * 3201.6 / ( 3201.6 + 20 t2 )
We can use 6 rpm as the initial angular speed. Then the final angular speed, in rpm, would be
final angular speed = 6 * 3201.6 / (3201.6 + 20 t2 )
Or
final angular speed = 19209.6 / (3201.6 + 20 t2 )
Moving at 0.5 m/s, it will take him 8 seconds to reach the edge of the platform (i.e. to go 4 meters). So the angular speed at that time will be
final angular speed = 19209.6 / (3201.6 + 20 * 82 ) = 4.29 rpm