physics help.....inertia?

Question

1.) The evenly spaced 4 spoke wheel has a diameter of 1.2 m. Its rim has a mass of 12 kg, and each spoke has a mass of 1.0 kg. Find the moment of inertia of the wheel.

2.) The seat of an adjustable piano stool has the mass (m) and the moment of inetria (I). If the seat is given an initial angular speed ot omega (w) in such a direction that the seat rises and there is no friction, how much higher is the seat when it stops turning?

Answer 1

I am gonna make number two and then number one, ok.

2) Here, I was thinking and thinking, and I got the idea, conservation of energy.

Actually the seat at the beginning has only rotational kinetic energy, you know it is :

E = I.w^2 / 2 >>> initial energy

At the end, the seat is gonna have a potential energy, but no kinetic energym because it stops turning, so the final energy will be :

E' = m*g*H

H is the altitude we have been asked, so then :

mgH = Iw^2 / 2

H = sqrt(2mg / I) meters

1) Ok, it's a rim, so the moment of inertia is : I = M.R^2

the moment of inertia of each spoke is : m.R^2, you can consider each spoke as a point mass.

Then, the moment of inertia will be :

I = 12*(0.6)^2 + 4*(1*(0.6)^2)

Note : I consider each spoke at the end of the diameters, so there will be 4 spokes, each one at the end and beginning of both diameters.

I = 4.32 + 1.44 = 5.76 (m^2.kg)

Hope that helped

Answer 2

1) you need to find the inertias for each spoke and wheel and then add them.
Wheel = I=mr^2 = (12x0.6^2) = 4.32
Spokes= I=4(1/2mL^2) = 4(1/2x1.0x1.2^2) = 2.88
I = 4.32+2.88 = 7.2

2) You have an inital speed, initial inertia, and final height. So...
h = (1/2V^2 + 1/4V^2) / g
you need to find the inital speed to find the height.