Question

A steel bicycle wheel (without the rubber tire) is rotating freely with an angular speed of 14.5 rad/s. The temperature of the wheel changes from -142 to 244 °C. No net external torque acts on the wheel, and the mass of the spokes is negligible. What is the angular speed of the wheel at the higher temperature?

Answer 1

Due to change in temp the wheel expands , its radius increases and the MI increases. There is no external troque , no work done , hence it KE remains same but the angular speed decreases

Let R be the initial radius of the wheel and L length of the rim

L = 2R

after expansion let the radius be R'

initially I = mR² ; after expansion I' = mR'²

I'/I =( R'/R)²  

conserving the energy

0.5I² = 0.5I''²  

'² = I/I' ² or ^' = R/R'  

length of the rim after expansion

L' =L + L * dT ( = 7.2e-6 /m/K - thermal expansion co-efficient of steel )

= L(1 + 7.2e-6 *386 )

= 2R'

R/R' =1/ 1.0028

' = 14.5/1.0028 = 14.4595 rads/s