Question

The rate at which a nebular cloud rotates increases as the cloud collapses to form systems of stars and planets. Consider a small segment of a nebular cloud with a mass ? of 1.9×1027 kg, tangential velocity ?initial equal to 6.8 km s−1 located at an orbital distance ?initial=2.5×104 km. After the cloud collapses, the same small segment is located at an orbital distance ?final=3.2×103 km. Calculate the change of the rotational velocity, Δ?, for the cloud segment, assuming perfectly circular orbits. Perform your work and report your solution using two significant figures.

Answer 1

initial tangential vel v_i = 6.8 km/s

angular velocity _i = v_i /r_i = 6.8/2.5E+4 = 2.72 E-4 rads/s

Kinetic energy of the cloud segment K = mv_i²/2

when the segment is at final radius r_f = 3.2E+3 m

K = I_f² /2 , energy is conserved , there is no loss of energy for the cloud segment

I = mr²  , we can approx. the cloud segment as a point mass

1/2 *mr_f² * _f² = 1/2 *mr_i² * _i²

final angular speed  _f = _i * r_i/r_f  

change in angular speed   =( _f - _i ) =  _i ( r_i/r_f -1)

= 2.72E-4 *( 25/3.2 -1) = 1.853 E-3 rads/s