The rate at which a nebular cloud rotates increases as the cloud collapses to form systems...

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.

Expert Solution

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

genius_generous answered 1 year ago

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