The 0.4-lb ball and its supporting cord are revolving about the vertical axis on the fixed smooth conical surface with an angular velocity of 4 rad /sec. the ball is held in the position b = 14 in. by the tension t in the cord. if the distance b is reduced to the constant value of 9 in. by increasing the tension t in the cord, compute the new angular velocity and the work w1-2 done on the system by t

Explanation:

In the whole process , angular momentum will be conserved because no external torque is acting on the system . The radius of circular path is reduced so angular velocity will be increased

I₁ x ω₁ = I₂ x ω₂

m r₁² x ω₁ = m r₁²ω₂

r₁² x ω₁ =  r₂²ω₂

ω₂ = r₁² x ω₁  /  r₂²

= ( 14 x 14 x 4) / ( 9 x 9 )

= 9.68 rad /s

Work done

= increase in rotational kinetic energy

= 1/2 x I₂ ω₂² - 1/2 x I₁ ω₁²        ( I = m r² )

=  1/2 x .4 x .454 x( 9 x 2.54 x 10⁻² x 9.68 )² - 1/2 x .4 x .454 x( 14 x 2.54 x 10⁻² x 4 )²

=   .4446 - 0.1837

= .261 J .

=