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

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.

=