Question

A merry-go-round rotates at the rate of 0.14 rev/s with an 84 kg man standing at a point 2.4 m from the axis of rotation. What is the new angular speed when the man walks to a point 0 m from the center

Answer 1

The given question is incomplete. The complete question is as follows.

A merry-go-round rotates at the rate of 0.14 rev/s with an 84 kg man standing at a point 2.4 m from the axis of rotation. What is the new angular speed when the man walks to a point 0.6 m from the center? Assume that the merry-go-round is a solid 25-kg cylinder of radius 2.0 m.

Explanation:

The given data is as follows.

    Initial angular velocity ($\omega_{1}$) = 0.14 rev/s

        $\omega_{1} = (0.14 \times 2 \pi) rad/s$

                        = 0.87 rad/s

Mass of man (M) = 84 kg,     Mass of cylinder (m) = 25 kg

Radius of the cylinder (r) = 2.4 m

Let $r_{1}$ be the initial distance of man from the axis of rotation = 2.0 m

Final distance of man from the axis of rotation ($r_{2}$) = 0.6 m

Formula to calculate the moment of inertia of cylinder is as follows.

        I = $\frac{Mr^{2}}{2}$  

And, moment of inertia of man = $mr^{2}$

Therefore, initial moment of inertia of the system is as follows.

      $I_{1} = \frac{Mr^{2}}{2} + mr^{2}_{1}$

                = $\frac{84 \times (2.4 m)^{2}}{2} + 25 \times (2)^{2}$

                = 241.92 + 100

                = 341.92 $kg m^{2}$

Now, final moment of inertia of the system will be calculated as follows.

           $I_{2} = \frac{Mr^{2}}{2} + mr^{2}_{2}$

                     = $\frac{84 \times (2.4)^{2}}{2} + 25 \times (0.6)^{2}$

                     = 241.92 + 9

                     = 250.92

Now, according to the law of conservation of angular momentum is as follows.

      Initial angular momentum = Final angular momentum

             $I_{1} \omega_{1} = I_{2} \omega_{2}$

or,   $\omega_{2} = \frac{I_{1}\omega_{1}}{I_{2}}$

                     = $\frac{341.92 \times 0.87}{250.92}$

                     = 1.18 rad/s

Thus, we can conclude that the new angular speed is 1.18 rad/s.