Question

A rigid spherical body is spinning around an axis without any external torque. Due to change in temperature, the volume increases by 1%. Its angular speed
A. will increase approximately by 1 %.
B. will decrease approximately by 1 %.
C. will decrease approximately by 0.67 %.
D. will decrease approximately by 0.33 %.

Answer 1

The above problem can be resolved using the angular momentum's concepts and fundamentals and its mathematical relationship with the angular speed. The volume for the given spherical cavity is given, for which the change in the radius is also calculated. This value for the change in the radius is used to calculate the change in angular speed for a constant change in angular momentum.

Complete step by step solution:
Let the volume of the spherical body is given by,
$V = \dfrac{4}{3}\pi {R^3}$
Here, R is the radius of the body.
Then the radius of body is,

\dfrac{{\Delta V}}{V} = 3\dfrac{{\Delta R}}{R}\\\ \dfrac{{\Delta R}}{R} = 0.33\% .........................................\left( 1 \right) \end{array}$$ There is no external torque applied on the body, therefore the momentum is conserved. The angular momentum is, $$L = \dfrac{2}{5}M{R^2}\omega $$ Here, M is the mass and $$\omega $$ is the angular speed of the body. Then the change in the angular speed is, $$\dfrac{{\Delta L}}{L} = 2\dfrac{{\Delta R}}{R} + \dfrac{{\Delta \omega }}{\omega }$$ As change in angular momentum is constant, then $$\dfrac{{\Delta L}}{L}$$ is equal to zero. Then solve by substituting the values as, $$\begin{array}{l} \dfrac{{\Delta L}}{L} = 2\dfrac{{\Delta R}}{R} + \dfrac{{\Delta \omega }}{\omega }\\\ 0 = 2\dfrac{{\Delta R}}{R} + \dfrac{{\Delta \omega }}{\omega }\\\ \dfrac{{\Delta \omega }}{\omega } = - 2\left( {0.33\;\% } \right)\\\ \dfrac{{\Delta \omega }}{\omega } \simeq - 0.67\% \end{array}$$ Negative sign shows the decrease in angular speed. **Therefore, angular speed will decrease approximately by 0.67 % and option (C) is correct.** **Note:** To resolve the above problem, one must understand the concepts and application of the angular momentum in calculating the percentage change in any object's angular speed. The angular momentum is observed for any analysis whenever there is some constant torque acting on the body during the rotational motion. Moreover, this will provide the relationship for the constant angular velocity of the body and the change in the volume of the object.