- #1

- 309

- 7

## Homework Statement

There is a dumbbell with two masses attached at each end, rotating about a point. The distance between the two masses is R, and the two masses do not have the same mass. m

_{1}< m

_{2}

If the angular velocity ω is held constant, by what factor must R change to double the rotational kinetic energy of the dumbbell?

## Homework Equations

[itex]I = mr^{2}[/itex]

[itex]r = \frac{R}{2}[/itex]

[itex]K_{rot} = \frac{1}{2}I\omega^{2}[/itex]

## The Attempt at a Solution

Since [itex]r = \frac{R}{2}[/itex] and [itex]I = (m_{1}+m_{2})r^{2}[/itex], then [itex]K_{rot} = \frac{1}{2}[(m_{1}+m_{2})((\frac{R}{2})^{2})]\omega^{2} =>\frac{1}{2}[(m_{1}+m_{2})(\frac{R^{2}}{4})]\omega^{2}[/itex]

If Rotational kinetic energy is doubled and angular velocity is to be constant, then:

[itex]2K_{rot} = 2(\frac{1}{2}[(m_{1}+m_{2})(\frac{R^{2}}{4})]\omega^{2})[/itex]

[itex]2K_{rot} = \frac{1}{2}[(m_{1}+m_{2})(\frac{2R^{2}}{4})]\omega^{2}[/itex]

[itex]2K_{rot} = \frac{1}{2}[(m_{1}+m_{2})(\frac{R^{2}}{2})]\omega^{2})[/itex]

Thus, by my conclusion, R increases by a factor of two if the rotational energy is doubled but angular velocity is to remain constant. My book's answer is that R increases by a factor of square root of 2. What did I do wrong?

Also, one more conceptual question: If an object is not rotating, does it have a moment of inertia? The answer is no, correct? Since a moment of inertia depends on the rotation of axis and changes when an object is rotating about a different point, thus if an object is not rotating, then it does not have a moment of inertia since moment of inertia is the rotational equivalent of mass.