Conceptual questions about rotational dynamics.

[-Dragoon-]

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. m₁ < m₂

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

$I = mr^{2}$
$r = \frac{R}{2}$
$K_{rot} = \frac{1}{2}I\omega^{2}$

The Attempt at a Solution

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

If Rotational kinetic energy is doubled and angular velocity is to be constant, then:
$2K_{rot} = 2(\frac{1}{2}[(m_{1}+m_{2})(\frac{R^{2}}{4})]\omega^{2})$
$2K_{rot} = \frac{1}{2}[(m_{1}+m_{2})(\frac{2R^{2}}{4})]\omega^{2}$
$2K_{rot} = \frac{1}{2}[(m_{1}+m_{2})(\frac{R^{2}}{2})]\omega^{2})$

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.

 

[Doc Al]

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?

Try this:
KE₁ = f(R₁)
KE₂ = f(R₂)
Set KE₂ = 2KE₁ and then solve for R₂ in terms of R₁.

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.

If an object isn't moving does it still have mass? Same idea here. Of course it has a moment of inertia (which depends on the axis you choose to rotate about).

 

[-Dragoon-]

Try this:
KE₁ = f(R₁)
KE₂ = f(R₂)
Set KE₂ = 2KE₁ and then solve for R₂ in terms of R₁.

The major mistake I made was to assume that R denoted the length, not the radius. In that case:
$\frac{1}{2}(m_{1} + m_{2})(R_{2})^{2}\omega^{2} =2[\frac{1}{2}(m_{1} + m_{2})(R_{1})^{2}\omega^{2}]$
Masses and angular velocity cancels out:
$\frac{1}{2}(R_{2})^{2} =2[\frac{1}{2}(R_{1})^{2}]$
$R_{2} = \sqrt{2R_{1}}$

Thanks for all the help. I just wanted to know, is this the preferred method of going about solving these types of questions? I would usually plug in numbers, but that method mislead me into thinking I had the right answer.

If an object isn't moving does it still have mass? Same idea here. Of course it has a moment of inertia (which depends on the axis you choose to rotate about).

Ah, I see. Thanks.

 

[Doc Al]

Thanks for all the help. I just wanted to know, is this the preferred method of going about solving these types of questions? I would usually plug in numbers, but that method mislead me into thinking I had the right answer.

It's how I would approach a problem like this. The main thing is to realize that the KE is proportional to R². So to double the KE you need to multiply R by √2.

But doing it systematically reduces the chance for error. (And may lead to a deeper understanding.)

 

[asteriatic]

Your solution is correct. The book's answer may be considering a different scenario, such as keeping the total mass (m1 + m2) constant while changing the distribution of mass between the two masses. This would result in a different factor for R.

To answer your second question, yes, an object that is not rotating does not have a moment of inertia. Moment of inertia is a measure of how an object's mass is distributed around an axis of rotation, so if an object is not rotating, there is no need for this measurement.