Oscillations of a ruler on a cylinder

[benf.stokes]

Homework Statement

You have ruler of length L and thickness 2d resting, in equilibrium , on a cylindrical body of radius r. Slightly unbalancing the ruler, and existing attrition between the surfaces prove that the ruler has a oscillatory motion of period:
$$T = 2\cdot \pi\cdot \sqrt{\frac{L^2}{12\cdot g\cdot (r-d)}}$$

Homework Equations

$$T=\frac{2\cdot \pi}{\omega}$$

$$\tau= F\cdot r\cdot \sin(\varphi)$$

The Attempt at a Solution

I can't wrap my mind about the idea that the ruler won't immediately begin to fall. I can't figure out why the ruler would do a simple harmonic motion in the first place.

 

[tiny-tim]

Hi benf.stokes!

… I can't wrap my mind about the idea that the ruler won't immediately begin to fall. I can't figure out why the ruler would do a simple harmonic motion in the first place.

If the ruler doesn't slip, then when it tilts to the right, its centre of mass will be on the left of the point of contact.

 

[benf.stokes]

Hi tiny-tim . Thanks for the help
I think I figured it out but I have three assumptions I'm not completely happy about and would like you to see if they're valid:

Initially the center of mass of the ruler is at position $$y_{c} = R+d$$ and after the ruler is displaced the center of mass of the ruler is at

$$y_{c'}=R\cdot \varphi\cdot \sin(\varphi)+(R+d)\cdot \cos(\varphi)$$

(I'm not really sure about the: $$R\cdot \varphi\cdot \sin(\varphi)$$, sometimes I think it should be $$(R+d)\cdot \varphi\cdot \sin(\varphi)$$ but if I do so I don't obtain the correct result any more, is this right?).

The frictional force in this case does no work since it's only role is to prevent slipping (I'm I correct?) and so the total energy of the ruler is conserved. And so:

$$m\cdot g\cdot y_{c}= m\cdot g\cdot y_{c'}+\frac{1}{2}\cdot I\cdot \omega^2$$
(This is my last uncertainty: is it ok to not take into account the velocity of the center of mass of the ruler?)

Then and using the small angle approximation :
$$\sin(\varphi)\approx \varphi \ and \ \cos(\varphi)\approx\ 1-\frac{\varphi^2}{2}$$

$$m\cdot g\cdot (R-d)\cdot \frac{\varphi^2}{2}+\frac{1}{2}\cdot I\cdot (\frac{d\varphi}{dt})^2=0$$

And derivating this expression with respect to time yields:

$$m\cdot g\cdot (R-d)\cdot \varphi\cdot \frac{d\varphi}{dt}+I\cdot \frac{d^2\varphi}{dt^2}\cdot \frac{d\varphi}{dt}=0$$

Which after some algebric manipulation yields:

$$\frac{m\cdot g\cdot (R-d)}{I}\cdot \varphi + \frac{d^2\varphi}{dt^2}=0$$

Which is the equation for SHM with the desired period.

Note:
$$I_{ruler}=\frac{1}{12}\cdot m\cdot l^2$$

 

[RoyalCat]

Hi tiny-tim.
I think I figured it out but I have three assumptions I'm not completely happy about and would like you to see if they're valid:

Initially the center of mass of the ruler is at position $$y_{c} = R+d$$ and after the ruler is displaced the center of mass of the ruler is at

$$y_{c'}=R\cdot \varphi\cdot \sin(\varphi)+(R+d)\cdot \cos(\varphi)$$
(I'm not really sure about the:
$$R\cdot \varphi\cdot \sin(\varphi)$$, sometimes I think it should be $$(R+d)\cdot \varphi\cdot \sin(\varphi)$$ but if I do so I don't obtain the correct result any more).

The frictional force in this case does no work since it's only role is to prevent slipping (I'm I correct?) and so the total energy of the ruler is conserved. And so:

$$m\cdot g\cdot y_{c}= m\cdot g\cdot y_{c'}+\frac{1}{2}\cdot I\cdot \omega^2$$
(This is my last uncertainty: is it ok to not take into account the velocity of the center of mass of the ruler?)

Then and using the small angle approximation :
$$\sin(\varphi)\approx \varphi \ and \ \cos(\varphi)\approx\ 1-\frac{\varphi^2}{2}$$

$$m\cdot g\cdot (R-d)\cdot \frac{\varphi^2}{2}+\frac{1}{2}\cdot (\frac{d\varphi}{dt})^2=0$$

The small angle approximation assumes that the ruler's center of mass moves only slightly, so neglecting the KE associated with the center of mass is fine (Though a more rigorous analysis is required to prove this point)

If you want to work around the pitfall of neglecting the KE of the CoM, look at the forces and torques acting on the center of mass of the ruler instead, there the approximations should be a bit more straightforward.

 

[tiny-tim]

Hi benf.stokes!

(have a phi: φ )

(I'm not really sure about the:
$$R\cdot \varphi\cdot \sin(\varphi)$$, sometimes I think it should be $$(R+d)\cdot \varphi\cdot \sin(\varphi)$$ but if I do so I don't obtain the correct result any more).

it's definitely Rφsinφ, not (R + d)φsinφ, because the distance along the bottom of the ruler is Rφ (because it's rolling).

The frictional force in this case does no work since it's only role is to prevent slipping (I'm I correct?) and so the total energy of the ruler is conserved.

That's absolutely right , but …

why use energy, with its awkward squared terms (and yes, you do need the c.o.m. KE also), when you can use the much simpler torque equation?

 

[benf.stokes]

Thanks. I tried to use a simple torque equation first but I just couldn't find an appropriate axis

 

[tiny-tim]

When you're dealing with rotating bodies, general rule is to use either the centre of mass or the instantaneous centre of rotation …

and in this case the latter will eliminate the unknown reaction force.

 

[benf.stokes]

But won't the instantaneous center of rotation introduce a new component in the moment of inertia?

 

[tiny-tim]

You'll need to use the parallel axis theorem to find the moment of inertia … is that what you meant?

 

[benf.stokes]

Yes. I tried that approach first but it the parallel axis theorem introduced an additional term I couldn't solve.

 

[benf.stokes]

Wait, I think I got it:

The moment of inertia about the new axis will be:

$$I_{ruler'}=\frac{1}{12}\cdot m\cdot l^2+m\cdot (R\cdot \varphi\cdot \sin(\varphi))^2$$

but since where're using the small angle approximation the second part will be very small compared to the first term?

If this is so the torque equation makes complete sense and yields the desired result

 

[tiny-tim]

(just finished watching the latest Dr Who )

… since where're using the small angle approximation the second part will be very small compared to the first term?

That's right!

(except technically you left out an md², but that will also be very small unless it's a horribly thick ruler )

 

[benf.stokes]

Yupi .Thanks you very much tiny-tim. Very helpful