# Homework Help: Inclined Plane Rotational Motion

1. Mar 4, 2014

### phoenixXL

1. The problem statement, all variables and given/known data
A cyclinder of mass m and radius r is rotated about its axis by an angular velocity ω and then lowered gently on an inclined plane as shown in figure. Then:

(a) It will start going upward.
(b) It will first go up and then downward.
(c) It will go downward just after it is lowered.
(d) It can never go upward.

2. The attempt at a solution

Taking torque that is due to sine component of weight about the point of contact,
$$\tau\ =\ I\alpha\\ \implies \tau\ =\ \frac{3}{2}mr^2(\alpha)\\ \implies r.mgsin(30°)\ =\ \frac{3}{2}mr^2(\alpha)\\ \implies \alpha\ =\ \frac{g}{3r}~~...(i)$$
The sense of rotation of angular acceleration and of angular velocity is opposite, therefore soon the cyclinder will stop rotating.

More Equations, from newtons laws of motion
$$mgsin(30°)\ -\ f\ =\ ma~~...(ii),~~~~f\ =\ frictional force,~~a\ =\ net acceleration.$$
Taking torque about center of mass
$$\tau\ =\ I\alpha\\ \implies f\ =\ \frac{1}{2}mr^2.\frac{g}{3r}~~~~from (i)\\ \implies f\ =\ \frac{mg}{6}~~...(iii)$$

Substituting (iii) in (ii), We get
$$a\ =\ \frac{g}{3}$$

Therefore, we can say that the body when lowered will begin to move down with its rotational motion getting reduced. Therefore, the answer should be (c)

But,
Solution Given in Book

(d) It can never go upward.

Reason: Since net force along the incline is zero, so cyclinder will remain in position till it stops rotating. After that it will start moving downwards.

2. Mar 4, 2014

### voko

You do not need torques to solve this problem. All you need to determine is the motion of the centre of mass. The motion of the centre of mass is determined by taking into account all the external forces that act on the system.

What are they?

3. Mar 4, 2014

### phoenixXL

The forces that act on the cyclinder are
- Normal due to Incline
- Weight of the cyclinder
- Friction by virtue of its rotation

But how do we combine them and get it where the body will move thereafter?

It can be seen that
$$\mu\ >\ \frac{c}{1+c}tan30°$$
Therefore it is a case of pure rolling, hence the velocity of the cyclinder would then be
$$v\ =\ \frac{\omega}{r}$$
Therefore it should go up, am I correct ?

Last edited: Mar 4, 2014
4. Mar 4, 2014

### voko

Yes, those are correct forces. I recommend that you write Newton's second law in the components parallel and perpendicular to the incline.

5. Mar 4, 2014

### phoenixXL

Parallel to incline
$$mgsin(30°)\ -\ f_s\ =\ ma$$
Perpendicular to incline
$$N\ =\ mgcos30°$$

Then with friction
$$f_s\ <=\ μN\\ \implies\ f_s\ <=\ μmgcos30°$$

6. Mar 4, 2014

### voko

And what acceleration does that give you?

Is it possible that $$\mu = \sqrt {\frac 1 3}$$?

7. Mar 4, 2014

### phoenixXL

Yes,
$$\mu\ =\ \frac{1}{\sqrt{3}}$$

It gives zero acceleration.
But what does this infer ?

Last edited: Mar 4, 2014
8. Mar 4, 2014

### voko

What is the initial velocity of the cylinder's centre of mass when it is lowered onto the incline?

9. Mar 4, 2014

### phoenixXL

There is no initial velocity of the centre of mass initially, it only has initial angular velocity.

10. Mar 4, 2014

### voko

So, if the centre of mass is initially stationary, and then the acceleration is zero, does it move anywhere?

11. Mar 4, 2014

### phoenixXL

But only one thing that pinchs, we haven't used the condition of the angular velocity ω.
The conditions we took would be the same as if the one when someone would simply keep a non-rotating cyclinder over the incline,

Additionally, the book says that it would down once it stops rotating. So, rotation should make some difference, isn't it ?

12. Mar 4, 2014

### voko

You did, but implicitly. Because of the rotation, the force of kinetic friction is up the incline.

No, that is not true. A non-rotating cylinder would begin to roll down immediately.

13. Mar 4, 2014

### phoenixXL

Right. Thank you so much

14. Mar 5, 2014

### Vibhor

@Voko : I think the cylinder will roll up the incline only if μ >1/√3 i.e say if μ= 0.7 . In that case it will roll up till the angular velocity reduces to zero .It will halt for a a moment and , after that it will roll down .

15. Mar 5, 2014

### voko

It will roll up when the coefficient of kinetic friction satisfies your condition. But it is not correct to say that as soon as its angular velocity is zero it will stop.

16. Mar 5, 2014

### Vibhor

The friction will gradually decrease as the cylinder rolls up the incline because as the angular speed decreases,there would be less slipping .Eventually friction would become zero when the cylinder stops rotating.After that again friction comes in the scene .This time cylinder rotates clockwise down the incline .

Is it correct ?

Last edited: Mar 5, 2014
17. Mar 5, 2014

### phoenixXL

I missed one point
As it is pure rolling, the contact point must be stationary, hence it must be static friction.

Moreover, if you we just keep a non-rotating cyclinder then also the friction of friction is up the incline as the body tends to move down due to its weight. Am I correct ?

Last edited: Mar 5, 2014
18. Mar 5, 2014

### voko

No, this is not correct. Friction does not behave like that.

19. Mar 5, 2014

### Vibhor

Fine .Thank you.

Then why would the cylinder move up the incline even when it stops rotating .Is it because of the momentum gained ?

When it stops rotating isn't the net force mgsinθ down the incline ? Please explain the motion of the cylinder under the condition μ >1/√3 .

20. Mar 5, 2014

### voko

How is it pure rolling? When you put the cylinder on the incline, the point of contact clearly has some non-zero velocity.

Generally, correct. Details are complicated, however.