You buy a bottle of water in the store and place it on the conveyor belt with the longitudinal axis perpendicular to the direction of movement of the belt. Initially, both the belt and the bottle are at rest. We can approach the bottle as one cylinder with radius $a$, mass $M$ and moment of inertia $I = M k^2$ ($k$ in units of length), in which the mass is not distributed uniformly. The speed of the belt at time $t$ is $V (t)$.

a) Find an expression for the speed $v(t)$ of the centre of mass of the bottle.

b) Explain why a bottle tends to start spinning on one moving band.

What I have tried:

a) Having both longitudinal axis and belt's velocity vector perpendicular to each other **on the same plane** means that there will be rolling motion if we assume that static friction will be overcome. Let's disregard slippery effects as well.

Here I used an approach based on the fact that **the bottle-belt-Earth system is nonisolated in terms of energy because the belt exerts an external force on the bottle, which means that it does work on the bottle.** Therefore:

$$W = \Delta K_{tr} + \Delta K_{rot}$$

$$(F-f_f)d =v_{CM}^2 (\frac{I_{CM}}{a^2} + M)$$

$$v_{CM} = \sqrt{ \frac{2(F-f_f)d}{\frac{I_{CM}}{a^2} + M}} = \sqrt{ \frac{(F-f_f)d}{ M}}$$

The result I got does not seem to be incorrect; I checked dimensions and got $\frac{L}{T}$ .**My question here is if I am right including the force F. I consider this force as a fictitious one, which is triggered by the non inertial frame in which the bottle is located: the conveyor belt.** Note I am analysing the scenario from an inertial reference frame: The Earth (we regard it as inertial for known reasons).

b) If the static friction force was equal to the fictitious force F the cylinder **would not spin**. It would just move transitionally along the belt. Actually, if we were located on the belt, we would not see the cylinder move at all.

**In order to spin, the magnitude of F must exceed the magnitude of the maximum force of static friction. This friction force is recalled as force of kinetic friction.**

**EDIT**

From second (translation) Newton's law:

$$a_o= \frac{f}{M}$$

From second (rotation) Newton's law:

$$\tau = I \alpha = fa$$

$$k^2 \alpha= a_o a$$

$$\alpha = \frac{a_o a}{k^2}$$

Where $a_o$ is the acceleration of the cylinder measured from the ground.

**Assuming that the acceleration of the belt is constant** ($a_b$):

$$ \frac{V}{t} = a_b$$

The acceleration of the cylinder with respect to the ground accounts for the acceleration of the belt and the angular acceleration of the cylinder (which has a negative sign because I considered the cylinder spinning clockwise i.e. the belt accelerating to the left and regarding that direction as the positive one). Therefore:

$$a_o = a_b - a\alpha$$

$$a_o = \frac{V}{t} -a\frac{a_o a}{k^2}$$

$$a_o = \frac{Vk^2}{(k^2 + a^2)t}$$

We know by kinematics that:

$$v_{cm} = \frac{Vk^2}{k^2 + a^2}$$

This makes sense to me. However I am confused because **I was suggested that I should not assume that the belt moves with constant acceleration** (please see kuruman's #2 comment https://www.physicsforums.com/threads/cylinder-lying-on-conveyor-belt.963471/ ); Then how could I solve this problem?

In an inertial frame the only horizontal force on the cylinder is the force of friction.

(b) Not true.

The fictitious force and the friction force do not act in the same straight line. The fictitious force acts at the COM of the cylinder, the friction force acts at the point of contact with the belt. If these forces are equal and in the opposite direction they form a couple which does rotate the cylinder. There would only be no rotation if they acted in the same straight line.

If there is no slipping at the contact point then the friction force is **static** not kinetic.

edited Dec 23, 2018 by sammy gerbil