Rotational analysis of chain and sprockets system

[spaghetti3451]

Homework Statement

Imagine a system consisting of a chain that runs over two sprockets. The chain rotates around the sprockets with a constant linear velocity (i.e. the chain is taut and rigid). The front sprocket has a radius r_front and an angular speed ω_front and the rear sprocket has a radius r_rear and an angular speed ω_rear.

(a) All points on the chain have the same linear speed. Is the magnitude of the linear acceleration also the same for all points on the chain? How are the angular accelerations of the two sprockets related? Explain.

(b) How are the radial accelerations of points at the teeth of the two sprockets related? Explain the reasoning behind your answer.

Homework Equations

The Attempt at a Solution

(a) Between the two sprockets, the chain moves in a straight line with a constant linear speed. In other words, these points move at a constant linear velocity. Therefore, these points do not have a linear acceleration.

The points which touch each sprocket when they move are rotating along the arc of a circle (that is defined by the sprocket). Therefore, these points have a radial acceleration towards the centre of the sprocket. The points, when they rotate along the arc, move with a constant linear speed. Therefore, these points do not have a tangential acceleration. Anyway, those points have a non-zero linear acceleration.

[The analysis of the points that touch the sprockets assumes that these points and the outer edges of the sprockets move at the same linear speed, i.e. the chain does not slip or stretch.]

Therefore, at any instant of time, all points on the chain do not have the same magnitude of the linear acceleration.


All points that touch the sprockets have the same linear speed. v = rω. Therefore, all those points have the same angular speed. The chain does not slip or stretch. Therefore, each point on the chain and the sprocket that it touches move at the same angular speed. Therefore, the sprockets have the same constant angular speed. Therefore, the sprockets do not have an angular accleration.

(b) a_rad = rω². Therefore, \frac{a_{rad,front}}{a_{rad,rear}} = \frac{r_{front}}{r_{rear}}.


Please point out any mistakes in my analysis. As always, thanks in advance for any comments.

 

[haruspex]

at any instant of time, all points on the chain do not have the same magnitude of the linear acceleration.

The question is unclear.
"Linear acceleration" is both a state of motion and the value of the acceleration for an object in that state. A part of the chain going around a sprocket is not in that state, so what does the question mean?
(It is sometimes used to mean the acceleration component in a specified direction, but no direction is specified here.)
If they mean tangential acceleration then it is zero everywhere.

The solution posted for b) wrongly assumes $\omega$ is the same for both sprockets.