What is the acceleration of two masses on pulleys with constant tension?

[Karol]

Homework Statement

All N+2 masses are m. what is the acceleration of the 2 masses at the ends.

Homework Equations

The rope on a weightless pulley has constant tension.

The Attempt at a Solution

The situation is the same as:

$$\left\{\begin{array}{l} Nmg-2T=Nma \\ T-mg=ma \end{array}\right.$$
$$\rightarrow~a=\frac{N-2}{N+2}g$$
$$N=1\rightarrow~a=-\frac{1}{3}g$$
And it's wrong

 

[gneill]

I wonder about your assumed equivalent system. Try the following thought experiment with the original system: Replace the two end masses with fixed supports so that the rope is fixed at those points and the system will be static. What would be the tension in the rope? Is it Nmg? Nmg/2? Something else?

 

[Chestermiller]

If the accelerations of the end masses is a, kinematicly, what is the acceleration of each of the N middle masses?

 

[Karol]

I will denote as x and a the vertical displacement and acceleration of the 2 masses at the ends, and with y the displacement of each of the N middle masses:
$$2x=2Ny~~\rightarrow~~y=\frac{x}{N}~~\rightarrow~~\ddot y=\frac{a}{N}$$
$$\left\{\begin{array}{l} Nmg-2T=Nm\frac{a}{N} \\ T-mg=ma \end{array}\right.~~\rightarrow~~a=\frac{N-2}{3}g$$

 

[Chestermiller]

I will denote as x and a the vertical displacement and acceleration of the 2 masses at the ends, and with y the displacement of each of the N middle masses:
$$2x=2Ny~~\rightarrow~~y=\frac{x}{N}~~\rightarrow~~\ddot y=\frac{a}{N}$$
$$\left\{\begin{array}{l} Nmg-2T=Nm\frac{a}{N} \\ T-mg=ma \end{array}\right.~~\rightarrow~~a=\frac{N-2}{3}g$$

That's not what I get. $$mg-2T=m\frac{a}{N}$$

 

[gneill]

Hi Karol. Instinctively it is tempting to think that the N masses must all contribute to the net force that the end masses experience. But if you look closely you'll see that by symmetry most of the N masses are balancing each other's contributions over the pulleys between them, and these upper pulleys are supported by the "ceiling". So in effect the "ceiling" is doing most of the "lifting".

I suggest that you focus on one general representative of the N masses to write your equation. I concur with @Chestermiller 's analysis.

 

[Karol]

The positive direction is upwards, a, the edge's acceleration is positive and a_y, the acceleration of the other masses is negative:
$$\left\{\begin{array}{l} 2T-mg=-ma_y=-m\frac{a}{N} \\ T-mg=ma \end{array}\right.~~\rightarrow~~a=-\frac{N}{2N+1}g$$
Not good, a<0.

 

[Chestermiller]

The positive direction is upwards, a, the edge's acceleration is positive and a_y, the acceleration of the other masses is negative:
$$\left\{\begin{array}{l} 2T-mg=-ma_y=-m\frac{a}{N} \\ T-mg=ma \end{array}\right.~~\rightarrow~~a=-\frac{N}{2N+1}g$$
Not good, a<0.

Surprisingly, this is actually correct. The two end masses move down, and all the other masses move up. This is a "lever" effect.

 

[Karol]

Thank you Haruspex and gneill. so only the 2 masses adjacent to the 2 outer masses contribute. the outer masses must hold half of the adjacent masses, $\frac{1}{2}mg$, plus the acceleration of all the inner N mases

 

[gneill]

Thank you Haruspex and gneill

You're welcome. But I think you meant to thank Chestermiller rather than Haruspex. Although Haruspex is often very helpful, too!

 

[Karol]

Sorry, true... but thank you all...