Conservation of String Exercise

[mancity]

Homework Statement

Explain why the acceleration of the left mass equals negative the average of the accelerations of the right two masses.

Relevant Equations

Conservation of string

I'm not quite sure how to apply conservation of string to this problem, so guidance would be appreciated. Normally as long as there isn't a "sub-pulley" I can do the problem fairly easily but this one tricks me up. Thanks

 

[jbriggs444]

Normally as long as there isn't a "sub-pulley" I can do the problem fairly easily but this one tricks me up. Thanks

The sub-pulley is symmetric, correct? That is, there is no difference between its right and left hand sides? They are mirror images of each other?

 

[haruspex]

The sub-pulley is symmetric, correct? That is, there is no difference between its right and left hand sides? They are mirror images of each other?

I take the masses as unknown.

I'm not quite sure how to apply conservation of string to this problem,

Write equations relating string (section) lengths to heights of masses, throwing in constants as necessary. Differentiate twice.

 

[BvU]

Homework Statement: Explain why the acceleration of the left mass equals negative the average of the accelerations of the right two masses.
Relevant Equations: Conservation of string

I'm not quite sure how to apply conservation of string to this problem, so guidance would be appreciated. Normally as long as there isn't a "sub-pulley" I can do the problem fairly easily but this one tricks me up. Thanks

'Conservation of string' is an unusual way to state that the length of the string is constant. So when you set up a set of equations for your exercise, one of them is a relationship between
$y_2$, the vertical position of the middle mass,
$y_3$, idem rightmost mass
$y_5$, the vertical position of the center of the pulley on the right:
$y_5-y_2+y_5-y_3=C$

Differentiation wrt time gives an equation for the vertical velocities; a second differentiation yields another for the accelerations.

$\$

 

[Chestermiller]

Homework Statement: Explain why the acceleration of the left mass equals negative the average of the accelerations of the right two masses.
Relevant Equations: Conservation of string

I'm not quite sure how to apply conservation of string to this problem, so guidance would be appreciated. Normally as long as there isn't a "sub-pulley" I can do the problem fairly easily but this one tricks me up. Thanks

Relative to the bottom pulley, the average acceleration of the bottom two masses is zero. The acceleration of the upper mass is minus the acceleration of the bottom pulley.

 

[haruspex]

Relative to the bottom pulley, the average acceleration of the bottom two masses is zero. The acceleration of the upper mass is minus the acceleration of the bottom pulley.

My impression is that the OP has been instructed to use conservation of string length to obtain the result. That suggests to me applying your framework to positions and then differentiating.