What is the Relationship Between Acceleration of a Block and a Pulley?

[imatreyu]

Homework Statement

A horizontal force F is applied to a frictionless pulley of mass m2. The horizontal surface is smooth. Show that the acceleration of the block of mass m1 is twice the acceleration of the pulley.

LOOKS LIKE THIS: http://cnx.org/content/m14060/latest/npq1.gif
But WITHOUT block B and its string.

Homework Equations

F=ma

The Attempt at a Solution

I drew separate force diagrams for m1 (the block) and m2 (the pulley. In the x direction, the block is only being acted on by T1 going in the pos. x direction. In the x direction, the pulley is being acted on by 2T1 and F. 2T1 is going in the neg. x direction. F, the opposite.

I have to show that a1= 2a2

So:

The pulley is in equilibrium:
F-2T1 = 0
m2a2 - 2(m1a1)=0

. . .and I don't know where to go from here. . . .I can't eliminate mass. . .


Thank you in advance!

 

[tsw99]

It is easy to see that the pulley is moving. Let s_1 be the distance between pulley and A, s_2 be the distance between pulley and B. During the pull, s_2 stays constant. For s_1, it is decreasing right? But that section goes to the upper side of the pulley, so we can set 2s_1 equals also a constant. the reason of 2s_1 comes from the initial situation, we ignore the upper portion that is left of the originial position of A.

Hence 2s_1+s_2=constant, differentiate twice yields your desired result.

 

[imatreyu]

Oh. IDK why I thought the pulley was in equilibrium.

Thank you so much!

 

[tsw99]

Oh. IDK why I thought the pulley was in equilibrium.

Thank you so much!

I made the same mistake as you when I was having a introductory mechanics class.
The method I present here is sometimes referred to no-stretch assumption. I don't know why the method is always not mentioned in the textbooks. Is it too obvious for the authors?

 

[imatreyu]

I suppose they think so! The textbook (College Physics, 3rd ed. Serway & Faughn) says nothing about using distances and time derivatives . . . xD

Thank you!