2 weights on a pully, show formula

[man_in_motion]

Homework Statement

The apparatus (see link to image) is used to measure the free fall acceleration g by measuring the acceleration of the two blocks connected by a string over a pulley. Asume a massless, frictionless pulley and a massless string.
http://img3.imageshack.us/img3/3459/phys.th.png
a) draw a free body diagram of each block
http://img16.imageshack.us/img16/3523/diagrams.th.png
b)use the free body diagrams and Newton's laws to show that the magnitude of acceleration of either block and tension in string are a=(m₁-₂)/(m₁+₂)
c)do these expressions give reasonable results if m₁=m₂ in the limit that m₁ >> m₂ and in the limit that m₁ << m₂ ? Explain

Homework Equations

F_net=ma
F_g=mg

The Attempt at a Solution

F=ma
T_1=-T_2
T_2=(m_2)g
T_1=(m_1)g
=>m_1(g)=-m_2(g)

 

[rock.freak667]

Assume block one moves downwards, so the resultant force on m₁ is

m₁a=mg-T

So what is the resultant force on m₂?

Now you have two equations in T and a, you can now eliminate T.

 

[tomcue]

=> (m_1+m_2)g=0 => (m_1+m_2)a=0 => a=0

The formula for the acceleration of either block in this system is a=(m1-m2)/(m1+m2). This can be derived by applying Newton's second law, Fnet=ma, to each block and solving for the acceleration. The free body diagrams show that the net force on each block is equal to the difference in the weights of the two blocks, divided by the total mass of the system. The tension in the string can also be calculated using this formula.

In the limit that m1 >> m2, the expression becomes a=(m1-m2)/m1, which simplifies to a=1- m2/m1. This means that as m1 becomes much larger than m2, the acceleration of the system approaches 1, which is the acceleration due to gravity. This is a reasonable result, as the larger mass will dominate and the system will behave similarly to a single block falling freely.

Similarly, in the limit that m1 << m2, the expression becomes a=(m1-m2)/m2, which simplifies to a=-1+ m1/m2. This means that as m2 becomes much larger than m1, the acceleration of the system approaches -1, which is the acceleration due to gravity in the opposite direction. This is also a reasonable result, as the smaller mass will be pulled down by the larger mass and will accelerate in the opposite direction.

In conclusion, the derived formula for the acceleration and tension in this system gives reasonable results in both scenarios, demonstrating the accuracy of Newton's laws and the usefulness of the free body diagrams in analyzing the forces at play.