- #1

krackers

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## Homework Statement

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Consider a half-atwood machine on a cart as below, with mass [itex]m_2[/itex] attached to [itex]M[/itex] via a frictionless track that keeps it pinned to M but allows it to move vertically. All surfaces (except between the wheels/ground) are frictionless, and the pulley and rope are massless.

If the system is released from rest, what will the acceleration of [itex]M[/itex] be?

## Homework Equations

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Not sure if the below are right —

[tex]T=m_{1}\left( a-a_{M} \right)[/tex]

[tex]T-m_{2}\text{g}=-m_{2}a[/tex]

Where the acceleration of [itex]m_1[/itex] is [itex]a - a_M[/itex] (where [itex]a[/itex] is the magnitude of the acceleration of [itex]m_2[/itex]) since [itex] m_1 [/itex] moves right while [itex]M[/itex] moves left.

Also possibly:

[tex]T = a_M\left(M + m_2 \right)[/tex]

since the tension must be enough to accelerate the combined mass of [itex]M[/itex] and [itex]m_2[/itex]

## The Attempt at a Solution

The force responsible for accelerating [itex]M[/itex] has to be the reaction to the tension in the [itex]x[/itex] direction. I tried to calculate the tension in the string while the cart and came up with two possible sets of equations. However, the solution to the first set does not match the second set so one is clearly not properly accounting for all forces.

Once the tension is found, the normal force between [itex]M[/itex] and [itex]m_2[/itex] also has to be accounted for — how is this expressed?

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