What Is the Acceleration of Mass m1 on a Frictionless Table Connected to m2?

[vertabatt]

Homework Statement

In the figure, find an expression for the acceleration of m1 (assume that the table is frictionless).

Homework Equations

F=ma

The Attempt at a Solution

My biggest problem here is that I don't know what variables I am supposed to answer in (my homework is done on a computer system that is picky about everything).

I know that because of acceleration constraints m1 and the string must move at the same acceleration. But must m2 move with the same acceleration?

The net force on m1 is the force of tension to the right. The net force on m2 is the sum of two tension forces upward and the weight downward.

The acceleration of m2 must be:

F = ma

weight - 2 (tension) = ma

(weight - 2 (tension)) / m = a

 

[vertabatt]

Is there anything I can add to get help on this? I am racking my brain here but I can't seem to figure out m1's acceleration relative to m2.

Intuition tells me that they can't be the same, but expressing this mathematically isn't coming to me so easily.

 

[ogb p]

The acceleration of m1 must be the same as m2, so:

a = (weight - 2 (tension)) / m

This is a good start, but there are a few things to consider here. First, it is important to clarify what the figure is showing. Is m1 the mass hanging from the string, and m2 the mass on the table? Or is m1 the mass on the table and m2 the hanging mass? This will affect how you approach the problem.

Assuming that m1 is the hanging mass and m2 is on the table, your solution is on the right track. The acceleration of m2 must be equal to the net force acting on it divided by its mass, as you have correctly written. However, you also need to take into account that the net force on m2 is equal to the tension in the string, which is pulling upwards, minus the weight of the mass, which is pulling downwards. So the equation should be:

a = (T - mg) / m

Where T is the tension in the string and mg is the weight of m2.

As for m1, it is constrained to move at the same acceleration as m2, so its acceleration will also be a. Since m1 is only acted on by the tension in the string, its equation would be:

a = T / m1

Where T is again the tension in the string, and m1 is the mass of m1.

Overall, the key concept here is that the acceleration of both masses is equal because they are connected by a string, and the net force on each mass is equal to the tension in the string. So the acceleration constraints can be expressed by setting the two equations for a equal to each other:

(T - mg) / m = T / m1

Solving for T, we get:

T = mg (1 + m/m1)

This is the expression for the tension in the string, which can then be used to calculate the acceleration of both masses using the equations for a that we found earlier.