Acceleration of Masses and Tension in the String

## Question:

A light string connects two masses of 3 kg situated on a frictionless horizontal surface. A force F = 12 N is exerted on one of the masses to the right. Determine the acceleration of the masses and the tension in the string.
Please help;

## Answer:

Here is the deal with connected masses. Taken separately and together they must obey Newton's 2nd law. The trick is to be careful with the forces, accelerations and the masses so as to keep it all straight. A free body diagram for each mass is a useful tool.

First let's nail down the forces. The right hand mass has a 12 N force pulling to the right and the string tension, T, pulling to the left. The other mass has only the string tension, T, pulling it to the right.

Now the accelerations. If the string does not stretch the distance between right and left masses remains fixed. That means that the acceleration on both is the same. Call it a.

Let's apply Newton's 2nd law to the right hand block. That gives us a=(12-T)/3. Then apply it to the left hand block, yielding T=3*a. Substitute for T in the first equation to get a=(12-3*a)/3 or a=4-a or 2*a=4 or a=2 m/s/s. Plug a back into the second equation to get T=6 N.

Now apply common sense to check the result. The 12 N force is applied to a total mass of 6 kg if the string does not stretch so a should be 2n/s/s, as we found. If a is 2 m/s/s then the force on the left block, which is the string tension, must be 3*2 N.