Solving for Tension & Acceleration in M1 & M2

[Harmony]

Assume there is negligible friction between the blocks and the table. Compute the tension in the cord and the acceleration of m2 if m1 = 300g, m2 = 200g, and F=0.4N

I figured out that T=m2*a2, F-2T=m1a1.
According to the answer, 2a1=a2. But I don't undestand why.

 

[Doc Al]

acceleration constraint

Your force equations are correct, but incomplete. You need to incorporate the constraint imposed by the fact that the masses are connected via a pulley and ropes--that's what will give you the additional equation a2 = 2 a1. To understand where that relationship comes from, answer this question: If m1 moves one meter to the right, how far does m2 move? (If you have trouble visualizing the motion, use a piece of string or a paper strip.)

 

[kyle8921]

I would like to commend you for your efforts in solving this problem and for recognizing the importance of understanding the underlying principles behind your calculations. Let me explain why 2a1=a2.

First, we need to understand that the tension in the cord (T) is the force that is transmitted through the cord from one block to the other. This means that the tension in the cord is the same for both blocks, as the cord is a continuous object and cannot have different tensions at different points along its length.

Now, let's consider the forces acting on each block. For m1, we have the force of gravity (mg) pulling it down and the tension in the cord (T) pulling it up. Similarly, for m2, we have mg pulling it down and T pulling it up. Since we are assuming negligible friction, the only horizontal force acting on the system is the applied force (F).

Using Newton's second law of motion (F=ma), we can set up equations for each block:

For m1: F-2T=m1a1
For m2: 2T-mg=m2a2

Since we know that T is the same for both blocks, we can set these equations equal to each other and solve for a1 and a2:

F-2T=m1a1 --> F=2T+m1a1
2T-mg=m2a2

Substituting 2T+m1a1 for F in the second equation, we get:

2T+m1a1=2T-mg=m2a2

Simplifying, we get:

m1a1=m2a2

Since we know the masses of each block (m1=300g and m2=200g), we can rearrange this equation to solve for a1 in terms of a2:

a1=(m2/m1)a2

Substituting the values given in the problem (m1=300g and m2=200g), we get:

a1=(200g/300g)a2

Simplifying, we get:

a1=(2/3)a2

This means that the acceleration of m1 is two-thirds of the acceleration of m2. In other words, for every unit of acceleration that m2 experiences, m1 will experience two-thirds of that acceleration in the opposite direction.

I hope this explanation helps you understand why