How Is Tension Calculated in a Pulley System with Unequal Masses?

[Notyou]

Homework Statement

Two objects are hanging from a pulley. One has a mass of 12kg and another has a mass of 10kg. Initially, the two masses are at rest and friction is ignored. Once in motion, find the tension in the string and the acceleration of each of the two masses.

Homework Equations

F=ma

The Attempt at a Solution

Ok... another tension problem. I'm really having an extremely hard time understanding the concept of tension. For this problem I drew the free body diagrams of both masses. The mass of 12kg is m1 and the mass of 10kg is m2.
For m1, the force in the negative y direction (pulling the object to the ground) is the weight, or the mass times acceleration. The mass pulling on m1 in the positive direction is the tension.
For m2, the force pulling in the negative y direction is the weight. The force pulling in the positive y direction is the tension and the weight of m1.
The net force acting on m1 would be... m1g-T=(m1 + m2)a
The net force acting on m2 would be... T+m1g=m2a

Something in that train of thought is wrong. I believe it is the net force acting on m2 or maybe even the force acting on m2 in the positive y direction. Any help would be greatly appreciated.

 

[Notyou]

Figured it out <_<. The force was acting on the objects independently. What I don't really understand though, is how come in this scenario the same force is acting independently on the objects but if I laid the two blocks on a flat surface and pulled the heavier block with a string the force would be acting on their combined weight?

 

[thomate1]

I can provide a response to the content by explaining the concept of tension and how it relates to this problem.

Tension is a force that is transmitted through a string, rope, or cable when it is pulled tight by forces acting on either end. In this problem, the two objects are connected by a string that runs over a pulley. The tension in the string is the force that is pulling the two objects together.

In order to solve this problem, we need to use Newton's Second Law, which states that the net force on an object is equal to its mass multiplied by its acceleration (F=ma). In this case, we have two objects, m1 and m2, with different masses hanging from the pulley.

Let's start by looking at m1. The only forces acting on m1 are its weight (mg) pulling it down and the tension in the string pulling it up. Since the objects are at rest, the net force on m1 is zero. This means that the tension in the string must be equal to the weight of m1 (T=m1g).

Now, let's look at m2. The forces acting on m2 are its weight (m2g) pulling it down and the tension in the string pulling it up. Since m2 is connected to m1 by the string, the tension in the string is also pulling m2 up. This means that the net force on m2 is equal to the difference between the tension and the weight of m2 (T-m2g).

Since the two objects are connected by the string, they will have the same acceleration. This means that the acceleration of m1 and m2 will be the same, and we can set the two net force equations equal to each other.

T=m1g
T-m2g=(m1+m2)a

Substituting the value of T from the first equation into the second equation, we get:

m1g-m2g=(m1+m2)a

Simplifying, we get:

g=(m1+m2)a

Solving for a, we get:

a=g/(m1+m2)

So, the acceleration of both objects will be equal to the acceleration due to gravity (g) divided by the sum of their masses (m1+m2).

To find the tension in the string, we can use either of the two