Allen aby sets up an atwood machine and wants to find the acceleration and the tension in the string. him. two blocks of masses m1 = 1.5 kg, and m2 = 2.4 kg are connected by a massless string which passes over a frictionless pulley of negligible mass, as shown above.

what is the acceleration of the system? m/s2

what is the magnitude of the tension in the string? n

Answers:

In order to solve this problem we will use Newton’s second Law, which is mathematically expressed after some simplifications as:

This can be read as: The Net Force of an object is equal to its mass multiplied by its acceleration .

We will also need to draw the Free Body Diagram of each block in order to know the direction of the acceleration in this system and find the Tension of the string (See figure attached).  

We already know is greater than , this means the weight of the block 2 is greater than the weight of the block 1 ; therefore the acceleration of the system will be in the direction of , as shown in the figure attached.

We also know by the information given in the problem that the pulley does not have friction and has negligible mass, and the string is massless.

This means that the tension will be the same along the string regardless of the difference of mass of the blocks.

Now that we have the conditions clear, let’s begin with the calculations:

1) Firstly, we have to find the weight of each block, in order to verify that block 2 is heavier than block 1.

This is done using equation (1), where the force of the weight is calculated using the acceleration of gravity  acting on the blocks:

For block 1:

  (3)

For block 2:

  (5)

Then, we are going to find the acceleration of the whole system:

  (7)

Where the Resulting Force  is equal to the sum of the weights and .  

In the figure attached, note that is in opposite direction to the acceleration , this means it must have a negative sing; while is in the same direction of .

Here we only have to isolate from equation (8) and substitute the values according to the conditions of the system:

Then:

2) For the second part of the problem, we have to find the tension of the string.

We can choose either the Free Body Diagram of block A or block B to make the calculations, the result will be the same.  

Let’s prove it:

For

we see in the free body diagram that the acceleration is in the same direction of the tension of the string, so:

  (9)

  (10)

Then;

For

we see in the free body diagram that the acceleration is in opposite direction of the tension of the string and must have a negative sign, so:

  (9)

  (10)

Then;

prolly a beacuse the momentuem the car has increases while going down hill