Two blocks connected by a string are pulled

across a rough horizontal surface by a force
applied to one of the blocks, as shown.
The acceleration of gravity is 9.8 m/s
Block 1 has a mass of 5 kg while block two has a mass of 8 kg. A string is tied to the 8 kg block pulling up at and angle of 44 degrees with a force of F
µ = 0.17
If each block has an acceleration of 2.8 m/s
2
to the right, what is the magnitude of the
applied force?
Answer in units of N

To find the magnitude of the applied force (F), we can use the concept of Newton's second law of motion which states that the net force acting on an object is equal to the product of its mass and acceleration.

Now, let's break down the forces acting on the system and analyze them:

1. The tension force (T) in the string: This force acts horizontally on both blocks and has the same magnitude for both blocks. We can find the tension force by analyzing block 1 or block 2.

2. The force of friction (f) between block 1 and the rough surface: This force opposes the motion and acts to the left.

3. The force of gravity (mg) acting on each block: This force acts vertically downwards.

The mass and acceleration information provided is as follows:

Mass of block 1 (m₁) = 5 kg
Mass of block 2 (m₂) = 8 kg
Acceleration of each block (a) = 2.8 m/s²

To solve this problem, we need to represent all forces in terms of their components. Let's define the x-axis as the direction of motion:

1. Tension force (T): It has both vertical and horizontal components. The vertical component T⋅sin(44°) is responsible for counteracting the force of gravity for block 2 (mg₂). The horizontal component T⋅cos(44°) is responsible for accelerating both blocks.

2. Force of friction (f): It acts to the left and opposes the motion. Since we know the coefficient of friction (µ) and the normal force (which is equal to the weight of block 1), we can calculate the force of friction as f = µ⋅(weight of block 1).

3. Force of gravity (mg): It acts vertically downwards and can be calculated as the product of the mass (m) and the acceleration due to gravity (9.8 m/s²).

Now, applying Newton's second law to each block, we have the following equations:

For block 1: 5⋅a = T⋅cos(44°) - f

For block 2: 8⋅a = T⋅sin(44°) + mg

Substituting the expressions for f (force of friction) and mg (force of gravity) into the equations, we get:

5⋅a = T⋅cos(44°) - µ⋅(weight of block 1)

8⋅a = T⋅sin(44°) + 8⋅9.8

Now, let's solve these equations simultaneously to find the tension force (T).