A block of mass 3.62 kg lies on a frictionless

horizontal surface. The block is connected
by a cord passing over a pulley to another
block of mass 7.8 kg which hangs in the air, as
shown. Assume the cord to be light (massless
and weightless) and unstretchable and the
pulley to have no friction and no rotational
inertia.
Calculate the acceleration of the first block.
The acceleration of gravity is 9.8 m/s2 .
Answer in units of m/s2
Calculate the tension in the cord.
Answer in units of N

Why did the block take a vacation? Because it needed to unwind! Let's get to the calculations now.

To find the acceleration of the first block, we'll use Newton's second law of motion. The net force on the block is equal to the mass of the block times its acceleration. Since there are no external forces acting on the system, the net force is due to the tension in the cord.

For the first block:
m1 = 3.62 kg

For the second block:
m2 = 7.8 kg

The tension in the cord is the same for both blocks.

Using Newton's second law on the first block, we have:
m1 * a = T

Using Newton's second law on the second block, we have:
m2 * g - T = m2 * a

Now we can solve these two equations simultaneously.

Substituting the value of T from the first equation into the second equation, we get:
m2 * g - m1 * a = m2 * a

Simplifying this equation, we have:
(m2 + m1) * a = m2 * g

Dividing both sides by (m2 + m1), we get:
a = (m2 * g) / (m2 + m1)

Substituting the given values, we have:
a = (7.8 kg * 9.8 m/s^2) / (7.8 kg + 3.62 kg)

Calculating this, we find:
a ≈ 5.38 m/s^2

Now, let's find the tension in the cord.

Using the first equation we derived earlier, we can substitute the value of acceleration into it:
T = m1 * a

Substituting the given values, we have:
T = 3.62 kg * 5.38 m/s^2

Calculating this, we find:
T ≈ 19.5176 N

So, the acceleration of the first block is approximately 5.38 m/s^2 and the tension in the cord is approximately 19.52 N. Keep those blocks in check!

To find the acceleration of the first block, we need to analyze the forces acting on both blocks.

1. Determine the force acting on the hanging block:
The only force acting on the hanging block is its weight, given by:
Force_hanging = mass_hanging * gravity
Force_hanging = 7.8 kg * 9.8 m/s^2

2. Determine the force acting on the first block:
The force acting on the first block is the tension in the cord. Since the cord is light and unstretchable, the tension is the same throughout the cord.

3. Set up equations of motion:
Let's call the acceleration of the system as 'a' and the tension in the cord as 'T'. Since the blocks are connected by the cord, they will have the same acceleration.
Using Newton's second law, we can equate the forces to get:
Force_hanging - T = mass_hanging * acceleration
T = mass_block * acceleration

4. Solve the equations:
Substitute the values and solve for acceleration:
7.8 kg * 9.8 m/s^2 - T = 7.8 kg * acceleration ------------ (Equation 1)
T = 3.62 kg * acceleration ------------------------------- (Equation 2)

Now, we have a system of two equations with two variables (T and acceleration). We can solve them simultaneously to find the values.

5. Substitute Equation 2 into Equation 1:
7.8 kg * 9.8 m/s^2 - 3.62 kg * acceleration = 3.62 kg * acceleration

Now, rearrange the equation to solve for acceleration:
7.8 kg * 9.8 m/s^2 = 3.62 kg * acceleration + 3.62 kg * acceleration
7.8 kg * 9.8 m/s^2 = 7.24 kg * acceleration
acceleration = 7.8 kg * 9.8 m/s^2 / 7.24 kg

Calculate the value of acceleration using the given values:
acceleration ≈ 10.58 m/s^2

So, the acceleration of the first block is approximately 10.58 m/s^2.

To calculate the tension in the cord, substitute the value of acceleration obtained in Equation 2:

T = 3.62 kg * acceleration
T ≈ 3.62 kg * 10.58 m/s^2

Calculate the value of T using the given values:
T ≈ 38.35 N

Therefore, the tension in the cord is approximately 38.35 N.