Two packing crates of masses m1 = 10.0 kg and m2 = 5.00 kg are connected by a light string that passes over a frictionless pulley as in the figure below. The 5.00 kg crate lies on a smooth incline of angle 40.0°. Find the acceleration of the 5.00 kg crate.

Find the tension on the string.

m1•a=m1•g –T

m2•a=T-m2•g•sinα
a(m1+m2)= m1•g –T+ T-m2•g•sinα,
a=g•(m1+m2•sin α)/(m1+m2) = ...
T= m1•(g-a)=...

124N

To find the acceleration of the 5.00 kg crate, we first need to determine the net force acting on it.

1. Start by calculating the gravitational force acting on the 10.0 kg crate.
The gravitational force can be determined using the equation:
F_gravity = m * g

where m is the mass of the object and g is the acceleration due to gravity (approximately 9.8 m/s^2).

F_gravity_10kg = (10.0 kg) * (9.8 m/s^2)

2. Next, find the gravitational force component acting along the incline for the 5.00 kg crate.
The gravitational force component can be determined using the equation:
F_gravity_component = m * g * sin(theta)

where m is the mass of the object, g is the acceleration due to gravity, and theta is the angle of the incline (40.0°).

F_gravity_component_5kg = (5.00 kg) * (9.8 m/s^2) * sin(40.0°)

3. Determine the net force acting on the 5.00 kg crate.
The net force acting on the crate is the difference between the gravitational force component and the force of friction.
Since the incline is smooth, the force of friction is negligible.

Net_force = F_gravity_component_5kg - F_friction

4. Calculate the acceleration of the 5.00 kg crate using Newton's second law.
Newton's second law states that the acceleration of an object is equal to the net force applied to it divided by its mass:
acceleration = Net_force / mass

acceleration = Net_force / (5.00 kg)

Now, let's move on to finding the tension in the string.

5. Determine the tension in the string.
The tension in the string can be found by considering the forces acting on the 10.0 kg crate. Since the 10.0 kg crate moves vertically, the tension force is equal in magnitude to the gravitational force acting on it.

Tension = F_gravity_10kg

Now you can plug in the calculated values to find the acceleration and tension.

To find the acceleration of the 5.00 kg crate and the tension on the string, we can apply Newton's second law of motion and use the principles of forces and acceleration.

Step 1: Identify the forces acting on each crate.

For the 10.0 kg crate (m1):
- The force due to gravity: Fg1 = m1 * g (where g is the acceleration due to gravity, approximately 9.8 m/s^2)

For the 5.00 kg crate (m2):
- The force due to gravity: Fg2 = m2 * g
- The force along the incline: F_parallel = m2 * g * sin(theta) (where theta is the angle of the incline)
- The normal force: N = m2 * g * cos(theta)

Step 2: Calculate the net force acting on each crate.

For the 10.0 kg crate (m1):
- The net force is equal to the tension in the string: Tension

For the 5.00 kg crate (m2):
- The net force is the difference between the force along the incline and the force due to gravity: F_net = F_parallel - Fg2

Step 3: Apply Newton's second law of motion to each crate.

For the 10.0 kg crate (m1):
Using F_net = m1 * a, where a is the acceleration:
- Tension = m1 * a1

For the 5.00 kg crate (m2):
Using F_net = m2 * a, where a is the acceleration:
- F_parallel - Fg2 = m2 * a2

Step 4: Relate the accelerations of both crates.

Since the crates are connected by a light string, they have the same acceleration:
- a1 = a2 = a

Step 5: Solve the set of equations.

Using the equations from steps 3 and 4, we can set up a system of equations:
- Tension = m1 * a
- F_parallel - Fg2 = m2 * a

Substituting the expressions for F_parallel and Fg2 from step 1:
- m2 * g * sin(theta) - m2 * g = m2 * a

Step 6: Solve for the acceleration (a).

Rearranging the equation:
- a = (m2 * g * sin(theta) - m2 * g) / m2

Step 7: Calculate the tension in the string.

Using the equation from step 3:
- Tension = m1 * a

Step 8: Substitute the given values and calculate the numeric results.

Substituting the given values:
- m1 = 10.0 kg
- m2 = 5.00 kg
- theta = 40.0°

We can now calculate the values for the acceleration (a) and the tension (Tension).