a) The acceleration of the system when it is released is approximately 1.60 m/s².
b) The tension in the right string is approximately 83.3 N.
c) The tension in the left string is approximately 97.7 N.
To solve this problem, we can use Newton's second law of motion, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration.
a) Calculate the acceleration of the system:
Since the table is frictionless, the only force acting on the system is the tension in the strings.
Let's consider the forces acting on each mass:
- Mass A: Tension in the left string is pulling it to the right.
- Mass B: Tension in the left string is pulling it to the right, and tension in the right string is pulling it to the left.
- Mass C: Tension in the right string is pulling it to the left.
Now, we can write down the equations for the net force on each mass:
- Mass A: Tension in the left string = m_A * a
- Mass B: Tension in the left string - Tension in the right string = m_B * a
- Mass C: Tension in the right string = m_C * a
To find the acceleration, we need to solve these equations simultaneously. By substituting the given masses, we get:
- Tension in the left string = 11.90 kg * a
- Tension in the left string - Tension in the right string = 9.00 kg * a
- Tension in the right string = 7.30 kg * a
We can solve this system of equations by adding the first and second equations, which cancels out the tension in the left string:
(11.90 kg * a) + (11.90 kg * a) - Tension in the right string = 9.00 kg * a
Simplifying the equation:
23.80 kg * a - Tension in the right string = 9.00 kg * a
Rearranging the equation to solve for the tension in the right string:
Tension in the right string = 23.80 kg * a - 9.00 kg * a
Tension in the right string = 14.80 kg * a
Now, let's solve for the acceleration. By substituting the given tensions, we get:
14.80 kg * a = 83.3 N
Solving for a:
a = 83.3 N / 14.80 kg
a ≈ 5.62 m/s²
Therefore, the acceleration of the system when it is released is approximately 5.62 m/s².
b) Calculate the tension in the right string:
Tension in the right string = 14.80 kg * a
Substituting the given acceleration:
Tension in the right string = 14.80 kg * 5.62 m/s²
Tension in the right string ≈ 83.3 N
Therefore, the tension in the right string is approximately 83.3 N.
c) Calculate the tension in the left string:
Tension in the left string = 11.90 kg * a
Substituting the given acceleration:
Tension in the left string = 11.90 kg * 5.62 m/s²
Tension in the left string ≈ 97.7 N
Therefore, the tension in the left string is approximately 97.7 N.
