A box of mangoes and a box of pears are connected by a light string that passes over a frictionless pulley as shown in Figure 2. Both boxes contain 2 kg of mangoes and 4 kg of pears. A 40 N horizontal force is applied to pull the box of mango to the left on a frictionless surface. Determine the tension in the string and the acceleration of the box of pears.
To determine the tension in the string and the acceleration of the box of pears, we can use Newton's second law of motion and the concept of an ideal pulley system.
1. Start by drawing a free-body diagram for each box:
- The box of mangoes will experience a force of 40 N to the left (due to the applied force) and a tension force coming from the string to the right.
- The box of pears will have the tension force coming from the string upward and its weight downward.
2. Apply Newton's second law of motion (F = ma) to each box:
- For the box of mangoes:
- Horizontal force: F = 40 N (applied force to the left)
- Acceleration: a
- Mass: m = 2 kg
- The net force in the horizontal direction is the difference between the applied force and the tension force:
- 40 N - Tension = m * a
- For the box of pears:
- Vertical force: F = Tension
- Acceleration: a (since the string is connected to both boxes, they move with the same acceleration)
- Mass: m = 4 kg
- The net force in the vertical direction is the difference between the weight and the tension force:
- Weight - Tension = m * a
- m * g - Tension = m * a (where g is the acceleration due to gravity)
3. Solve the two equations simultaneously to find the tension and acceleration:
- 40 N - Tension = 2 kg * a
- 4 kg * g - Tension = 4 kg * a
Rearrange the equations and solve:
- Tension = 40 N - 2 kg * a
- Tension = 4 kg * g - 4 kg * a
Equate the two tensions and simplify:
40 N - 2 kg * a = 4 kg * g - 4 kg * a
40 N - 4 kg * g = -2 kg * a + 4 kg * a
40 N - 4 kg * g = 2 kg * a
2 kg * a = 40 N - 4 kg * g
a = (40 N - 4 kg * g) / 2 kg
Substituting the value of acceleration into one of the equations for tension, we can find the tension force:
Tension = 40 N - 2 kg * [(40 N - 4 kg * g) / 2 kg]
4. Finally, calculate the tension in the string and the acceleration of the box of pears using the given values for acceleration due to gravity (g = 9.8 m/s^2):
Tension = 40 N - 2 kg * [(40 N - 4 kg * 9.8 m/s^2) / 2 kg]
Tension = 40 N - 2 kg * (40 N - 39.2 N)
Tension = 40 N - 2 kg * 0.8 N
Tension = 40 N - 1.6 N
Tension = 38.4 N
a = (40 N - 4 kg * 9.8 m/s^2) / 2 kg
a = (40 N - 39.2 N) / 2 kg
a = 0.8 N / 2 kg
a = 0.4 m/s^2
Therefore, the tension in the string is 38.4 N and the acceleration of the box of pears is 0.4 m/s^2.