An object with mass m1 = 5.00 kg, rests on a frictionless horizontal table and is connected to a cable that passes over a pulley and is then fastened to a hanging object with mass m2 = 10.0 kg, as shown in the figure below. Find the acceleration of each object and the tension in the cable.
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To find the acceleration of each object and the tension in the cable, we can use Newton's second law of motion and the equation for the tension in a rope or cable.
1. Start by drawing a free-body diagram for each object.
- For the object with mass m1 (resting on the table), the only force acting on it is the tension in the cable.
- For the hanging object with mass m2, the forces acting on it are the tension in the cable and the force of gravity.
2. Write the equations for the net force on each object.
- For the object with mass m1, the net force is equal to the tension T:
ΣF1 = T = m1a (equation 1)
- For the hanging object with mass m2, the net force is the difference between the tension T and the force of gravity:
ΣF2 = T - m2g = m2a (equation 2)
(Note: g is the acceleration due to gravity, approximately 9.8 m/s²)
3. Solve the system of equations.
- Substitute equation 1 into equation 2 to eliminate the tension T:
T - m2g = m2a
Substituting T = m1a:
m1a - m2g = m2a
- Rearrange the equation to solve for the acceleration a:
a = m2g / (m1 + m2) (equation 3)
4. Substitute the given values into equation 3 to calculate the acceleration.
- m1 = 5.00 kg, m2 = 10.0 kg, g = 9.8 m/s²
- Plug in the values:
a = (10.0 kg)(9.8 m/s²) / (5.00 kg + 10.0 kg)
- Calculate the acceleration:
a ≈ 6.53 m/s²
5. Calculate the tension in the cable using equation 1.
- Plug in the values:
T = (5.00 kg)(6.53 m/s²)
- Calculate the tension:
T ≈ 32.7 N
So, the acceleration of each object is approximately 6.53 m/s², and the tension in the cable is approximately 32.7 N.