Two masses are connected by a light cord over a frictionless pulley as shown. If M1 = 8.9 kg and M2 = 5.7 kg, calculate the tension in the cable (in N). Take the coefficient of kinetic friction between M1 and the horizontal surface to be 0.3 and assume the acceleration due to gravity to be 9.81 m/s2.
To calculate the tension in the cable, we need to consider the forces acting on the masses and apply Newton's second law of motion.
Step 1: Determine the forces acting on M1 and M2.
- M1 experiences the force of tension (T) in the cable and the force of kinetic friction (fk) on the horizontal surface.
- M2 experiences the force of tension (T) in the cable and the force of gravity (mg) pulling it downward.
Step 2: Write down the equations for the forces acting on M1 and M2.
- For M1:
- In the horizontal direction: T - fk = M1 * a
- In the vertical direction: None (since M1 is on a horizontal surface).
- For M2:
- In the horizontal direction: None (since M2 is hanging vertically).
- In the vertical direction: T - mg = M2 * a
Step 3: Solve the system of equations.
- Substituting the given values: M1 = 8.9 kg, M2 = 5.7 kg, coefficient of kinetic friction (μk) = 0.3, and acceleration due to gravity (g) = 9.81 m/s^2:
- For M1: T - 0.3 * M1 * g = M1 * a
- For M2: T - M2 * g = M2 * a
Step 4: Simplify the equations.
- For M1: T - 2.647 * 8.9 = 8.9 * a
- For M2: T - 5.7 * 9.81 = 5.7 * a
Step 5: Solve for Tension (T).
- Subtract the equations to eliminate a: T - 2.647 * 8.9 - (T - 5.7 * 9.81) = 0
- Simplify: -23.573 + T - T + 56.157 = 0
- Combine like terms: 32.584 = 0
Step 6: Calculate Tension (T).
- T = 32.584 N (rounded to three decimal places).
Therefore, the tension in the cable is approximately 32.584 N.