2 objects are connected on an inclined table. The objects have masses 8.0kg, 4.0kg, and 2.0kg, and the pulleys are frictionless. (The one that is 8.0 is suspended from a pulley off the left side of the table, 4.0kg is on the table, and 2.0kg is suspended from the pulley of the right side.) The tabletop is rough, with a coefficient of kinetic friction of 0.47, and makes an angle of 20 degrees with the horizontal.
-Draw a free-body diagram for each object.
-Determine the acceleration and direction of motion of the system.
-Determine the tensions of the 2 cords.
To solve this problem, we need to analyze the forces acting on each object to determine the acceleration and the tensions in the cords. Let's follow these steps:
1. Draw a free-body diagram for each object:
- For the 8.0kg mass suspended from the left pulley, the forces acting on it are:
- Tension force (upward) from the cord.
- Weight force (downward) due to gravity.
- Normal force (perpendicular to the table).
- Frictional force (opposes motion on the rough table).
- For the 4.0kg mass on the table, the forces acting on it are:
- Weight force (downward) due to gravity.
- Normal force (perpendicular to the table).
- Frictional force (opposes motion on the rough table).
- For the 2.0kg mass suspended from the right pulley, the forces acting on it are:
- Tension force (upward) from the cord.
- Weight force (downward) due to gravity.
- Normal force (perpendicular to the table).
2. Determine the acceleration and direction of motion of the system:
- Now, consider the forces acting on the entire system. Since the objects are connected, they will have the same acceleration.
- The forces causing acceleration are the tension in the left cord (pulling to the right) and the tension in the right cord (pulling to the left).
- The force opposing motion is the frictional force on the table.
- To find the net force, subtract the frictional force from the sum of the tension forces.
- Apply Newton's second law: net force = mass × acceleration.
- Solve for acceleration.
3. Determine the tensions in the two cords:
- Once we know the acceleration, we can find the tension in each cord.
- Consider the object with smaller mass (2.0kg) to calculate the tension in the right cord first.
- Apply Newton's second law to this object: net force = mass × acceleration.
- The net force for this object is the difference between its weight and the tension force.
- Solve for tension.
- To find the tension in the left cord, use the same process with the object of larger mass (8.0kg). However, the net force will now be the sum of its weight and the tension force.
- Solve for tension.
By following these steps, we can find the free-body diagrams, determine the acceleration and direction of motion, and calculate the tensions in the two cords.