A block-and-pulley system is set up as shown in the diagram below. There is no friction between the 2 kg and 3 kg blocks, but there is friction between the 3 kg block and the table; the coefficient of friction here is 0.28. Draw a free-body diagram for each mass and determine the acceleration of the system and the tension in the strings, T1 and T2.
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To determine the acceleration of the system and the tension in the strings, we need to analyze the forces acting on each mass separately and use Newton's second law of motion. Let's break it down step by step.
1. Start by drawing a free-body diagram for each mass:
For the 2 kg block:
- Draw an arrow pointing downward to represent the weight (mg) of the block.
- Draw an arrow pointing to the right to represent the tension in the string (T1).
- Since there is no friction acting on the 2 kg block, there are no other forces involved.
For the 3 kg block:
- Draw an arrow pointing downward to represent the weight (mg) of the block.
- Draw an arrow pointing to the left to represent the tension in the string (T2).
- Draw an arrow pointing to the right to represent the friction force (F_friction) between the 3 kg block and the table.
2. Now, let's determine the acceleration of the system:
- Since the 2 kg and 3 kg blocks are connected by a string, they will have the same acceleration.
- The total force acting on the system is the difference between the tensions in the two strings: T1 - T2.
- From Newton's second law, F = ma, we can now write the equation: T1 - T2 = (2 kg + 3 kg) * a, where a is the acceleration.
- Rearranging the equation, we get: a = (T1 - T2) / 5 kg.
3. Next, let's determine the tension in the strings, T1 and T2:
- For the 2 kg block, examining the free-body diagram, the only force acting on it is T1.
- Using Newton's second law, we have T1 - mg = ma.
- Plugging in the values: T1 - (2 kg * 9.8 m/s^2) = 2 kg * a.
- For the 3 kg block, examining the free-body diagram, the forces acting on it are T2, F_friction, and m3 * g (weight of the block).
- The friction force F_friction can be calculated as the product of the coefficient of friction (μ) and the normal force (mg).
- The normal force is equal to the weight of the block, so F_friction = μ * m3 * g.
- Using Newton's second law, we have T2 + F_friction - mg = m3 * a.
- Plugging in the values: T2 + (0.28 * 3 kg * 9.8 m/s^2) - (3 kg * 9.8 m/s^2) = 3 kg * a.
Now, we have two equations with two unknowns (acceleration and tension). Solve the system of equations to find the values of acceleration and tensions, T1 and T2.