•Box A weighs 50 N and rests on a table. A rope that connects Eboxes A and B drapes over a pulley so that box B hangs 2 above the table, as shown in the figure. The pulley and rope are massless, and the pulley is frictionless.
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What force n, does the table exert on box A if box B weighs
20 N? Assume that the positive direction is upwards.
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What force n2 does the table exert on box A if box B weighs
40 N? Assume that the positive direction is upwards.
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What force n3 does the table exert on box A if box B weighs
60 N? Assume that the positive direction is upwards.

To solve the problem, let's analyze the forces acting on box A in different scenarios based on the weight of box B.

### Step-by-step Analysis:
1. **System Overview:**
   - Box A rests on a table, and a rope connects it to box B, which hangs off the table through a pulley.
   - The weight of box A is given as 50 N.
   - The system is in equilibrium (no acceleration), meaning that the net force on each box is zero.

2. **Calculating the Normal Force \( n \) on Box A:**
   - The weight of Box A \( W_A \) = 50 N acts downward.
   - The tension \( T \) in the rope, which is equal to the weight of box B \( W_B \), acts horizontally on box A.
   - Since there is no horizontal motion, the horizontal forces on Box A cancel out.
   - The vertical forces acting on box A include:
     - The weight of Box A (\( W_A = 50 \) N) acting downward.
     - The normal force \( n \) from the table acting upward.
     - If box B is heavier or lighter, the rope will either pull box A slightly downwards or upwards, affecting the net force on the table.

3. **Force Equilibrium for Each Case:**

#### Case 1: Box B Weighs 20 N
- The vertical forces on box A are:
  - Weight of Box A: \( 50 \, \text{N} \) (downward)
  - Table’s normal force \( n_1 \) (upward)

Because box B is pulling downwards (20 N), the tension in the rope reduces the effective weight of box A on the table:
\[
n_1 = W_A - W_B = 50 \, \text{N} - 20 \, \text{N} = 30 \, \text{N}
\]

#### Case 2: Box B Weighs 40 N
- Using the same analysis:
\[
n_2 = W_A - W_B = 50 \, \text{N} - 40 \, \text{N} = 10 \, \text{N}
\]

#### Case 3: Box B Weighs 60 N
- This time, the weight of Box B is greater than Box A, which means it will pull box A slightly upwards, reducing the force the table exerts:
\[
n_3 = W_A - W_B = 50 \, \text{N} - 60 \, \text{N} = -10 \, \text{N}
\]

### Final Answers:
- **\( n_1 = 30 \, \text{N} \)** (for \( W_B = 20 \, \text{N} \))
- **\( n_2 = 10 \, \text{N} \)** (for \( W_B = 40 \, \text{N} \))
- **\( n_3 = -10 \, \text{N} \)** (for \( W_B = 60 \, \text{N} \))

The negative sign indicates that in the third case, the table is not exerting a normal force on Box A; instead, the table is being pulled slightly upwards by Box A.

If you'd like further clarification or a detailed diagram, feel free to ask!

### Related Questions:
1. What is the tension in the rope for each case?
2. How would the normal force change if the mass of the pulley is not negligible?
3. What is the acceleration of the system if Box B is heavier than Box A?
4. How would friction between the table and Box A affect the system?
5. What would happen if the pulley had a non-negligible mass?

### Tip:
Always consider both vertical and horizontal force components separately when dealing with systems involving pulleys and connected masses.

Box A weighs 50 N and rests on a table. A rope that connects boxes A and B drapes over a pulley so that box B hangs above the table. The pulley and rope are massless, and the pulley is frictionless. What force does the table exert on box A if box B weighs 20 N, 40 N, and 60 N? Assume that the positive direction is upwards.

This problem explores the normal force exerted by the table on Box A when connected to Box B through a pulley system. The problem uses Newton's laws of motion to calculate the normal force when Box B weighs 20 N, 40 N, and 60 N. Suitable for students in Grades 9-12.

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