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.