The key to solving forces on multiple objects is to find the net force on each individual object, and then solve the system of equations.

Examples:

1. In the diagram below, find the acceleration of the system and the tension of each rope, assuming that the system is frictionless.

Answer: Separate each block to find the forces acting on it.

Net force on the 5-kg block: 100 — T₁ = 5a

Net force on the 3-kg block: T₁ — T₂= 3a

Net force on the 2-kg block: T₂ = 2a

Solve these three equations to get a = 10 m/s², T₁ = 50 N, and T₂= 20 N.

2. In the diagram below, find the acceleration of the system and the tension of the rope, assuming that the system is frictionless.

Answer: Net force on the 4-kg block: 4 × 9.8 — T = 4a

Net force on the 1-kg block:

Solve these two equations to get a = 5.88 m/s² and T = 15.68 N.

3. In the diagram below, find the acceleration of the system and the tension of each rope. Do NOT assume that the system is frictionless.

Answer:

F_r1= N× μ = mgμ = 5 × 9.8 × 0.2 = 9.8 N

F_r2= N× μ = mgμ = 3 × 9.8 × 0.2 = 5.88 N

F_r3= N× μ = mgμ = 2 × 9.8 × 0.2 = 3.92 N

Net force on the 5-kg block (circle the 5-kg block): 100 — T₁ —F_r1= 5a

Net force on the 3-kg block (circle the 3-kg block): T₁ — T₂—F_r2= 3a

Net force on the 2-kg block (circle the 2-kg block): T₂ —F_r3= 2a

Solve these three equations above to get a = 8.04 m/s², T₁ = 50 N, and T₂= 20 N.

4. In the diagram below, find the acceleration of the system and the tension of the rope.

Answer: Net force on the 5-kg block: T — Fr = T — 5 × 9.8 × 0.4 = 5a

Net force on the 8-kg block: 8 × 9.8 — T = 8a

Solve these two equations above to get a = 4.52 m/s²and T = 42.22 N.

5. In the diagram below, find the acceleration of the system and the tension of the rope.

Answer: Net force on the 1-kg block:

Net force on the 3-kg block (on the inclined plane) =

3×9.8×

Solve these two equations above to get a = 0.196 m/s² and T = 10.0 N.