Section 2.5: Tension and Horizontal Motion

This section explains how **tension affects objects moving horizontally**, either on frictionless surfaces or with friction. It covers forces in strings, ropes, and pulleys for horizontal motion.

Horizontal Tension (No Friction):

For a block of mass \( m \) pulled by a force \( F \) via a rope on a frictionless surface:

  • Acceleration: \( a = \frac{F}{m} \)
  • Tension in a string connecting two blocks: \( T = \frac{m_1}{m_1 + m_2} F \)
Horizontal Tension (With Friction):

If the surface has kinetic friction \( f_k = \mu_k m g \):

  • Net force: \( F_{net} = F - f_k \)
  • Acceleration: \( a = \frac{F - f_k}{m} \)

Example 1

Two blocks of masses 5 kg and 10 kg are connected by a rope on a frictionless horizontal surface. A force of 30 N pulls the 10 kg block. Find the tension in the rope and acceleration.

Total mass: \( 5+10=15\text{ kg} \)

Acceleration: \( a = F/m_{total} = 30/15 = 2\text{ m/s²} \)

Tension: \( T = m_1 a = 5*2 = 10\text{ N} \)

Example 2

A 8 kg block is pulled horizontally with 40 N. The coefficient of kinetic friction is 0.2. Find the acceleration.

Friction: \( f_k = \mu_k m g = 0.2*8*9.8 \approx 15.68\text{ N} \)

Net force: \( F_{net} = 40 - 15.68 \approx 24.32\text{ N} \)

Acceleration: \( a = F_{net}/m = 24.32/8 \approx 3.04\text{ m/s²} \)

Practice Problems

  1. Two blocks 3 kg and 6 kg connected over a frictionless surface. Pulling 18 N on 6 kg block, find acceleration and tension.
  2. 5 kg block pulled by 20 N on a surface with μk = 0.1. Find acceleration.
  3. Blocks 4 kg and 8 kg connected. 24 N pulls 8 kg block. Compute tension in rope.
  4. 10 kg block pulled on horizontal surface, μk = 0.25, force = 50 N. Find acceleration.
  5. Three blocks, 2 kg, 3 kg, 5 kg, in series. Force 30 N on last block, frictionless. Find acceleration and tensions between blocks.
  6. 7 kg block pulled horizontally with 28 N, μk = 0.2. Determine acceleration.
  7. Two blocks 6 kg and 9 kg on frictionless surface. Pulled with 45 N. Compute tension in connecting rope.
  8. 5 kg block pulled on horizontal surface, μk = 0.3, F = 25 N. Find acceleration.
  9. 8 kg and 12 kg blocks connected. Pull 60 N on 12 kg block, frictionless. Determine tension.
  10. 6 kg block on surface with μk = 0.15. Pulled with 30 N. Find acceleration.
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