Section 2.4: Tension and Vertical Motion
This section introduces **tension in ropes or strings** and the **motion of objects moving vertically** under gravity. Key concepts include forces on hanging objects, pulleys, and the acceleration of vertically moving bodies.
For a mass \( m \) suspended from a rope:
- At rest: \( T = mg \)
- Moving upward with acceleration \( a \): \( T = m(g + a) \)
- Moving downward with acceleration \( a \): \( T = m(g - a) \)
For two masses \( m_1 \) and \( m_2 \) connected over a frictionless pulley:
- Acceleration: \( a = \frac{m_2 - m_1}{m_1 + m_2} g \) (assuming \( m_2 > m_1 \))
- Tension: \( T = \frac{2 m_1 m_2}{m_1 + m_2} g \)
Example 1
A 10 kg mass is suspended from a rope. Find the tension when it is at rest, moving upward at 2 m/s², and moving downward at 2 m/s².
At rest: \( T = mg = 10*9.8 = 98 \text{ N} \)
Upward: \( T = m(g+a) = 10*(9.8+2) = 118 \text{ N} \)
Downward: \( T = m(g-a) = 10*(9.8-2) = 78 \text{ N} \)
Example 2
Two masses, \( m_1 = 5 \text{ kg} \) and \( m_2 = 8 \text{ kg} \), are connected over a frictionless pulley. Find acceleration and tension.
Acceleration: \( a = \frac{8-5}{5+8}*9.8 \approx 2.26 \text{ m/s²} \)
Tension: \( T = \frac{2*5*8}{5+8}*9.8 \approx 60.3 \text{ N} \)
Practice Problems
- A 12 kg mass is suspended. Find tension when moving up with 1.5 m/s² acceleration.
- 10 kg and 15 kg connected over a pulley. Find acceleration and tension.
- 5 kg block hanging, moving downward at 2 m/s². Find tension.
- Two blocks, 6 kg and 9 kg, on a frictionless pulley. Compute acceleration and tension.
- 8 kg mass at rest. What is tension in the rope?
- A 7 kg object is pulled upward with 3 m/s². Find tension.
- Two masses 4 kg and 10 kg connected over a pulley. Find tension.
- 10 kg hanging mass accelerates down at 1 m/s². Compute tension.
- 5 kg and 12 kg masses, frictionless pulley. Determine acceleration.
- 15 kg mass moving upward at 2 m/s². Find tension.