Section 2.4: Tension & Vertical Motion
When objects move vertically under gravity, tension in ropes or cables balances or supplements weight. Newton’s second law applies along vertical direction.
Vertical Forces:
\[
\sum F_y = ma \quad \Rightarrow \quad T - mg = ma \quad \text{(upward positive)}
\]
Free-Fall: For objects dropped without air resistance:
\[
a = g, \quad F_{\text{net}} = mg
\]
Acceleration of Pulley Systems:
\[
a = \frac{m_2 - m_1}{m_1 + m_2} g \quad \text{(for two masses over a pulley)}
\]
Example 1
A 10 kg mass hangs from a rope. Find the tension if held at rest.
\( T = mg = 10*9.8 = 98 \text{ N} \)
Example 2
A 5 kg mass is pulled upward with acceleration 2 m/s². Find the tension in the rope.
\( T - mg = ma \Rightarrow T = m(g+a) = 5*(9.8+2) = 59 \text{ N} \)
Practice Problems
- A 12 kg mass is pulled upward with 3 m/s². Compute tension.
- Two masses (4 kg and 6 kg) connected over frictionless pulley. Find acceleration and tensions.
- A 15 kg crate slides vertically downward; find acceleration if rope tension is 100 N.
- Block hangs from spring scale. Determine reading when block is stationary.
- A 20 kg mass is lifted with constant velocity. Find rope tension.