Section 2.8: Systems of Connected Objects
When multiple objects are connected by strings, pulleys, or other connectors, they form a system. We analyze such systems using Newton’s laws, tension forces, and acceleration constraints.
- All objects in a string move with the same magnitude of acceleration if the string is inextensible.
- Apply Newton’s 2nd law to each object: \( \sum F = m a \).
- Tension acts along the string and is the same throughout a massless string.
- Identify the forces acting on each object separately, then combine equations to solve.
Example 1
Two blocks, \(m_1 = 5 \text{ kg}\) and \(m_2 = 3 \text{ kg}\), are connected over a frictionless pulley. Find acceleration and tension.
Let \(T\) be tension and \(a\) acceleration. Equations:
- \( m_1 g - T = m_1 a \)
- \( T - m_2 g = m_2 a \)
Adding: \( m_1 g - m_2 g = (m_1 + m_2) a \Rightarrow a = \frac{5-3}{5+3} \cdot 9.8 \approx 2.45 \text{ m/s²} \)
Then \( T = m_2 g + m_2 a = 3*9.8 + 3*2.45 \approx 36.7 \text{ N} \)
Example 2
Three blocks connected in series on a frictionless surface. Find tension between blocks and acceleration.
Use Newton’s law for each block, sum forces, and solve systematically. Each block shares the same acceleration \(a\). Tension can be found from \(T = m a\) for the block(s) ahead.
Practice Problems
- Two blocks \(10 \text{ kg}\) and \(6 \text{ kg}\) connected by a string over a pulley. Find acceleration and tension.
- Two blocks on an incline connected by a string over a pulley. Coefficients of friction: \(\mu_1 = 0.2\), \(\mu_2 = 0.1\). Calculate acceleration and tension.
- Three blocks connected in series on a horizontal surface, \(m_1=2\), \(m_2=3\), \(m_3=5\) kg, pulled by 30 N force. Find acceleration and tensions.
- A block hangs off a frictionless pulley and is connected to another block on a horizontal surface. Find motion and tension.
- Two blocks of 5 kg each connected over a frictionless pulley. One block placed on incline 30°. Determine acceleration and tension.