Section 2.6: Advanced Tension Problems
Advanced tension problems often involve multiple connected objects, pulleys, or forces acting at angles. The key is to apply Newton's second law to each object separately and relate tensions using constraints from pulleys or ropes.
- Draw free-body diagrams for each object.
- Identify all forces including tension, weight, friction, and applied forces.
- Apply \(\sum F = ma\) for each object along the direction of motion.
- Use rope/pulley constraints to relate accelerations.
- Solve the resulting system of equations for tension and acceleration.
Example 1
Two blocks, m1=5 kg and m2=3 kg, are connected by a rope over a frictionless pulley. Find the acceleration and tension.
\( m_1 a = T - m_1 g \sin \theta \) (if inclined, else \( m_1 a = T \))
\( m_2 a = m_2 g - T \)
Solving: \( a = 2.5 \text{ m/s²}, T = 22.5 \text{ N} \)
Example 2
A horizontal rope connects 3 blocks on a frictionless surface. A force F=30 N pulls the first block. Find tensions in the ropes.
Total mass \( M = m_1 + m_2 + m_3 \)
Acceleration \( a = F / M \)
Tensions found by applying \( T = m a \) for connected sections.
Practice Problems
- Two blocks on inclined planes connected over a pulley. Find tension and acceleration.
- Three blocks on horizontal frictionless surface, connected by ropes. A force pulls the first. Find tensions in each rope.
- Block on a table connected to hanging mass over a pulley. Determine acceleration and rope tension.
- System of pulleys lifting weights. Find tensions in all ropes.
- Blocks connected at angles via pulleys. Compute accelerations and tensions.