5.3 Dynamics of Connected Objects

Connected systems in physics involve objects linked by ropes, pulleys, or other mechanisms. These setups transmit forces between components, creating complex interactions that can be analyzed using Newton's laws and free-body diagrams.

Understanding connected systems is crucial for solving real-world problems like elevators, cranes, and simple machines. By breaking down these systems and applying force analysis techniques, we can predict motion and calculate important quantities like tension and acceleration.

Forces and Motion in Connected Systems

Forces on connected objects

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• Common forces in connected systems include tension in strings or ropes transmits force along length, normal forces from surfaces perpendicular to contact, friction opposes motion, gravity pulls objects downward
• Ideal strings and pulleys assumed massless and inextensible strings maintain constant length, frictionless pulleys redirect force without loss
• Force transmission through connections maintains equal tension throughout ideal string, pulleys change direction of applied force without altering magnitude

Newton's laws for connected motion

• Newton's First Law analyzes static equilibrium when net force is zero (hanging mobile)
• Newton's Second Law determines accelerations using $\sum F = ma$ (accelerating elevator)
• Newton's Third Law considers action-reaction pairs equal and opposite (tug-of-war)
• Create separate free-body diagrams for each object showing all forces
• Write force equations for each object based on diagrams
• Solve systems of equations to find unknown quantities like accelerations or tensions

Acceleration and tension analysis

• Draw accurate free-body diagrams for each object showing all forces
• Choose consistent coordinate system typically with positive y-axis upward
• Write force equations $\sum F_x = ma_x$ and $\sum F_y = ma_y$ for each object
• Identify relationships between accelerations of connected objects often equal magnitude
• Solve system of equations to find accelerations and tensions in pulleys or ropes

Mechanical advantage in simple machines

• Mechanical advantage ratio of output force to input force measures force amplification
• Trade-off between force and distance in simple machines larger force over shorter distance
• Pulley system mechanical advantages:
  • Single fixed pulley MA = 1 changes direction only
  • Single movable pulley MA = 2 doubles force halves distance
  • Compound pulley systems MA increases with more pulleys (block and tackle)
• Mechanical advantage relates to efficiency real machines less than 100% due to friction

Equilibrium of connected systems

• Equilibrium conditions: $\sum F = 0$ for translational and $\sum \tau = 0$ for rotational
• Identify all forces acting on system including tensions, normal forces, friction
• Write equilibrium equations for forces and torques
• Solve for unknowns masses or forces needed to balance system (seesaw)
• Consider friction effects in real-world problems reduces efficiency requires larger input force