Analyze a pulley system with multiple components and derive kinematic equations describing the motion and string length | Step-by-Step Solution

What We're Solving:

We're analyzing a pulley system where a string of total length L connects different points (A to O, around a pulley of radius R, then from C to D). We need to understand the kinematic relationships between the moving parts and derive equations that describe how their accelerations are related.

The Approach:

The key insight here is the constraint of constant string length! When you have a string or rope in a pulley system, its total length never changes. This means that if one part of the string gets longer, another part must get shorter by exactly the same amount. We'll use calculus to express this constraint in terms of velocities and accelerations.

Step-by-Step Solution:

Step 1: Understand the String Length Constraint The equation AO + πR + CD = L tells us:

• AO = length of string from point A to the pulley center O
• πR = length of string wrapped around the pulley (half the circumference)
• CD = length of string from point C to point D
• L = total constant string length

Step 2: Apply the Constraint Principle Since L is constant, when we take the derivative with respect to time:

• First derivative: d/dt(AO) + d/dt(πR) + d/dt(CD) = 0
• Since R is constant: d/dt(AO) + 0 + d/dt(CD) = 0
• Therefore: d/dt(AO) + d/dt(CD) = 0

Step 3: Find the Acceleration Relationship Taking the derivative again (second derivative gives us acceleration):

• d²/dt²(AO) + d²/dt²(CD) = 0

This means the accelerations are equal in magnitude but opposite in direction!

Step 4: Interpret the Second Equation The equation d²/dt²(AO) + d²/dt²(CE) = 0 suggests there's another constraint involving point E, likely indicating that CE represents another segment of the string system with the same constraint relationship.

The Answer:

The equations you've derived are kinematic constraint equations that show:

• 1. The acceleration of segment AO is always equal and opposite to the acceleration of segment CD
• 2. Similarly, the acceleration of segment AO is equal and opposite to the acceleration of segment CE
• 3. These relationships ensure the total string length remains constant during motion

Memory Tip:

Think of it like a seesaw! 🎪 When one side goes up (gets longer), the other side must come down (get shorter) by the same amount. In pulley systems, "what goes in must come out" - if you pull 2 cm of string on one side, exactly 2 cm must appear somewhere else in the system. The same principle applies to velocities and accelerations!

Great work identifying these constraint relationships! Understanding how string length constraints create kinematic relationships is fundamental to solving complex pulley problems. Keep practicing with these concepts - you're building excellent physics intuition! 💪