Motion of Two Particles Connected by Pulleys

Overview

• Focus on the motion of one particle depending on another (pulleys and cables).
• Examples will clarify concepts.

Example 1: Finding Speed of Block B

1. Draw a Datum
   • A fixed reference line; for this example, use a wall-fixed pulley.
2. Position Coordinates
   • Identify moving parts: SA, SB, SC.
   • Only SB is drawn to the middle of the pulley as the length of rope across it remains constant.
3. Length Equation
   • Total length: SA + SB + SB + SC.
4. Differentiate for Velocity
   • Derivative leads to: VA + VB + VB + VC = 0.
   • Given: VA = 2 m/s (down), VC = -1 m/s (up).
   • Solve to get VB = -0.5 m/s (implying upward movement of 0.5 m/s).

Example 2: Speed of Block B Rising

1. Draw a Datum
   • Place at the top pulley (fixed).
2. Position Coordinates
   • Identify moving parts: SA, SB, SC.
3. Equation for Two Cables
   • First cable: SA + SC + SC = Length 1.
   • Second cable: SB + SB - SC = Length 2.
4. Considering Movement with Pulley
   • Block B moves with the pulley.
5. Differentiate for Velocity
   • Substitute and solve, yielding 0.5 m/s upwards.

Example 3: Time to Gain Speed at Block B

1. Identify System
   • Motor pulls cable right, moving the load upwards.
2. Two Datums
   • One for horizontal, one for vertical movement (both at the top pulley).
3. Position Coordinates
   • SA, SB, SC.
4. Equations for Cables
   • First cable: SA + SC = Length 1.
   • Second cable: SB + 2(SB - SC) = Length 2.
5. Acceleration Consideration
   • Given motor acceleration: AA = 3 m/s^2.
   • Differentiate again to find acceleration for AB: -2 m/s^2 (upwards).
6. Using Kinematics
   • From rest to 10 m/s at 2 m/s^2, solving gives 5 seconds.

Example 4: Fixed Lengths in Pulley System

1. Draw Datum
   • At the top pulley.
2. Identify Fixed Lengths
   • A bar attached to the small pulley that does not move.
3. Position Coordinates
   • SA, SB, and constant h.
4. Equation for Length
   • SA + SB + 2(SA - a) = Total length.
5. Differentiate for Velocity
   • Solve for VA using VB = 4 m/s, yielding 1.33 m/s upwards.

Key Takeaways

• Establish a datum first.
• Write position equations for moving parts.
• Account for fixed lengths when necessary.
• Differentiate equations to find velocities and accelerations.

Conclusion

• Understanding the dynamics of connected particles through pulleys can be streamlined by clearly defining references, positions, and relationships between components.
• Further examples can be found in the provided links.