Constrained motion

🏗️ Motion of Connected Particles  

1. Introduction

Constrained motion arises when the motion of two or more particles is related through a geometric constraint, such as an inextensible rope or cable. Instead of analysing the motion of each particle independently, the constraint provides relationships between their positions, velocities, and accelerations.

In this example, two blocks are connected by an inextensible rope passing over a moving pulley. Rather than writing and differentiating the total rope-length equation, a more systematic approach is to analyse the mechanism one pulley at a time. This local approach extends naturally to more complicated pulley systems.

To establish the kinematic constraint for a pulley, consider two material points A and B located on either side of the pulley P. Since the rope segment between A and B is inextensible, its length remains constant throughout the motion. Consequently, any shortening of one segment must be accompanied by an equal lengthening of the other. This leads to the relative velocity constraint

(𝐯A − 𝐯P) · 𝐝A + (𝐯B − 𝐯P) · 𝐝B = 0

where 𝐝A and 𝐝B are unit vectors directed along the rope away from the pulley.

2. Animation – Relative Velocity Constraint

Objective: Visualize how the motion of two particles connected by an inextensible rope satisfies the velocity constraint.

Vectors Displayed:

• Velocity of particle A, 𝐯A

• Velocity of particle B, 𝐯B

• Velocity of the moving pulley 𝐯P

• Rope direction vectors 𝐝A and 𝐝B

Additional Feature: An inset illustrates the lengths of the rope segments AP and PB, together with their constant sum, highlighting the inextensibility of the rope.

Learning Objective: Understand the relative velocity constraint

(𝐯A − 𝐯P) · 𝐝A + (𝐯B − 𝐯P) · 𝐝B = 0

3. Kinematic Relations

4. Key Observations

• The total length of the rope remains constant.

• The rope direction changes continuously as the pulley moves.

• Only the velocity components along the rope contribute to the kinematic constraint.

• The constraint is formulated using relative velocities with respect to the moving pulley, making the method readily applicable to more complex pulley systems.

5. Watch For

• Observe how material points A and B move along the rope.

• Notice that the rope segments AP and PB change continuously while their total length remains constant.

• Compare the velocity vectors of the particles with that of the moving pulley.

• Verify that the projected relative velocities always satisfy the kinematic constraint.