Determine the clearance distance d between two rectangles based on the rotation angle θ in a clevis joint | Step-by-Step Solution

Clevis Joint Problem 🔧

What We're Solving:

We need to find the clearance distance `d` between two rectangles in a clevis joint when one rotates by angle `θ`. This is crucial for mechanical design to prevent binding or interference!

The Approach:

Think of this like a door hinge - when the "door" (rectangle B) swings, it needs enough clearance so it doesn't hit the "frame" (rectangle A). We'll use geometry to track how point P moves when rectangle B rotates, then find where the interference occurs.

Step-by-Step Solution:

Step 1: Visualize the Setup

• Rectangle A is stationary (our reference)
• Rectangle B rotates about a pivot point by angle θ
• Point P is at height h/2 from the edges of both rectangles
• We need to find distance `d` where they would intersect

Step 2: Understand the Rotation When rectangle B rotates by angle θ:

• Point P moves in a circular arc
• The key insight: we need to track how far point P "swings out" horizontally from its original position

Step 3: Apply Trigonometry

• Original position of P: at distance h/2 from the edge
• After rotation by θ: P moves to a new position
• The horizontal displacement of P = (h/2) × sin(θ)
• This is because sin(θ) gives us the horizontal component of the circular motion

Step 4: Calculate the Clearance The clearance distance d must account for:

• The original spacing
• Plus the additional space needed due to rotation

Therefore: d = (h/2) × sin(θ)

The Answer:

The clearance distance is: d = (h/2) sin(θ)

This makes intuitive sense:

• When θ = 0° (no rotation), sin(0°) = 0, so d = 0 (no extra clearance needed)
• When θ = 90° (quarter turn), sin(90°) = 1, so d = h/2 (maximum clearance)

Memory Tip:

Remember "Sine for Sideways motion" - when something rotates, sine gives you the sideways (horizontal) displacement component. The bigger the angle, the more clearance you need! 🎯

Great job tackling this mechanical design problem! Understanding these clearance calculations is essential for creating joints that move smoothly without binding.