Velocity ratio (VR) of an ideal inclined plane: For lifting a load along a smooth plane inclined at angle θ to the horizontal, what is the velocity ratio?

Introduction / Context:
Velocity ratio (VR) is a measure of kinematic advantage in a machine: VR = distance moved by effort / distance moved by load. For an ideal (frictionless) inclined plane, VR links the plane geometry to mechanical advantage and effort calculations.

Given Data / Assumptions:

• Smooth (frictionless) plane inclined at angle θ.
• Load moves a vertical height h while effort moves along the plane length l.
• Effort is applied parallel to the plane.

Concept / Approach:
For an inclined plane, l and h are related by sin θ = h/l. Therefore VR = distance moved by effort / distance moved by load = l / h = 1 / sin θ = cosec θ. This result is purely geometric and independent of load magnitude (for an ideal plane).

Step-by-Step Solution:

Verification / Alternative check:
Mechanical advantage MA (ideal) equals VR. For θ = 30°, VR = 2, matching the familiar result that a 30° plane halves the required ideal effort (ignoring friction).

Why Other Options Are Wrong:

• sin θ, cos θ, tan θ, sec θ: do not match l/h for the plane’s rise and run.

Common Pitfalls:
Using cos θ instead of sin θ (confusing run and rise), or mixing VR with mechanical advantage in the presence of friction (MA < VR with friction).

Final Answer:
cosec θ