Explain the concept of velocity ratio in a pulley system and derive its formula.

A pulley system allows you to lift a load by applying effort. In a system with multiple pulleys, the distance the effort moves increases in a pattern of powers of two for each additional pulley, making it easier to lift heavy loads.

Let $X$ be the distance the load moves.

• For a system with one pulley, the effort moves $X$.
• For a system with two pulleys, the effort moves $2X$.
• For a system with three pulleys, the effort moves $2(2X) = 2^2X$.
• For a system with four pulleys, the effort moves $2(2^2X) = 2^3X$. This pattern continues, meaning for a system with $n$ pulleys, the distance moved by the effort is $2^n X$.

The Velocity Ratio (VR) is the ratio of the distance moved by the effort to the distance moved by the load.

For a pulley system with $n$ pulleys, where the effort moves $2^n X$ and the load moves $X$:

Canceling $X$ from the numerator and denominator:

So, the final formula for the Velocity Ratio in a pulley system with $n$ pulleys is:

Mechanical Advantage (MA) is the ratio of the load lifted to the effort applied.

For an ideal pulley system (one with no friction or energy loss), the Mechanical Advantage is equal to the Velocity Ratio.
Therefore, for an ideal pulley system:

Efficiency ($\eta$) is a measure of how effectively a machine converts input work into output work. It is calculated as the ratio of Mechanical Advantage to Velocity Ratio.