Pulleys

Theoretical Mechanical Advantage

The theoretical mechanical advantage (TMA) is the ratio of the useful load force to the applied effort, assuming no friction:

\[ \text{TMA} = \frac{\text{Load}}{\text{Effort}} \]

Velocity Ratio

Velocity ratio (VR) is the ratio of the distance moved by the effort to the distance moved by the load:

\[ \text{VR} = \frac{\text{Distance moved by effort}}{\text{Distance moved by load}} \]

Work Done

The work done by a constant force \(F\) acting through a displacement \(s\) is:

\[ W = Fs \]

One Wheel

Single fixed pulley diagram

A single fixed pulley changes the direction of the force but does not reduce the effort.

Single pulley force direction change

Pulling down 1 m of rope raises the load by 1 m.

The velocity ratio is:

\[ \text{VR} = 1 \]

Single pulley VR demonstration

Loaded Pulley

Loaded pulley diagram

The tension \(T\) is the same throughout the string.

Tension in pulley rope

What happens next depends on the values of M₁ and M₂

Case 1: \(M_1 > M_2\)

Pulley with M1 heavier than M2

Mass \(M_1\) moves down, pulling \(M_2\) up.

This causes downwards acceleration of load \(M_1\) and upwards acceleration of load \(M_2\)

\[ F = ma \] \[ M_1 g - T = M_1 a \] \[ T - M_2 g = M_2 a \]

\[ \text{Adding} \] \[ M_1 g - M_2 g = M_1 a + M_2 a \] \[ a = \frac{M_1 g - M_2 g}{M_1 + M_2} \] \[ a = \frac{g\,(M_1 - M_2)}{M_1 + M_2} \]

Case 2: \(M_2 > M_1\)

Pulley with M2 heavier than M1

Mass \(M_2\) moves down, pulling \(M_1\) up.

This causes downwards acceleration \( a_2 \) and upwards acceleration \( a_1 \)

\[ T - M_1 g = M_1 a \] \[ M_2 g - T = M_2 a \] \[ \text{Adding} \] \[ M_2 g - M_1 g = M_1 a + M_2 a \] \[ a = \frac{M_2 g - M_1 g}{M_1 + M_2} \] \[ a = \frac{g\,(M_2 - M_1)}{M_1 + M_2} \]

Example

Find the downward acceleration of the block:

Example pulley problem diagram

\[ a = \frac{(M_1 - M_2)g}{M_1 + M_2} \]

\[ a = \frac{g(3 - 1)}{3 + 1} \] \[ = \frac{2g}{4} \] \[ = \frac{g}{2} \]

Alternatively, from the diagram:

\[ 3g - T = 3a \] \[ T - g = a \] \[ \text{Adding} \] \[ 3g - g = 3a + a \] \[ a = \frac{3g - g}{3 + 1} \] \[ a = \frac{2g}{4} = \frac{g}{2} \]

Two Wheels

Two-wheel pulley system

Pulling down 1 m of rope raises the load by 0.5 m.

Two-wheel pulley VR demonstration

\[ \text{VR} = 2 \] \[ \text{Effort} = \frac{1}{2}\text{Load} \]

Effort reduction in two-wheel pulley

Youtube example

If a pulley system is perfectly efficient, the mechanical advantage and velocity ratio equal the number of pulleys.

Example

Calculate the acceleration of the load and the tension in the system.

Four-pulley example diagram

\[ 3g - T = 3a \] \[ 2T - 2g = 2a \] \[ \text{So} \] \[ 6g - 2T = 6a \] \[ 2T - 2g = 2a \] \[ \text{Adding } \] \[ 4g = 8a \] \[ a = \frac{4g}{8} \] \[ a = \frac{g}{2}\ \text{ms}^{-2} \]

\[ 3g - T = 3a \] \[ T = 3g - 3a \] \[ \text{substitute } a = \frac{g}{2} \] \[ T = 3g - 3\left(\frac{g}{2}\right) \] \[ = \frac{6g - 3g}{2} \] \[ = \frac{3g}{2}\,N \]

Three Wheels

Three-wheel pulley system

\[ \text{VR} = 3 \]

\[ M_1 g - T = M_1 a_1 \] \[ 2T - xg = x\left(\frac{a_1 + a_2}{2}\right) \] \[ M_2 g - T = M_2 a_2 \]

Example

Calculate the acceleration of the moveable pulley and the tension in the rope.

Three-wheel example diagram

\[ 3g - T = 3a_1 \] \[ 2T - 4g = 4\left(\frac{a_1 + a_2}{2}\right) \] \[ 3g - T = 3a_2 \]

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\[ 2T - 4g = 2a_1 + 2a_2 \] \[ \text{Multiply first equation by 2} \] \[ 6g - 2T = 6a_1 \] \[ \text{Adding} \] \[ 2g = 8a_1 + 2a_2 \]

\[ 2T - 4g = 2a_1 + 2a_2 \] \[ 3g - T = 3a_2 \] \[ \text{Multiply second equation by 2} \] \[ 6g - 2T = 6a_2 \] \[ \text{Adding} \] \[ 2g = 2a_1 + 8a_2 \]

\[ 2g = 8a_1 + 2a_2 \] \[ 2g = 2a_1 + 8a_2 \] \[ \text{subtracting} \] \[ -6g = -30a_2 \] \[ a_2 = \frac{1}{5}g \]

\[ \text{substitute } a_2 = \frac{1}{5}g \] \[ 2g = 8a_1 + \frac{2}{5}g \] \[ \frac{8}{5}g = 8a_1 \] \[ a_1 = \frac{1}{5}g \]

\[ a_{\text{pulley}} = \frac{a_1 + a_2}{2} \] \[ = \frac{\frac{1}{5}g + \frac{1}{5}g}{2} \] \[ = \frac{1}{5}g \]

\[ 3g - T = 3a_2 \] \[ T = 3g - 3a_2 \] \[ = 3g - \frac{3}{5}g \] \[ T = \frac{12}{5}g \]