Wheel and Axle

Wheel and axle

A wheel and axle is a simple machine that consists of a wheel and an axle mounted with the same axis of rotation.

The radius of the wheel is always greater than that of the axle.

Consider the diagram below:

Mechanical advantage, velocity ratio and efficiency of a wheel and axle

Formulas:

Mechanical Advantage: $\text{M.A} = \dfrac{\text{Load}}{\text{Effort}}$

Velocity Ratio: $\text{V.R} = \dfrac{R}{r}$, where $R$ is the radius of the wheel and $r$ is the radius of the axle

Efficiency: $\eta = \dfrac{\text{M.A}}{\text{V.R}} \times 100\%$

Example 8

A wheel and axle of efficiency 80% is used to raise a load of 2000N. If the radius of the wheel is 50 cm, and that of the axle is 2 cm, calculate:

(i) The velocity ratio and mechanical advantage of the machine

(ii) The effort required to overcome the load.

Solution:

Given:

Efficiency $\eta = 80\%$

Load $= 2000\, \text{N}$

Radius of wheel $R = 50\, \text{cm}$

Radius of axle $r = 2\, \text{cm}$

Step 1: Calculate the Velocity Ratio (V.R)

$$\text{V.R} = \dfrac{R}{r} = \dfrac{50}{2} = 25$$

Step 2: Use efficiency to find Mechanical Advantage (M.A)

$$\eta = \dfrac{\text{M.A}}{\text{V.R}} \times 100\%$$ $$\dfrac{80}{100} = \dfrac{\text{M.A}}{25}$$ $$\text{M.A} = \dfrac{80}{100} \times 25 = 20$$

Step 3: Use M.A to calculate the Effort

$$\text{M.A} = \dfrac{\text{Load}}{\text{Effort}}$$ $$20 = \dfrac{2000}{\text{Effort}}$$ $$\text{Effort} = \dfrac{2000}{20} = 100\, \text{N}$$

Final Answers:

Velocity Ratio $= 25$

Mechanical Advantage $= 20$

Effort Required $= 100\, \text{N}$

Application of wheel and axle

1. Bicycles: The wheel and axle mechanism is used in bicycles, where the large wheel allows for easy motion over long distances, while the axle transfers the force to the wheel, enabling movement.
2. Washing Machines: In washing machines, the wheel and axle mechanism helps in the spinning of the drum. The motor turns the axle, which then spins the drum to clean clothes effectively.
3. Cars: Cars use wheels and axles to move. The axle transfers the rotational force from the engine to the wheels, allowing the car to move. The larger wheels make it easier to cover distance.
4. Cranes and Hoists: In cranes, a wheel and axle system is used to raise and lower heavy loads. A small effort applied to the wheel is multiplied by the axle, lifting large loads with ease.
5. Capstans on Ships: Capstans are used to raise anchors and pull ropes on ships. The large wheel allows sailors to apply a small effort over a long period of time, while the axle pulls the rope with a much higher force.
6. Gears in Machines: In machinery, wheels and axles (gears) are used to transfer motion and force. The larger wheel or gear turns more slowly but applies greater force, while the smaller gear turns faster but with less force.