Simple Machines – Mechanical Advantage and Velocity Ratio

Simple machines transmit or modify force and motion to perform mechanical work. They can provide force multiplication, change the direction of an applied force, or exchange force for movement.

This page summarises the mechanical advantage, velocity ratio and ideal relationships for the six traditional simple machines: the inclined plane, wedge, lever, wheel and axle, pulley and screw.

Contents

• Mechanical Advantage, Velocity Ratio and Efficiency
• Inclined Plane
• Wedge
• Lever
• Wheel and Axle
• Pulley Systems
• Screw

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Simple Machine Principles

A simple machine is an idealised mechanical device that transmits or modifies force and motion. An applied effort produces an output force that acts against a load or resistance.

The advantage of a machine is not the creation of energy. Ideally, a reduction in the required input force is obtained at the expense of a greater input movement. Conversely, a mechanism may increase output movement or speed at the expense of output force.

The six traditional simple machines are:

1. Inclined plane
2. Wedge
3. Lever
4. Wheel and axle
5. Pulley
6. Screw

Mechanical Advantage

The actual mechanical advantage of a machine is the ratio of the output load or resistance W to the applied effort F:

$$\mathrm{MA}=\frac{W}{F}$$

A mechanical advantage greater than one indicates force multiplication.

Velocity Ratio

The velocity ratio is the ratio of the distance moved by the effort to the corresponding distance moved by the load:

$$\mathrm{VR}=\frac{\text{distance moved by effort}}{\text{distance moved by load}}$$

Equivalently, for compatible instantaneous velocities:

$$\mathrm{VR}=\frac{v_{\text{effort}}}{v_{\text{load}}}$$

Velocity ratio is determined by the geometry and kinematics of the machine. For an ideal, rigid, frictionless machine:

$$\mathrm{MA}=\mathrm{VR}$$

Efficiency

Machine efficiency is the ratio of useful output work to input work:

$$\eta =\frac{\text{useful output work}}{\text{input work}}$$

For a machine operating through corresponding effort and load movements:

$$\eta =\frac{\mathrm{MA}}{\mathrm{VR}}$$

or, as a percentage:

$$\eta \text{ (\%)}=\frac{\mathrm{MA}}{\mathrm{VR}}\times 100$$

Real machines have mechanical advantage below the ideal value because energy is dissipated by friction, deformation and other losses.

Law of a Machine

For some real machines, over a useful operating range, the relationship between effort and load can be represented empirically by the linear law of the machine:

$$F=a+bW$$

where F is the effort, W is the load, and a and b are constants determined experimentally for the particular machine and operating conditions.

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Inclined Plane

An inclined plane allows a load to be raised through a vertical distance by moving it through a greater distance along a slope.

For an incline at angle θ to the horizontal, movement through a distance x along the slope produces a vertical rise:

$$h=x\sin \theta$$

Therefore the velocity ratio is:

$$\mathrm{VR}=\frac{x}{h}=\frac{1}{\sin \theta }$$

For a frictionless incline with the effort applied parallel to the slope, equilibrium or constant-speed motion requires:

$$F=W\sin \theta$$

and hence:

$$\mathrm{MA}=\mathrm{VR}=\frac{1}{\sin \theta }$$

With friction present, a greater effort is required and the actual mechanical advantage is lower than the ideal value.

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Wedge

A wedge is effectively a moving inclined plane. An input movement along the length of the wedge produces a smaller transverse movement and can therefore generate a larger transverse force.

Single Wedge

For a single wedge with wedge angle θ as shown:

$$\mathrm{VR}=\frac{1}{\tan \theta }$$

Symmetrical Double Wedge

For a symmetrical double wedge having total included angle θ, each face is inclined at θ/2 to the centreline. If the output displacement being considered is the total separation of the two opposing surfaces:

$$\mathrm{VR}=\frac{1}{2\tan \left(\frac{\theta }{2}\right)}$$

Friction is particularly important in practical wedges. It reduces mechanical advantage and may make a wedge self-locking under suitable geometry and friction conditions.

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Lever

A lever is a rigid member that rotates about a fulcrum. Its ideal mechanical advantage is determined by the perpendicular moment arms of the effort and load about the fulcrum.

For equilibrium of an ideal lever:

$$F{L}_F=W{L}_W$$

where L_F and L_W are the perpendicular moment arms of the effort and load respectively.

Therefore:

$$\mathrm{MA}=\mathrm{VR}=\frac{L_F}{L_W}$$

Lever Classes

Using dimensions A and B as shown in the original diagram:

A third-class lever has a mechanical advantage below one in the ideal case, but provides increased output displacement and velocity.

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Wheel and Axle

A wheel and axle consists of two effective radii rotating together about a common axis. If the effort acts tangentially at radius R and the load acts at radius r:

$$\mathrm{VR}=\frac{R}{r}$$

For the ideal machine:

$$\mathrm{MA}=\frac{R}{r}$$

This follows directly from equality of moments:

$$FR=Wr$$

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Pulley Systems

Block and Tackle

A pulley system can change the direction of an applied force and, when moving pulleys are used, provide force multiplication.

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Screw

A screw can be regarded kinematically as an inclined plane wrapped around a cylinder. Rotation of the screw converts circumferential movement into axial movement.

If the effort is applied at radius R, one complete revolution moves the effort through:

$$s_{\text{effort}}=2\pi R$$

If the screw advances axially by lead L during one revolution, the velocity ratio is:

$$\mathrm{VR}=\frac{2\pi R}{L}$$

For a single-start screw, the lead equals the thread pitch p, giving:

$$\mathrm{VR}=\frac{2\pi R}{p}$$

For a multi-start screw:

$$L=np$$

where n is the number of thread starts.

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Related RoyMech Mechanics References

• Vectors
• Statics
• Kinematics
• Kinetics and Dynamics
• Mechanical Linkages
• Component Loading
• Basic Strain Theory
• Failure Modes
• Buckling Notes
• Mohr's Circle
• Thick Cylinders
• Rotating Discs and Cylinders
• Torsion
• Stress-Strain Curve
• Loaded Flat Plates

Related RoyMech Machine-Element References

• Machine Element Selection Guide
• Power Screws
• Mechanical Mechanisms
• Cam and Follower Mechanisms

External Simple Machine References

1. Simple Machines in Appliances, Gadgets, and Everyday Life – examples of simple machines in practical devices.
2. Simple Machines – University of Arkansas – introductory reference material.
3. Machines – J. B. Calvert – detailed notes on machines and mechanical principles.
4. Simple Machine – Wikipedia – general reference material and further links.

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