Machines: Use Effort and Load Distances to Check MA, VR and Efficiency

Follow both the force and the movement

A machine can let a smaller effort raise a larger load, but the effort usually moves farther. To judge its performance, compare both parts of the trade-off. Mechanical advantage compares forces; velocity ratio compares movements; efficiency compares useful work with work supplied.

The examples use steady lifting with constant force magnitudes, movement along each force’s line of action, and no release of previously stored energy. They are idealised calculations, not load ratings or advice for constructing lifting equipment.

1. Label load, effort and their distances

The load L is the resisting force associated with the useful task, measured here in newtons. The effort E is the input force. Mechanical advantage MA = L/E. It is a ratio of like units, so MA has no unit. MA = 3 means the load force is three times the effort force in that stated operation.

The effort distance d_E is how far the effort point travels. The load distance d_L is how far the load travels during the same motion. Velocity ratio VR = d_E/d_L. The traditional name refers to the corresponding speed ratio when the movement relationship is constant; it does not mean divide whichever two velocities happen to appear in a question.

Do not mix load mass with effort force. If a problem gives a 12 kg load and g = 10 N/kg, its weight is 120 N. That conversion belongs before MA is calculated. It is unnecessary when the load is already stated in newtons.

2. Derive the efficiency shortcut

Under the steady-force assumptions, useful output work = Ld_L and input work = Ed_E. Dividing gives efficiency as a fraction = (L/E)/(d_E/d_L) = MA/VR. Therefore percentage efficiency = (MA/VR) × 100%.

This derivation explains why the two efficiency methods should agree. It also explains why a force advantage is not free energy: a smaller effort can act through a longer distance. In an ideal lossless version of this model, MA = VR and efficiency is 100%. In a real dissipative machine performing the same defined task, MA is smaller than VR.

Worked case 1: check one machine by two routes

A machine raises a 360 N load through 0.60 m. The effort is 120 N and its point of application travels 2.40 m. In this case, the machine itself has no net change in stored energy between the endpoints. Find MA, VR and efficiency, then account for the work difference.

Force and distance route: MA = 360/120 = 3.0. VR = 2.40/0.60 = 4.0. Efficiency = (3.0/4.0) × 100% = 75%.

Work route: Output = 360 × 0.60 = 216 J. Input = 120 × 2.40 = 288 J. Efficiency = (216/288) × 100% = 75%, agreeing with the ratio route. The difference is 288 − 216 = 72 J.

Under the stated steady-operation assumptions, that 72 J is energy transferred into non-useful forms for this task, such as heating due to friction. It has not disappeared. A high effort force alone would not reveal efficiency; the input travel is equally necessary.

3. Read pulley movement from the rope path

Start with the attachment points and ask which rope lengths change when the load moves. The diagrams below use an effectively inextensible rope, no slip and vertical straight segments. Wheel rotation and rope direction can change without adding another supporting segment to the moving load.

Native diagram alternative, fixed pulley: A wheel is attached to an overhead support and its centre does not move. A single rope passes over it. One end is attached to the load on the left; the other is the free effort end on the right. Pulling the effort end downward by 0.30 m raises the load by 0.30 m. Thus d_E = d_L and VR = 1. The pulley redirects the effort.

Native diagram alternative, single movable pulley: The left rope end is fixed to an overhead support. The rope descends around a wheel attached to the load and rises to the free effort end on the right. That free end is pulled upward. If the free end were held at its original height while the wheel rose 0.30 m, each supporting span would shorten by 0.30 m. To keep the rope taut and its total length unchanged, the free end instead moves upward by 0.60 m. Thus VR = 2. There is no extra fixed redirecting wheel in this drawing.

Count moving rope spans, not pictures of wheels. In this simple movable arrangement, two spans support the moving assembly. In the fixed arrangement, the wheel’s support carries its axle and only one rope end is connected to the load. Different attachment paths can produce different ratios even when the number of wheels looks similar.

Worked case 2: a real movable-pulley result

For the single movable arrangement, a useful load of 180 N rises 0.25 m while the effort end moves 0.50 m. The measured effort is 120 N. Any pulley weight and friction are included in the measured input; useful output here means the load’s gain only.

Solution: VR = 0.50/0.25 = 2.0. MA = 180/120 = 1.5. Efficiency = (1.5/2.0) × 100% = 75%. Output work = 180 × 0.25 = 45 J; input work = 120 × 0.50 = 60 J; their ratio confirms 75%.

The two supporting spans establish the displacement ratio under the rope assumptions. They do not guarantee actual MA = 2. That force result would require the additional idealisations of negligible pulley weight and no losses, or a carefully defined load that includes the moving assembly.

Worked case 3: large MA does not prove high efficiency

Machine A has MA = 4 and VR = 8. Machine B has MA = 3 and VR = 4. Which has greater mechanical advantage, and which is more efficient?

Solution: A has greater MA because 4 exceeds 3. However, A’s efficiency is (4/8) × 100% = 50%, while B’s is (3/4) × 100% = 75%. B is more efficient. The two questions concern different comparisons, so they need not have the same winner.

For an equal load, A requires less effort force than B, but A’s effort point travels farther per unit load rise. Neither ratio says how quickly the work is performed; a statement about power would also require time. This prevents force, work and power from being used as interchangeable descriptions.

4. Test an impossible-looking answer

If a passive machine in this steady model gives efficiency above 100%, first inspect the data and the definition of useful output. Common errors include reversing VR, using centimetres in one distance and metres in the other, treating a load mass as a force, or omitting an energy input.

A released spring or descending counterweight can supply previously stored energy. If such a source is present but omitted from input bookkeeping, an apparent efficiency above 100% does not establish energy creation. The exercises here explicitly exclude that complication. A ratio formula should be used only after its assumptions fit the problem.

Three tasks with complete solutions

Task 1: calculate all three ratios

A 250 N load rises 0.40 m when a 100 N effort moves 1.60 m. Find MA, VR, input work, output work and efficiency.

Solution: MA = 250/100 = 2.5; VR = 1.60/0.40 = 4.0. Input = 100 × 1.60 = 160 J; output = 250 × 0.40 = 100 J. Efficiency = 100/160 × 100% = 62.5%, also equal to 2.5/4 × 100%.

Task 2: find the required effort

A machine has VR = 5 and efficiency 80%. It raises a 400 N load steadily. Find MA and effort; then check the work for a 0.20 m rise.

Solution: As a fraction, efficiency is 0.80, so MA = 0.80 × 5 = 4. Effort = L/MA = 400/4 = 100 N. Effort distance = 5 × 0.20 = 1.00 m. Output = 80 J; input = 100 J, confirming 80%.

Task 3: a fixed-pulley check

A fixed pulley raises a 90 N load by 0.35 m while a 100 N effort moves 0.35 m. Calculate the efficiency and explain the machine’s useful role.

Solution: VR = 1, MA = 90/100 = 0.90 and efficiency = 90%. Output = 31.5 J and input = 35 J. The pulley changes the direction of effort even though this real arrangement has no force advantage. “Useful” does not necessarily mean MA is greater than one.

Frequently asked questions

Do MA and VR have units?

No. Each is a ratio of quantities with the same units. Convert the two forces or the two distances to compatible units before dividing.

Why is VR based on distances?

The effort and load movements reveal the geometrical trade-off. For a constant movement relationship, their distance ratio over the same interval equals their speed ratio.

Does a large MA guarantee high efficiency?

No. Efficiency depends on MA divided by VR under the stated work assumptions. A high force advantage can be accompanied by a much larger effort distance.

Does a fixed pulley always save effort force?

No. A single ideal fixed pulley has VR = 1 and redirects the force. Friction can make the required effort larger than the useful load force.

Why not count pulley wheels to find VR?

Rope attachments and which spans change length determine the movement relationship. Wheel count alone does not specify how far the effort and load move.

Can efficiency be above 100% in these examples?

No. The model includes all energy input and no release of stored energy. A larger result requires checking units, reversed ratios, omitted inputs or incorrect data.

Is there one efficiency value for every machine?

No. Efficiency depends on the machine, its operating conditions and the defined useful task. Use the supplied forces and distances rather than assuming a universal percentage.

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Browse the Physics topical revision collection; individual resource access may require membership. Reference checks: OpenStax on force advantage in simple machines and input work, output work and efficiency. The datasets and diagrams here are original.