About Mechanical Engineering
Mechanical engineering questions in a recruitment paper are the strength of materials, the mechanics of beams and levers, the flow of fluids and the basics of heat engines - each one a single relation applied to the numbers given. This topic covers all of those areas on one page, so that the subject can be practised as a whole rather than in fragments.
What you need to understand
- Direct stress is the load divided by the area carrying it: stress = F / A. With the load in newtons and the area in square millimetres the answer is in N/mm squared, which is the same as MPa.
- Strain is how much something stretches compared with how long it was: strain = change in length / original length. It has no units, and it is the original length that is divided by - not the new one.
- Young's modulus is the stiffness of a material: E = stress / strain. Steel is about 200 kN/mm squared and aluminium about 70, which is why a steel bar of the same size stretches far less.
- The moment of a force about a point is the force multiplied by the perpendicular distance from the point: M = F x d, in newton metres.
- A beam on two supports is in equilibrium, so the reactions at the supports add up to the total load, and the moments about either support balance.
- A point load at the centre of a simply supported beam gives half the load at each support. Away from the centre the nearer support takes more, in proportion to the load distance from the far support.
- The flow rate through a pipe does not change when the pipe narrows, so the area times the velocity is the same on both sides: A1 V1 = A2 V2. A smaller area therefore means a faster flow.
- Pressure in a fluid column is the density times gravity times the depth: p = rho g h. It depends only on the depth, not on the shape or the width of the vessel.
- Work is force times the distance moved along the direction of the force, and power is work divided by the time taken: P = F d / t.
- A heat engine takes heat from a hot reservoir, turns part of it into work and rejects the rest to a cold one. Its best possible efficiency is one minus the cold temperature over the hot, both in kelvin.
- Heat to change the temperature of a substance is Q = m c dT, where c is the specific heat capacity. No temperature change means no sensible heat, however much is supplied.
- In a gear or pulley train the speeds are in inverse proportion to the diameters or the numbers of teeth: the smaller wheel always turns faster.
How to work through these questions
- Write down the given quantities with their units, and identify the single relation that links them to what is asked. Most questions here need exactly one.
- Check the units before substituting. Stress and modulus are normally wanted in N/mm squared, so loads in kilonewtons have to become newtons and areas have to be in square millimetres.
- For anything to do with a beam, take moments about the support you are not trying to find. The other reaction then drops out of the equation.
- Check the answer against common sense: a reaction cannot exceed the total load, water speeds up in a narrower pipe rather than slowing down, and an efficiency has to be below 100 per cent.
- Where a temperature difference is involved, subtract the smaller from the larger and keep both in kelvin. A ratio of temperatures is only meaningful on the absolute scale.
Mistakes that cost marks
- Dividing the stress by the extension instead of by the strain. Strain is the extension as a fraction of the original length, so the original length belongs in the sum.
- Multiplying the load by the area to find a stress, or dividing the area by the load.
- Taking moments about the support you are trying to find, which leaves both unknowns in the equation.
- Assuming a point load always splits equally between two supports. That is true only when the load is at the centre.
- Putting the narrower area and the wider velocity on the same side of A1 V1 = A2 V2. The area and the velocity on each side belong to the same section.
- Forgetting to convert degrees Celsius to kelvin before working out an efficiency.
- Reporting work as power, or power as work. Dividing by the time is what turns one into the other.
Worked example
A steel bar of length 500 mm and cross-sectional area 200 mm squared carries an axial load of 40 kN. It stretches by 0.5 mm. What is the stress in the bar, and what is the modulus of elasticity of the steel?
- Turn the load into newtons: 40 kN = 40 000 N. A load left in kilonewtons divided by an area in square millimetres gives a stress a thousand times too small, which is the first trap in the question.
- Stress is the load over the area: 40 000 / 200 = 200 N/mm squared.
- Strain is the stretch as a fraction of the original length: 0.5 / 500 = 0.001. It is the original length that is divided by.
- Young's modulus is the stress over the strain: 200 / 0.001 = 200 000 N/mm squared.
- That is 200 kN/mm squared, the figure usually quoted for steel, so the answer is of the right size as well as correctly worked out.
Answer: Stress 200 N/mm squared, modulus of elasticity 200 000 N/mm squared
Practice questions with answers
A few Mechanical Engineering questions with the full solution shown, so you can see
how the method is applied before you attempt the timed set.
Question 1
A load of 12555 N acts on a bar of cross-sectional area 135 mm^2. What is the stress, in N/mm^2?
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A
1694925
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B
12690
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C
1/93
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D
93
Answer: Option D — with explanation
Direct stress is the load divided by the area it acts on: stress = F / A, and with the load in newtons and the area in square millimetres the answer comes out in N/mm^2, which is the same as MPa.
12555 / 135 = 93 N/mm^2. Multiplying the load by the area, or dividing the area by the load, are the two slips that produce the other numbers offered.
Common mistakes
- 1694925 is not the answer: 1694925
- 12690 is not the answer: 12690
- 1/93 is not the answer: 1/93
Question 2
A bar of length 400 mm stretches by 100 mm under a tensile stress of 25 N/mm^2. What is the modulus of elasticity, in N/mm^2?
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A
2500
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B
100
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C
10000
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D
200
Answer: Option B — with explanation
Young's modulus is stress divided by strain, and strain is the stretch as a fraction of the original length - not the stretch itself.
The strain is 100 / 400 = 1/4, so the modulus is 25 / 1/4 = 100 N/mm^2. Dividing the stress by the extension instead of by the strain is the mistake that gives 100.
Common mistakes
- 200 is not the answer: 200
- 10000 is not the answer: 10000
- 2500 is not the answer: 2500
Question 3
A beam 8 m long is simply supported at both ends. A point load of 6 kN acts at 4 m from the left-hand support. What is the reaction at the left-hand support, in kN?
Answer: Option C — with explanation
For a beam at rest, take moments about the right-hand support. The load times its distance from that support must equal the left reaction times the whole length.
The load is 4 m from the right support, so the reaction is 6 x 4 / 8 = 3 kN. Putting the load's distance from the left support into the same fraction gives 3 kN, which is the reaction at the other end - the two must add up to the load, and 3 + 3 = 6, so this one is right.
Common mistakes
- 8 is not the answer: 8
- 9 is not the answer: 9
- 6 is not the answer: 6
Question 4
Water flows at 4 m/s through a pipe whose cross-sectional area is 99 cm^2. The pipe then narrows to a cross-sectional area of 33 cm^2. What is the velocity in the narrower section, in m/s?
Answer: Option B — with explanation
The same volume of water passes each section every second, so the area times the velocity is the same on both sides of the narrowing: A1 V1 = A2 V2.
That gives V2 = 99 x 4 / 33 = 12 m/s. The water speeds up as the pipe narrows, so any answer smaller than 4 m/s has the ratio the wrong way up.
Common mistakes
- 24 is not the answer: 24
- 4/3 is not the answer: 4/3
- 4 is not the answer: 4
Question 5
A heat engine works between a source at 850 K and a sink at 450 K. What is the greatest efficiency it can have, as a percentage? Give the answer to two decimal places.
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A
800/17
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B
225/2
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C
900/17
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D
970/17
Answer: Option A — with explanation
The best any heat engine can do between two temperatures is the Carnot efficiency, one minus the sink temperature over the source temperature, with both temperatures in kelvin.
1 - 450/850 = 8/17, which as a percentage is 800/17%. Using the ratio the other way up gives 900/17%, which is the fraction of the heat that has to be thrown away, not the fraction turned into work.
Common mistakes
- 225/2 is not the answer: 225/2
- 900/17 is not the answer: 900/17
- 970/17 is not the answer: 970/17
Frequently asked questions
What is the difference between stress and pressure?
Both are a force spread over an area, and both are measured in the same units. Stress is the word used inside a material and pressure the word used for a fluid or a gas. The arithmetic is identical.
Why does the water speed up when a pipe narrows?
Because the same volume has to get through in the same time. The flow rate is the area times the velocity, so if the area falls the velocity has to rise to keep the product the same.
Do I have to use kelvin in an efficiency calculation?
Yes. The efficiency works out as a ratio of temperatures and the zero has to be absolute zero, otherwise the ratio is meaningless. Add 273 to a Celsius temperature first.
When is a point load shared equally between two supports?
Only when it acts at the mid-point. Anywhere else the nearer support takes the larger share, and that share is found by taking moments about the other support.
The question gives the diameter of a pipe rather than the area. What then?
Work the area out first, from pi d squared over four. Leave both areas in the same units so that the ratio is a pure number, and the velocity comes out in whatever unit the question gave it in.
