NDA (I) 2017 General Ability Test · Question 108 of 149

Correct — B, Stress. Pressure is force divided by area, so its SI unit is the newton per square metre, given the name pascal. Stress is defined in exactly the same way — the internal restoring force set up in a deformed body, divided by the cross-sectional area over which it acts — so it is measured in newton per square metre as well, and shares the dimensions of pressure, ML⁻¹T⁻². The two ideas differ in physics rather than in units: pressure is exerted on a body from outside and acts along the normal to a surface, while stress is the body's own internal response and can be tensile, compressive or shearing depending on the direction of the load.

• (a)Angular momentum — Angular momentum is the moment of linear momentum, so it is measured in kilogram metre squared per second, the same unit as the joule second. That is nowhere near force per unit area.
• (c)Strain — Strain is a ratio of two lengths, or of two volumes — a change in dimension divided by the original dimension. Being a ratio of like quantities it has no unit at all, which is the opposite of what the question is asking for. Its constant companionship with stress is exactly why it is placed here.
• (d)Work — Work is force multiplied by displacement, so it is measured in newton metre, called the joule. Pressure divides force by an area instead of multiplying it by a distance, so the two units are quite different.

Pressure, stress and the elastic moduli all share one unit, the pascal, because all three are a force divided by an area. Strain sits alongside them without a unit, being a pure ratio. Young's modulus is stress divided by strain, and because the denominator carries no unit, the modulus is measured in pascal as well — which is why elastic constants, atmospheric pressure and the strength of a steel rod are all quoted in the same unit even though they describe very different things.

Read the printed stem as it stands — it says 'Which one of the following physical quantity has', where 'quantities have' would be the usual agreement; the slip is the booklet's and it changes nothing about the physics being asked. The method for an item like this is to write each option as a formula rather than to recall its unit. Force over area, force times distance, mass times velocity times distance, length over length — once written that way the odd one out is obvious, and you have also protected yourself against the strain trap, since a ratio of like quantities can never carry the unit of anything.

• The SI unit of pressure is the pascal, which is one newton per square metre.
• Stress is force per unit area, so it is measured in pascal and has the dimensions ML⁻¹T⁻².
• Strain is a ratio of like quantities and is therefore dimensionless and unitless.
• Work and energy are measured in joule, which is one newton metre; angular momentum is measured in kilogram metre squared per second.

• Choosing strain because it is always paired with stress — strain has no unit.
• Confusing the joule (force times distance) with the pascal (force divided by area).
• Assuming that quantities sharing a unit are the same quantity; pressure is external, stress is the body's internal response.

NDA asks which quantity shares a unit or a dimension with another, or picks out the one quantity in a list that is dimensionless.

• practice — not a real PYQ

  Which one of the following quantities is dimensionless?

  • (a)Stress
  • (b)Strain
  • (c)Pressure
  • (d)Young's modulus

  Answer(b) Strain — it is a ratio of two lengths or two volumes, so it has neither dimension nor unit, while the other three are all measured in pascal.

• practice — not a real PYQ

  The SI unit of Young's modulus is the same as that of

  • (a)force
  • (b)energy
  • (c)pressure
  • (d)power

  Answer(c) pressure — Young's modulus is stress divided by strain, and since strain has no unit the modulus keeps the pascal.