CDS (I) 2020 General Knowledge · Question 13 of 120

Correct — B, N/m. The force constant of a spring comes from Hooke's law, which says the restoring force is proportional to the extension: F equals k times x. Rearranged, k equals F divided by x, so its unit is the unit of force divided by the unit of length — newton per metre. Read physically, k is the force needed to stretch or compress the spring by one metre, so a stiff spring has a large k and a slack one a small k. Every unit question of this kind is settled the same way: write the defining equation, make the wanted quantity the subject, and substitute units for symbols.

• (a)N-m — Newton times metre is force multiplied by distance, which is work or energy — the joule — and also the unit of torque. It comes from multiplying where the definition calls for division.
• (c)N-m² — Newton metre squared appears in constants such as the gravitational constant, written N m² per kg squared. Nothing in Hooke's law produces a squared length.
• (d)N/m² — Newton per square metre is the pascal, the unit of pressure and of stress. The area in the denominator is the giveaway; a spring constant involves a length, not an area.

Hooke's law states that the force needed to extend or compress a spring by some distance scales linearly with that distance, F equal to k times x, where k is the spring or force constant. The law holds up to the elastic limit of the material. The energy stored in a stretched spring is one half k x squared, and a mass m on a spring oscillates with a period of two pi times the square root of m over k.

All four options are built from newtons and metres, so no amount of recall about springs will separate them — the discrimination comes from the algebra. Force divided by length gives newton per metre, and only option (b) has the length in the denominator to the first power. It helps to have the three neighbouring units labelled in your head, because they are the standing distractors in every question of this type: newton metre is energy or torque, newton per square metre is pressure, and newton per metre is a force constant. There is a fourth member of that family worth adding — newton per metre is also the unit of surface tension, for the same dimensional reason.

• Hooke's law gives F equal to k times x, so k equals force divided by extension.
• The SI unit of the spring constant is the newton per metre, N/m.
• Newton metre is the joule, the unit of work and energy, and also the unit of torque.
• Newton per square metre is the pascal, the unit of pressure and of stress.
• Potential energy stored in a spring stretched by x is one half k x squared.

• Multiplying instead of dividing and landing on the newton metre.
• Confusing newton per metre with newton per square metre; one is a force constant or a surface tension, the other a pressure.
• Assuming a stiffer spring has a smaller constant; k rises with stiffness.

As the unit of a named quantity, or in reverse — which physical quantity has the unit newton per metre.

• practice — not a real PYQ

  A force of 20 N stretches a spring by 0.5 m. The force constant of the spring is

  • (a)10 N/m
  • (b)20 N/m
  • (c)40 N/m
  • (d)100 N/m

  Answer(c) 40 N/m — k equals F divided by x, so 20 divided by 0.5.

• practice — not a real PYQ

  Which one of the following physical quantities has the same unit as the force constant of a spring?

  • (a)Pressure
  • (b)Surface tension
  • (c)Torque
  • (d)Work

  Answer(b) Surface tension — force per unit length, newton per metre, the same combination as a spring constant.