Dimensions of the following three quantities are the same: [PMT/NEET-1997]
a. Work, energy, force
b. Velocity, momentum, impulse
c. Potential energy, kinetic energy, momentum
d. Pressure, stress, coefficient of elasticity

Solution

• Work
  $
  W = F \cdot d = (MLT^{-2})(L) = ML^{2}T^{-2}
  $
• Energy
  Same as work:
  $
  E = ML^{2}T^{-2}
  $
• Force
  $
  F = MLT^{-2}
  $
• Velocity
  $
  v = LT^{-1}
  $
• Momentum
  $
  p = mv = M(LT^{-1}) = MLT^{-1}
  $
• Impulse
  $
  J = F \cdot t = (MLT^{-2})(T) = MLT^{-1}
  $
• Potential energy
  $
  PE = ML^{2}T^{-2}
  $
• Kinetic energy
  $
  KE = \tfrac{1}{2}mv^2 = M(LT^{-1})^{2} = ML^{2}T^{-2}
  $
• Momentum
  $
  p = MLT^{-1}
  $
• Pressure
  $
  P = \frac{F}{A} = \frac{MLT^{-2}}{L^{2}} = ML^{-1}T^{-2}
  $
• Stress
  Stress = Force/Area, so
  $
  = ML^{-1}T^{-2}
  $
• Coefficient of elasticity (Bulk modulus, Young’s modulus, etc.)
  $
  E = \frac{\text{Stress}}{\text{Strain}} = \frac{ML^{-1}T^{-2}}{1} = ML^{-1}T^{-2}
  $

All have same dimensions here.

Answer
d. Pressure, stress, coefficient of elasticity