Question

Which of the following pairs has the same dimensions?

• A. specific heat and latent heat
• B. Impulse and momentum
• C. surface tension and force
• D. moment of Inertia and torque

Answer 1

Answer: (B)

B) Impulse and momentum

Answer 2

To determine which pair of quantities has the same dimensions, we need to find the dimensional formula for each quantity and then compare them.

Let's analyze each option:

A) Specific heat and latent heat

• Specific heat ($c$): It is defined by the formula $Q = mc\Delta T$, where $Q$ is heat energy, $m$ is mass, and $\Delta T$ is change in temperature. So, $c = \frac{Q}{m\Delta T}$. The dimensions are:

  • Heat energy ($Q$) = $[ML^2T^{-2}]$ (same as work or energy)
  • Mass ($m$) = $[M]$
  • Temperature ($\Delta T$) = $[K]$ (Kelvin) Dimension of specific heat = $\frac{[ML^2T^{-2}]}{[M][K]} = [L^2T^{-2}K^{-1}]$

• Latent heat ($L$): It is defined by the formula $Q = mL$, where $Q$ is heat energy and $m$ is mass. So, $L = \frac{Q}{m}$. The dimensions are:

  • Heat energy ($Q$) = $[ML^2T^{-2}]$
  • Mass ($m$) = $[M]$ Dimension of latent heat = $\frac{[ML^2T^{-2}]}{[M]} = [L^2T^{-2}]$

• Comparison: $[L^2T^{-2}K^{-1}]$ vs $[L^2T^{-2}]$. These dimensions are different.

B) Impulse and momentum

• Impulse ($J$): It is defined as the product of force ($F$) and the time interval ($\Delta t$) for which the force acts. Formula: $J = F\Delta t$. The dimensions are:

  • Force ($F$) = $[MLT^{-2}]$
  • Time ($\Delta t$) = $[T]$ Dimension of impulse = $[MLT^{-2}][T] = [MLT^{-1}]$

• Momentum ($p$): It is defined as the product of mass ($m$) and velocity ($v$). Formula: $p = mv$. The dimensions are:

  • Mass ($m$) = $[M]$
  • Velocity ($v$) = $[LT^{-1}]$ Dimension of momentum = $[M][LT^{-1}] = [MLT^{-1}]$

• Comparison: $[MLT^{-1}]$ vs $[MLT^{-1}]$. These dimensions are the same.

C) Surface tension and force

• Surface tension ($\gamma$): It is defined as force ($F$) per unit length ($L$) acting tangentially on the surface of a liquid. Formula: $\gamma = \frac{F}{L}$. The dimensions are:

  • Force ($F$) = $[MLT^{-2}]$
  • Length ($L$) = $[L]$ Dimension of surface tension = $\frac{[MLT^{-2}]}{[L]} = [MT^{-2}]$

• Force ($F$): It is defined as mass ($m$) times acceleration ($a$). Formula: $F = ma$. Dimension of force = $[M][LT^{-2}] = [MLT^{-2}]$

• Comparison: $[MT^{-2}]$ vs $[MLT^{-2}]$. These dimensions are different.

D) Moment of Inertia and torque

• Moment of Inertia ($I$): For a point mass, it is defined as mass ($m$) times the square of the distance ($r$) from the axis of rotation. Formula: $I = mr^2$. The dimensions are:

  • Mass ($m$) = $[M]$
  • Distance ($r$) = $[L]$ Dimension of moment of inertia = $[M][L^2] = [ML^2]$

• Torque ($\tau$): It is defined as the product of force ($F$) and the perpendicular distance ($r$) from the axis of rotation to the line of action of the force. Formula: $\tau = Fr$. The dimensions are:

  • Force ($F$) = $[MLT^{-2}]$
  • Distance ($r$) = $[L]$ Dimension of torque = $[MLT^{-2}][L] = [ML^2T^{-2}]$

• Comparison: $[ML^2]$ vs $[ML^2T^{-2}]$. These dimensions are different.

Based on the analysis, only Impulse and Momentum have the same dimensions.