Dimensions of Physical Quantities

Understanding the dimensions of physical quantities is crucial in physics. Dimensions refer to the fundamental physical quantities (like mass [M], length [L], time [T], etc.) that make up a derived quantity. Dimensional analysis helps verify the correctness of equations and understand relationships between physical variables.

Analyzing Pairs of Quantities

We need to find the pair of physical quantities from the given options that do not have identical dimensions.

Option 1: Momentum and Impulse

• Momentum ($p$): Momentum is defined as the product of mass ($m$) and velocity ($v$).
  • Formula: $p = mv$
  • Dimensions Calculation: $[p] = [M] \times [L][T]^{-1} = [M][L][T]^{-1}$
• Impulse ($J$): Impulse is defined as the change in momentum, or the product of average force ($F$) and the time interval ($\Delta t$) over which the force acts.
  • Formula: $J = F \Delta t$
  • Dimensions Calculation: Since Force $F = ma$, its dimensions are $[M][L][T]^{-2}$. Therefore, $[J] = [M][L][T]^{-2} \times [T] = [M][L][T]^{-1}$.
• Comparison: The dimensions of momentum and impulse are both $[M][L][T]^{-1}$. They are identical.

Option 2: Torque and Energy

• Torque ($\tau$): Torque is the rotational equivalent of linear force. It's calculated as the product of force ($F$) and the perpendicular distance ($r$) from the pivot point to the line of action of the force.
  • Formula: $\tau = F \times r$
  • Dimensions Calculation: $[F] = [M][L][T]^{-2}$ and $[r] = [L]$. So, $[\tau] = [M][L][T]^{-2} \times [L] = [M][L]^2[T]^{-2}$.
• Energy ($E$): Energy, whether kinetic or potential, represents the capacity to do work. For example, Kinetic Energy ($KE$) is given by $KE = \frac{1}{2}mv^2$.
  • Formula: $KE = \frac{1}{2}mv^2$
  • Dimensions Calculation: $[m] = [M]$ and $[v] = [L][T]^{-1}$. So, $[KE] = [M] \times ([L][T]^{-1})^2 = [M][L]^2[T]^{-2}$.
• Comparison: The dimensions of torque and energy are both $[M][L]^2[T]^{-2}$. They are identical.

Option 3: Kinetic energy and potential energy

• Kinetic Energy ($KE$): As shown above, its dimensions are $[M][L]^2[T]^{-2}$.
• Potential Energy ($PE$): Potential energy is stored energy due to position or configuration. For example, gravitational potential energy ($PE$) is $PE = mgh$, where $m$ is mass, $g$ is acceleration due to gravity, and $h$ is height.
  • Formula: $PE = mgh$
  • Dimensions Calculation: $[m] = [M]$, $[g] = [L][T]^{-2}$, and $[h] = [L]$. So, $[PE] = [M] \times [L][T]^{-2} \times [L] = [M][L]^2[T]^{-2}$.
• Comparison: The dimensions of kinetic energy and potential energy are both $[M][L]^2[T]^{-2}$. They are identical.

Option 4: Moment of a force and angular momentum

• Moment of a force: This is another term for torque, which we calculated earlier. Its dimensions are $[M][L]^2[T]^{-2}$.
• Angular momentum ($L$): Angular momentum is the measure of the rotation of a body, calculated as the product of moment of inertia ($I$) and angular velocity ($\omega$).
  • Formula: $L = I \omega$
  • Dimensions Calculation: Moment of inertia ($I = mr^2$) has dimensions $[M][L]^2$. Angular velocity ($\omega$) has dimensions $[T]^{-1}$. So, $[L] = [M][L]^2 \times [T]^{-1} = [M][L]^2[T]^{-1}$.
• Comparison: The dimensions of the moment of a force ($[M][L]^2[T]^{-2}$) and angular momentum ($[M][L]^2[T]^{-1}$) are not identical. They differ in the power of time.

Dimension Comparison Table

Based on the analysis, the pair that does not have identical dimensions is the moment of a force and angular momentum.