Question

Choose the physical quantity pair with the same dimensions

• A. angular momentum and work
• B. work and torque
• C. potential energy and linear momentum
• D. kinetic energy and velocity

Answer 1

Answer: (B)

Dimensional formula of work,
$[W]=[F]\times [s]$
$=\left[ML{T}^{-2}\right][L]=\left[M{L}^2{T}^{-2}\right]$
Similarly, dimensional formulae of
Angular momentum $[L]=[m][v][r]=[M]\left[L{T}^{-1}\right][L]$
$=\left[M{L}^2{T}^{-1}\right]$
Torque $[\tau ]=[F]\times [r]$
$=\left[ML{T}^{-2}\right][L]=\left[M{L}^2{T}^{-2}\right]$
Potential energy $[U]=[m][g][h]$
$=[M]\left[L{T}^{-2}\right][L]$
$=\left[M{L}^2{T}^{-2}\right]$
Linear momentum $[p]=[m][v]=[M]\left[L{T}^{-1}\right]$
$=\left[ML{T}^{-1}\right]$
Kinetic energy $[K]=\frac{1}{2}[m]\left[v^2\right]$
$=[M]{\left[L{T}^{-1}\right]}^2=\left[M{L}^2{T}^{-2}\right]$
Velocity $[v]=\left[L{T}^{-1}\right]$
Clearly, dimensional formulae of work and torque are same