We analyze the dimensions of each pair of physical quantities to determine which have the same dimensions.

Option A: Angular Momentum and Planck's Constant

Angular momentum (\(L\)):
\(L=mvr=\text{mass}\times \text{velocity}\times \text{distance}\) \(=[M][L{T}^{−1}][L]=[M{L}^{2}{T}^{−1}]\)
Planck's constant (\(h\)):
Planck’s constant is defined in the equation:
\(E=hf\)
where \(E\) is energy and \(f\) is frequency.
Since energy has dimension \([M{L}^{2}{T}^{−2}]\) and frequency \([{T}^{−1}]\), we get:
\([h]=\frac{[E]}{[f]}=\frac{[M{L}^{2}{T}^{−2}]}{[{T}^{−1}]}=[M{L}^{2}{T}^{−1}]\)

Both have the same dimensions: \([M{L}^{2}{T}^{−1}]\).

Option B: Torque and Moment of Inertia

Torque (\(\tau\)):
\(\tau =r\times F=[L]\times [ML{T}^{−2}]=[M{L}^{2}{T}^{−2}]\)
Moment of inertia (\(I\)):
\(I=m{r}^{2}=[M][{L}^{2}]=[M{L}^{2}]\)

Different dimensions.

Option C: Impulse and Surface Tension

Impulse (\(J\)):
\(J=\text{Force}\times \text{time}=[ML{T}^{−2}][T]=[ML{T}^{−1}]\)
Surface tension (\(T\)):
\(T=\frac{\text{Force}}{\text{Length}}=\frac{[ML{T}^{−2}]}{[L]}=[M{T}^{−2}]\)

Different dimensions.

Option D: Momentum and Work Done

Momentum (\(p\)):
\(p=mv=[M][L{T}^{−1}]=[ML{T}^{−1}]\)
Work Done (\(W\)):
\(W=\text{Force}\times \text{Distance}=[ML{T}^{−2}][L]=[M{L}^{2}{T}^{−2}]\)

Different dimensions.

Final Answer:

\(A\ \text{Angular momentum and Planck’s constant}\)

Which of the following pairs of physical quantities have the same dimensions?(Shift - I Memory Based)