Given below are two statements : Statement (I) : Planck's constant and angular momentum have same dimensions. Statement (II) : Linear momentum and moment of force have same dimensions. In the light of the above statements, choose the correct answer from the options given below :

Statement I is true but Statement II is false

Both Statement I and Statement II are false

Both Statement I and Statement II are true

Statement I is false but Statement II is true

Units and Measurements PYQ — JEE Main 2024 · 27 Jan · Shift 1 · Q77 with Solution

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Correct answer: A

Planck's constant $h$ has dimensions of energy $\times$ time.

$[h] = [E]\,[t] = (ML^2T^{-2})(T) = ML^2T^{-1}$

Angular momentum is:

$L = r \times p$

where $[r] = L$ and linear momentum $[p] = MLT^{-1}$ .

So,

$[\text{angular momentum}] = L \cdot MLT^{-1} = ML^2T^{-1}$

Thus,

$[h] = [\text{angular momentum}]$

So Statement (I) is true.

Linear momentum has dimensions:

$[p] = MLT^{-1}$

Moment of force (torque) is:

$\tau = F \times r$

Since force has dimensions

$[F] = MLT^{-2}$

therefore,

$[\tau] = (MLT^{-2})(L) = ML^2T^{-2}$

Now compare:

These are not the same.

So Statement (II) is false.

Therefore, the correct option is:

$\boxed{\text{A}}$