Question

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

• A. Both Statement I and Statement II are false
• B. Statement I is true but Statement II is false
• C. Both Statement I and Statement II are true
• D. Statement I is false but Statement II is true

Answer 1

The Correct Option is B

Analysis of the given statements requires examining the dimensions of the quantities involved.

1. Planck's constant and Angular momentum:
   • Planck's constant (\(h\)): This fundamental constant possesses the dimensions of action, represented dimensionally as \([M][L]^2[T]^{-1}\), where [M], [L], and [T] denote Mass, Length, and Time, respectively.
   • Angular momentum (\(L\)): Defined as the cross product of the radius vector (\({\bf r}\)) and linear momentum (\({\bf p}\)) (\({\bf r} \times {\bf p}\)), its dimensional formula is also \([M][L]^2[T]^{-1}\).
   • Consequently, Statement I is confirmed as true due to the identical dimensions of Planck's constant and angular momentum.
2. Linear momentum and Moment of force:
   • Linear momentum (\(p\)): The product of mass and velocity, its dimensional formula is \([M][L][T]^{-1}\).
   • Moment of force (Torque, \(\tau\)): Calculated as the product of force and the perpendicular distance from the pivot, its dimensional formula is \([M][L]^2[T]^{-2}\).
   • It is evident that linear momentum and moment of force have distinct dimensions, rendering Statement II false.

In summary, the accurate conclusion is: Statement I is true but Statement II is false.