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

To analyze the given statements, we need to examine the dimensions of the quantities mentioned. 

1. Planck's constant and Angular momentum:
   • Planck's constant (\(h\)): This is a fundamental constant with dimensions of action. Its dimensional formula is given by \([M][L]^2[T]^{-1}\) where:
     • [M] stands for Mass
     • [L] stands for Length
     • [T] stands for Time
   • Angular momentum (\(L\)): The angular momentum of a particle is given by \({\bf r} \times {\bf p}\) where \({\bf r}\) is the radius vector, and \({\bf p}\) is the linear momentum. Its dimensional formula is also \([M][L]^2[T]^{-1}\).
   • Therefore, Statement I is true as Planck's constant and angular momentum have the same dimensions.
2. Linear momentum and Moment of force:
   • Linear momentum (\(p\)): It is defined as the product of mass and velocity, and its dimensional formula is \([M][L][T]^{-1}\).
   • Moment of force (Torque, \(\tau\)): It is the product of force and the perpendicular distance from the pivot point. Its dimensional formula is \([M][L]^2[T]^{-2}\).
   • It is clear that linear momentum and moment of force have different dimensions. Thus, Statement II is false.

In conclusion, the correct answer is: Statement I is true but Statement II is false.

Answer 2

Dimensions of Planck’s Constant (h):

Planck’s constant has dimensions of action (energy × time), which is equivalent to angular momentum:

\[ [h] = ML^2T^{-1} \]

Dimensions of Linear Momentum and Moment of Force:

Linear momentum has dimensions:

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

Moment of force (torque) has dimensions:

\[ [\tau] = ML^2T^{-2} \]

These are different, so Statement II is false.

Therefore, Statement I is true, and Statement II is false.