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:
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:
Both Statement I and Statement II are false
Statement I is true but Statement II is false
Both Statement I and Statement II are true
Statement I is false but Statement II is true
The Correct Option is B
Approach Solution - 1
To analyze the given statements, we need to examine the dimensions of the quantities mentioned.
- 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.
- 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:
- 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.
Approach Solution -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.
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Concepts Used:
Momentum
It can be defined as "mass in motion." All objects have mass; so if an object is moving, then it is called as momentum.
the momentum of an object is the product of mass of the object and the velocity of the object.
Momentum = mass • velocity
The above equation can be rewritten as
p = m • v
where m is the mass and v is the velocity.
Momentum is a vector quantity and the direction of the of the vector is the same as the direction that an object.