Question 102: Units , Dimensions and Error Analysis

Question

Match List I with List II

LIST I

LIST II

A.
Spring constant
I.
$$\mathrm{[T^{-1}]}$$

B.
Angular speed
II.
$$\mathrm{[MT^{-2}]}$$

C.
Angular momentum
III.
$$\mathrm{[ML^2]}$$

D.
Moment of inertia
IV.
$$\mathrm{[ML^2T^{-1}]}$$

Choose the correct answer from the options given below:

Accepted Answer

Let's analyze each item in List I and find the corresponding dimensional formula in List II.
A. Spring constant (k)
The spring constant is related to Hooke's Law, which states that the force exerted by a spring is proportional to its displacement: $$F = kx$$. The dimensional formula for force is $$\mathrm{[MLT^{-2}]}$$, and for displacement is $$\mathrm{[L]}$$.
Dimensional formula of spring constant, k:
$$\mathrm{[k]} = \frac{\mathrm{[MLT^{-2}]}}{\mathrm{[L]}} = \mathrm{[MT^{-2}]}$$
So, A matches with II.
B. Angular speed (ÃÂ)
Angular speed is the rate of change of angular displacement with respect to time. The dimensional formula for angular displacement is the same as the plane angle, which is dimensionless. Therefore, the dimensional formula for angular speed is the reciprocal of the dimensional formula for time.
Dimensional formula of angular speed, ÃÂ:
$$\mathrm{[\omega]} = \mathrm{[T^{-1}]}$$
So, B matches with I.
C. Angular momentum (L)
Angular momentum is the product of the moment of inertia (I) and the angular velocity (ÃÂ). The dimensional formula for the moment of inertia is $$\mathrm{[ML^2]}$$, and the dimensional formula for angular velocity is $$\mathrm{[T^{-1}]}$$.
Dimensional formula of angular momentum, L:
$$\mathrm{[L]} = \mathrm{[ML^2]}\cdot\mathrm{[T^{-1}]} = \mathrm{[ML^2T^{-1}]}$$
So, C matches with IV.
D. Moment of inertia (I)
The moment of inertia is a measure of an object's resistance to rotational motion. It depends on the mass of the object and its distribution around the axis of rotation.
Dimensional formula of moment of inertia, I:
$$\mathrm{[I]} = \mathrm{[ML^2]}$$
So, D matches with III.
The correct matching is A-II, B-I, C-IV, D-III

Answer

A-IV, B-I, C-III, D-II

Answer

A-II, B-III, C-I, D-IV

Answer

A-II, B-I, C-IV, D-III

Answer

A-I, B-III, C-II, D-IV

Units , Dimensions and Error Analysis

Match List I with List II LIST I LIST II A. Spring constant I. $\mathrm{[T^{-1}]} B. Angular speed II. \mathrm{[MT^{-2}]} C. Angular momentum III. \mathrm{[ML^2]} D. Moment of inertia IV. \mathrm{[ML^2T^{-1}]}$ Choose the correct answer from the options given below:

Choose the correct answer: