Question

\( [L^0 M^0 T^{-1}] \) is the dimensional formula for

• A. angular velocity
• B. activity of radioactive substance
• C. time period of oscillation
• D. half life period of a radioactive substance
• E. impulse of the force

Hint:

Activity of radioactive substance has dimension of frequency.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The dimensional formula \([L^0 M^0 T^{-1}]\) represents a physical quantity that has the dimension of inverse time, or frequency (unit: s\(^{-1}\)). We need to check the dimensions of each option.
Step 2: Detailed Explanation:
Let's analyze the dimensions of each given quantity:
(A) Angular velocity (\(\omega\)): Defined as angle per unit time (\(\omega = \Delta\theta / \Delta t\)). Since angle is dimensionless, the dimension is \([T^{-1}]\). This matches.
(B) Activity of a radioactive substance (R): Defined as the rate of decay (\(R = -\Delta N / \Delta t\)). This is the number of decays per unit time. The unit is Becquerel (Bq), which is s\(^{-1}\). The dimension is \([T^{-1}]\). This also matches.
(C) Time period of oscillation (T): This is a measure of time. Its dimension is \([T]\). This does not match.
(D) Half-life period (T\(_{1/2}\)): This is the time taken for a quantity to reduce to half its initial value. Its dimension is \([T]\). This does not match.
(E) Impulse of the force (J): Defined as force multiplied by time (\(J = F \cdot \Delta t\)). The dimension of force is \([MLT^{-2}]\). So, the dimension of impulse is \([MLT^{-2}] \cdot [T] = [MLT^{-1}]\). This does not match.
Both angular velocity and activity of a radioactive substance have the dimensional formula \([T^{-1}]\). Since a multiple-choice question should ideally have only one correct answer, the question is flawed.
Step 3: Final Answer:
The question is marked as cancelled because both options (A) and (B) have the correct dimensional formula.