Question

Given that T stands for time period and l stands for the length of simple pendulum. If g is the acceleration due to gravity, then which of the following statements about the relation \[{T^2} = \left( {l/g} \right)\] is correct?
A. It is correct both dimensionally as well as numerically.
B. It is neither dimensionally correct nor numerically.
C. It is dimensionally correct but not numerically.
D. It is numerically correct but not dimensionally.

Answer 1

Hint:The above problem can be resolved using the mathematical relation or the time period of simple pendulum, along with dimensional analysis of the formulation.


Complete step-by-step answer
Given: The units of the variables used in the formula are used to predict the dimensional accuracy, where l and g has its usual meanings.
The relation for the time period is, \[{T^2} = \left( {l/g} \right)\]
The length of the pendulum is, l.
The mathematical relation for the time period of the simple pendulum is given as,
\[T = 2\pi \sqrt {\dfrac{l}{g}} \]
Here, l and g are the length of the pendulum and gravitational acceleration respectively possessed by the pendulum during the oscillations.
The above relation shows that there is deviation in terms of mathematical formula with respect to the given relation. Therefore, the given relation is numerically incorrect. But dimensionally the given relation matches with that of the actual relation of the time taken by the pendulum to possess oscillation.
Hence, the given statement is numerically incorrect, but it is dimensionally correct. And option C is correct.


Note: To resolve the given condition the appropriate analysis of the concepts under the simple pendulum is to be taken out. Moreover, the correct formula for the time taken by the simple pendulum is to be remembered, along with calculation for the dimensional formula is to be carried out.