Question

In the case of measurement of g, if the error in measurement of the length of the pendulum is 2%, the percentage error in the time period is 1%. The maximum error in the measurement of g is
A. 1%
B. 2%
C. 4%
D. No error

Answer 1

Hint: This question is based on the simple harmonic motion of a pendulum. You need to use the formula for the time period for the simple harmonic motion to calculate the error when the length is errored.

Complete step by step answer:
The oscillation of the pendulum is simple harmonic motion, and in this type of motion, the displacement is proportional to the acceleration. And by using this relationship, the time period is calculated for the oscillations of the pendulum.
The force in oscillation is is written as $F = - kx$ and in the pendulum motion $F = mg\sin \theta $ so, and we can write $I\alpha = mg\sin \theta $ where I is the moment of inertia and is angular acceleration. By using this relation, we can finally deduce the expression of the time period.
The time period of the oscillation of the pendulum is given by $T = 2\pi \sqrt {\dfrac{l}{g}} $
Where T is the time period, and l is the length of the string from which the pendulum is attached.
We can find the expression of acceleration due to gravity from the expression of the time period
i.e $g = \dfrac{{4{\pi ^2}l}}{{{T^2}}}$
error in g can be written as
\[\dfrac{{\Delta g}}{g} \times 100 = \left( {\dfrac{{\Delta l}}{l} \times 100} \right) + \left( {2\dfrac{{\Delta T}}{T} \times 100} \right)\]
We know the values of \[\dfrac{{\Delta l}}{l}\]and\[\dfrac{{\Delta T}}{T}\] from the question. So, putting these values in the error expression, we get
\[\dfrac{{\Delta g}}{g} \times 100 = 2\% + \left( {2 \times 1\% } \right) = 4\% \]
Hence, the error in acceleration due to gravity g is 4%.

Additional Information:
Pendulum problem is the most used problem of simple harmonic motion. And generally, the motion is shown by sine or cosine expression. The time period and frequency are important parameters of this kind of motion. By analyzing the pendulum's motion, we form different expressions to calculate the various parameters of the simple harmonic motion.

So, the correct answer is “Option C”.

Note:
Here we calculated the error in the gravity acceleration by differentiating the expression of g and putting the length error value and time period error value in it. You may go wrong in the differentiation expression of the time period.