The period of oscillation of a simple pendulum is $T =2 \pi \sqrt{\frac{ L }{ g }}$. Measured value of 'L' is $1.0 m$ from meter scale having a minimum division of $1 mm$ and time of one complete oscillation is 1.95 s measured from stopwatch of $0.01 s$ resolution. The percentage error in the determination of 'g' will be :
- $1.13 \%$
- $1.03 \%$
- $1.33 \%$
- $1.30 \%$
The Correct Option is A
Solution and Explanation
To find the percentage error in the determination of the acceleration due to gravity \(g\), given the period of oscillation of a simple pendulum, we start by using the formula for the period of a simple pendulum:
\(T = 2 \pi \sqrt{\frac{L}{g}}\)
Rearranging to solve for \(g\), we get:
\(g = \frac{4 \pi^2 L}{T^2}\)
To find the percentage error in \(g\), we need the errors in both length \(L\) and time period \(T\).
- The measured length \(L\) is \(1.0 \, m\) with a least count of \(1 \, mm\) (or \(0.001 \, m\)), so the percentage error in \(L\) is:
\(\frac{0.001}{1.0} \times 100 = 0.1\%\)
- The time period \(T\) is \(1.95 \, s\), with a stopwatch resolution of \(0.01 \, s\). The percentage error in \(T\) is:
\(\frac{0.01}{1.95} \times 100 \approx 0.51\%\)
Since \(g\) depends on \(L\) and \(T^2\), the error in \(T^2\) is twice the error in \(T\):
\(2 \times 0.51\% = 1.02\%\)
Using the formula for the combined percentage error in \(g\):
\(\% \text{ error in } g = \% \text{ error in } L + 2 \times \% \text{ error in } T\)
Substitute the errors:
\(= 0.1\% + 1.02\% = 1.12\%\)
Since the closest option is \(1.13\%\), the correct answer is \(1.13\%\). Thus, the percentage error in the determination of \(g\) is approximately \(1.13\%\).
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Concepts Used:
Simple Harmonic Motion
Simple Harmonic Motion is one of the most simple forms of oscillatory motion that occurs frequently in nature. The quantity of force acting on a particle in SHM is exactly proportional to the displacement of the particle from the equilibrium location. It is given by F = -kx, where k is the force constant and the negative sign indicates that force resists growth in x.
This force is known as the restoring force, and it pulls the particle back to its equilibrium position as opposing displacement increases. N/m is the SI unit of Force.
Types of Simple Harmonic Motion
Linear Simple Harmonic Motion:
When a particle moves to and fro about a fixed point (called equilibrium position) along with a straight line then its motion is called linear Simple Harmonic Motion. For Example spring-mass system
Conditions:
The restoring force or acceleration acting on the particle should always be proportional to the displacement of the particle and directed towards the equilibrium position.
- – displacement of particle from equilibrium position.
- – Restoring force
- - acceleration
Angular Simple Harmonic Motion:
When a system oscillates angular long with respect to a fixed axis then its motion is called angular simple harmonic motion.
Conditions:
The restoring torque (or) Angular acceleration acting on the particle should always be proportional to the angular displacement of the particle and directed towards the equilibrium position.
Τ ∝ θ or α ∝ θ
Where,
- Τ – Torque
- α angular acceleration
- θ – angular displacement