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 \(1.13\%\).
Top Questions on simple harmonic motion
A particle is subjected to simple harmonic motions as: $ x_1 = \sqrt{7} \sin 5t \, \text{cm} $ $ x_2 = 2 \sqrt{7} \sin \left( 5t + \frac{\pi}{3} \right) \, \text{cm} $ where $ x $ is displacement and $ t $ is time in seconds. The maximum acceleration of the particle is $ x \times 10^{-2} \, \text{m/s}^2 $. The value of $ x $ is:
- JEE Main - 2025
- Physics
- simple harmonic motion
Two simple pendulums having lengths $l_{1}$ and $l_{2}$ with negligible string mass undergo angular displacements $\theta_{1}$ and $\theta_{2}$, from their mean positions, respectively. If the angular accelerations of both pendulums are same, then which expression is correct?
- JEE Main - 2025
- Physics
- simple harmonic motion
- Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A): Knowing initial position \( x_0 \), and initial momentum \( p_0 \) is enough to determine the position and momentum at any time \( t \) for a simple harmonic motion with a given angular frequency \( \omega \).
Reason (R): The amplitude and phase can be expressed in terms of \( x_0 \) and \( p_0 \).
In the light of the above statements, choose the correct answer from the options given below:- JEE Main - 2025
- Physics
- simple harmonic motion
- A particle is executing simple harmonic motion with a time period of 2 s and amplitude 1 cm. If \( D \) and \( d \) are the total distance and displacement covered by the particle in 12.5 s, then the ratio \( \frac{D}{d} \) is:
- JEE Main - 2025
- Physics
- simple harmonic motion
- At 27°C temperature, the mean kinetic energy of the atoms of an ideal gas is \( E_1 \). If the temperature is increased to 327°C, then the mean kinetic energy of the atoms will be
- KCET - 2025
- Physics
- simple harmonic motion
Questions Asked in JEE Main exam
Method used for separation of mixture of products (B and C) obtained in the following reaction is:
- JEE Main - 2026
- p -Block Elements
- Consider light travelling from a medium A to medium B separated by a plane interface. If the light undergoes total internal reflection during its travel from medium A to B and the speed of light in media A and B are \(2.4 \times 10^8\) m/s and \(2.7 \times 10^8\) m/s, respectively, then the value of critical angle is :
- JEE Main - 2026
- Newtons Laws of Motion
In the following \(p\text{–}V\) diagram, the equation of state along the curved path is given by \[ (V-2)^2 = 4ap, \] where \(a\) is a constant. The total work done in the closed path is:
- JEE Main - 2026
- Thermodynamics
Let \( ABC \) be a triangle. Consider four points \( p_1, p_2, p_3, p_4 \) on the side \( AB \), five points \( p_5, p_6, p_7, p_8, p_9 \) on the side \( BC \), and four points \( p_{10}, p_{11}, p_{12}, p_{13} \) on the side \( AC \). None of these points is a vertex of the triangle \( ABC \). Then the total number of pentagons that can be formed by taking all the vertices from the points \( p_1, p_2, \ldots, p_{13} \) is ___________.
- JEE Main - 2026
- Coordinate Geometry
- A voltage regulating circuit consisting of a Zener diode having breakdown voltage of $10\,\text{V}$ and maximum power dissipation of $0.4\,\text{W}$ is operated at $15\,\text{V}$. The approximate value of protective resistance in this circuit is ___ $\Omega$.
- JEE Main - 2026
- Semiconductor electronics: materials, devices and simple circuits
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