A Simple pendulum experiment is performed for the value of 'g',the acceleration due to the earth's gravity. The measured value of length of the pendulum is 25 cm with an accuracy of 1 mm and the measured time for 100 oscillations is found to be 100 sec with an accuracy of 1 sec. The percentage uncertainty in the determination of 'g' is:
9.8
0.98
4.8
2.4
1.4
The Correct Option is D
Approach Solution - 1
Given parameters:
- Length \( L = 25 \, \text{cm} = 0.25 \, \text{m} \) ± 1 mm
- Time for 100 oscillations \( t = 100 \, \text{s} \) ± 1 s
Pendulum equation: \[ T = 2\pi \sqrt{\frac{L}{g}} \Rightarrow g = \frac{4\pi^2 L}{T^2} \]
Uncertainty calculations: \[ \frac{\Delta L}{L} = \frac{0.001}{0.25} = 0.004 \] \[ \frac{\Delta T}{T} = \frac{1/100}{1} = 0.01 \] \[ \frac{\Delta g}{g} = \frac{\Delta L}{L} + 2\frac{\Delta T}{T} = 0.004 + 0.02 = 0.024 \]
Percentage uncertainty: \[ 0.024 \times 100\% = 2.4\% \]
Thus, the correct option is (D): 2.4.
Approach Solution -2
1. Recall the formula for the period of a pendulum:
The period (T) of a simple pendulum is given by:
\[T = 2\pi\sqrt{\frac{l}{g}}\]
where:
- l is the length of the pendulum
- g is the acceleration due to gravity
2. Express g in terms of T and l:
Squaring both sides and rearranging:
\[g = \frac{4\pi^2l}{T^2}\]
Since we measure time for 100 oscillations the expression for g is: \[g = \frac{4\pi^2 l \times 100^2}{t^2} = \frac{40000 \pi^2 l}{t^2} \] where t is the time for 100 oscillations.
3. Calculate the percentage uncertainty in l:
\[\frac{\Delta l}{l} = \frac{1 \, mm}{25 \, cm} = \frac{0.1 \, cm}{25 \, cm} = 0.004 = 0.4\%\]
4. Calculate the percentage uncertainty in t:
\[\frac{\Delta t}{t} = \frac{1 \, s}{100 \, s} = 0.01 = 1\%\]
5. Calculate the percentage uncertainty in g:
The percentage uncertainty in g is given by:
\[\frac{\Delta g}{g} = \frac{\Delta l}{l} + 2\frac{\Delta t}{t}\]
(We multiply the percentage uncertainty in t by 2 because it is squared in the formula for g.)
\[\frac{\Delta g}{g} = 0.4\% + 2(1\%) = 0.4\% + 2\% = 2.4\%\]
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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