Choose the correct length \(( L )\) versus square of time period (T2) graph for a simple pendulum executing simple harmonic motion
Show Hint
For graph-based questions, establish the mathematical relationship between the variables involved. This will help identify the correct graph representing that relationship.
The Correct Option is C
Solution and Explanation
Step 1: Recall the Formula for the Time Period of a Simple Pendulum
The time period (\(T\)) of a simple pendulum is given by: \[ T = 2\pi \sqrt{\frac{\ell}{g}} \] where \(\ell\) is the length of the pendulum and \(g\) is the acceleration due to gravity.
Step 2: Find the Relationship between \(T^2\) and \(\ell\)
Squaring both sides of the equation, we get: \[ T^2 = 4\pi^2 \frac{\ell}{g} \] Since \(4\pi^2\) and \(g\) are constants, we can write: \[ T^2 \propto \ell \] This indicates a linear relationship between \(T^2\) and \(\ell\).
Step 3: Determine the Correct Graph
The graph of \(T^2\) versus \(L\) should be a straight line passing through the origin, representing a direct proportionality.
Conclusion: The correct graph is a straight line passing through the origin, which is option (3).
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Concepts Used:
Oscillations
Oscillation is a process of repeating variations of any quantity or measure from its equilibrium value in time . Another definition of oscillation is a periodic variation of a matter between two values or about its central value.
The term vibration is used to describe the mechanical oscillations of an object. However, oscillations also occur in dynamic systems or more accurately in every field of science. Even our heartbeats also creates oscillations. Meanwhile, objects that move to and fro from its equilibrium position are known as oscillators.
Read More: Simple Harmonic Motion
Oscillation- Examples
The tides in the sea and the movement of a simple pendulum of the clock are some of the most common examples of oscillations. Some of examples of oscillations are vibrations caused by the guitar strings or the other instruments having strings are also and etc. The movements caused by oscillations are known as oscillating movements. For example, oscillating movements in a sine wave or a spring when it moves up and down.
The maximum distance covered while taking oscillations is known as the amplitude. The time taken to complete one cycle is known as the time period of the oscillation. The number of oscillating cycles completed in one second is referred to as the frequency which is the reciprocal of the time period.