Equation of SHM
Equation of Simple Harmonic Motion (SHM)
Simple Harmonic Motion (SHM) is a type of periodic motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement. The motion is sinusoidal in time and demonstrates a single resonant frequency.
Basic Equation of SHM
The basic equation that governs SHM is derived from Newton's second law, which states that the force acting on a body is equal to the mass of the body multiplied by its acceleration (F = ma). For SHM, the force is also proportional to the displacement (x) from the equilibrium position but in the opposite direction, leading to the equation:
[ F = -kx ]
where:
- ( F ) is the restoring force,
- ( k ) is the spring constant (or force constant),
- ( x ) is the displacement from the equilibrium position.
The negative sign indicates that the force acts in the direction opposite to the displacement.
Using Newton's second law, we can write:
[ ma = -kx ]
or
[ a = -\frac{k}{m}x ]
Since acceleration is the second derivative of displacement with respect to time, we can write:
[ \frac{d^2x}{dt^2} = -\frac{k}{m}x ]
This is the differential equation for SHM, where the solution is a function that describes the position of the oscillating object as a function of time.
Solution to the Differential Equation
The general solution to the differential equation of SHM is:
[ x(t) = A \cos(\omega t + \phi) ]
where:
- ( x(t) ) is the displacement as a function of time,
- ( A ) is the amplitude of the motion,
- ( \omega ) is the angular frequency,
- ( \phi ) is the phase constant,
- ( t ) is the time.
The angular frequency ( \omega ) is related to the spring constant ( k ) and the mass ( m ) by:
[ \omega = \sqrt{\frac{k}{m}} ]
The period ( T ) of the SHM, which is the time taken for one complete cycle of motion, is given by:
[ T = \frac{2\pi}{\omega} = 2\pi \sqrt{\frac{m}{k}} ]
The frequency ( f ), which is the number of cycles per second, is the reciprocal of the period:
[ f = \frac{1}{T} = \frac{\omega}{2\pi} = \frac{1}{2\pi} \sqrt{\frac{k}{m}} ]
Differences and Important Points
Here is a table summarizing the key differences and important points of SHM:
| Feature | Description |
|---|---|
| Restoring Force | The force that brings the system back to equilibrium; proportional to displacement and opposite in direction. |
| Displacement | The distance from the equilibrium position; varies sinusoidally with time. |
| Amplitude | The maximum displacement from the equilibrium position; denoted by ( A ). |
| Period | The time taken for one complete cycle of motion; denoted by ( T ). |
| Frequency | The number of cycles per second; denoted by ( f ). |
| Angular Frequency | The rate of change of the phase of the sinusoidal function; denoted by ( \omega ). |
| Phase Constant | A constant that determines the initial angle at ( t = 0 ); denoted by ( \phi ). |
Examples
Example 1: Finding the Period of SHM
A mass of 0.5 kg is attached to a spring with a spring constant of 200 N/m. Find the period of the SHM.
Using the formula for the period:
[ T = 2\pi \sqrt{\frac{m}{k}} ]
[ T = 2\pi \sqrt{\frac{0.5}{200}} ]
[ T = 2\pi \sqrt{0.0025} ]
[ T = 2\pi \times 0.05 ]
[ T = 0.1\pi ]
[ T \approx 0.314 \text{ seconds} ]
Example 2: Displacement as a Function of Time
Given an SHM system with an amplitude of 0.2 m, an angular frequency of 5 rad/s, and a phase constant of ( \frac{\pi}{3} ), find the displacement at ( t = 2 ) seconds.
Using the displacement equation:
[ x(t) = A \cos(\omega t + \phi) ]
[ x(2) = 0.2 \cos(5 \times 2 + \frac{\pi}{3}) ]
[ x(2) = 0.2 \cos(10 + \frac{\pi}{3}) ]
[ x(2) = 0.2 \cos(\frac{10\pi}{3} + \frac{\pi}{3}) ]
[ x(2) = 0.2 \cos(\frac{11\pi}{3}) ]
Now, calculate the cosine value and multiply by the amplitude to find the displacement at ( t = 2 ) seconds.
Understanding the equation of SHM is crucial for analyzing systems that exhibit simple harmonic motion, such as pendulums, springs, and other oscillating systems. It is a fundamental concept in physics that describes a wide range of periodic phenomena.