A simple pendulum acts like a harmonic oscillator with a period dependent only on L and g for sufficiently small amplitudes.
A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting equilibrium position, it is subject to a restoring force; after it reaches its highest point in its swing, gravity will accelerate it back toward the equilibrium position. When released, the restoring force combined with the pendulum’s mass causes it to oscillate about the equilibrium position, swinging back and forth.
Simple Pendulum: A simple pendulum has a small-diameter bob and a string that has a very small mass but is strong enough not to stretch appreciably. The linear displacement from equilibrium is s, the length of the arc. Also shown are the forces on the bob, which result in a net force of −mgsinθ toward the equilibrium position—that is, a restoring force.
For small displacements, a pendulum is a simple harmonic oscillator. A simple pendulum is defined to have an object that has a small mass, also known as the pendulum bob, which is suspended from a wire or string of negligible mass, such as shown in the illustrating figure. Exploring the simple pendulum a bit further, we can discover the conditions under which it performs simple harmonic motion, and we can derive an interesting expression for its period.
Pendulums: A brief introduction to pendulums (both ideal and physical) for calculus-based physics students from the standpoint of simple harmonic motion.
We begin by defining the displacement to be the arc length s. We see from the figure that the net force on the bob is tangent to the arc and equals −mgsinθ. (The weight mg has components mgcosθ along the string and mgsinθ tangent to the arc. ) Tension in the string exactly cancels the component mgcosθ parallel to the string. This leaves a net restoring force drawing the pendulum back toward the equilibrium position at θ = 0.
Now, if we can show that the restoring force is directly proportional to the displacement, then we have a simple harmonic oscillator. In trying to determine if we have a simple harmonic oscillator, we should note that for small angles (less than about 15º), sinθ≈θ (sinθ and θ differ by about 1% or less at smaller angles). Thus, for angles less than about 15º, the restoring force F is
\[\mathrm{F≈−mgθ.}\]
The displacement s is directly proportional to θ. When θ is expressed in radians, the arc length in a circle is related to its radius (L in this instance) by:
s=Lθs=Lθ
so that
\[\mathrm{θ=sL.}\]
For small angles, then, the expression for the restoring force is:
\[\mathrm{F≈\dfrac{mgL}{s}.}\]
This expression is of the form of Hooke’s Law:
\[\mathrm{F≈−kx}\]
where the force constant is given by k=mg/L and the displacement is given by x=s. For angles less than about 15º, the restoring force is directly proportional to the displacement, and the simple pendulum is a simple harmonic oscillator.
Using this equation, we can find the period of a pendulum for amplitudes less than about 15º. For the simple pendulum:
\[\mathrm{T=2π\sqrt{\dfrac{m}{k}}=2π\sqrt{\dfrac{m}{\dfrac{mg}{L}}}.}\]
Thus,
\[\mathrm{T=2π\sqrt{\dfrac{L}{g}}}\]
or the period of a simple pendulum. This result is interesting because of its simplicity. The only things that affect the period of a simple pendulum are its length and the acceleration due to gravity. The period is completely independent of other factors, such as mass. Even simple pendulum clocks can be finely adjusted and accurate. Note the dependence of T on g. If the length of a pendulum is precisely known, it can actually be used to measure the acceleration due to gravity. If θ is less than about 15º, the period T for a pendulum is nearly independent of amplitude, as with simple harmonic oscillators. In this case, the motion of a pendulum as a function of time can be modeled as:
\[\mathrm{θ(t)=θ_o \cos (\dfrac{2πt}{T})}\]
For amplitudes larger than 15º, the period increases gradually with amplitude so it is longer than given by the simple equation for T above. For example, at an amplitude of θ0 = 23° it is 1% larger. The period increases asymptotically (to infinity) as θ0approaches 180°, because the value θ0 = 180° is an unstable equilibrium point for the pendulum.
As we know, many forms of energy like light and sound travel in waves. A wave is defined through various characteristics like frequency, amplitude and speed. In wave mechanics, any given wave enfolds parameters like – frequency, time period, wavelength, amplitude etc. This article lets us understand and learn in detail about frequency, time period, and angular frequency.
Frequency definition states that it is the number of complete cycles of waves passing a point in unit time. The time period is the time taken by a complete cycle of the wave to pass a point. Angular frequency is angular displacement of any element of the wave per unit of time.
Consider the graph shown below. It represents the displacement y of any element for a harmonic wave along a string moving in the positive x-direction with respect to time. Here, the string element moves up and down in simple harmonic motion.
Read More: Angular Displacement
The relation describing the displacement of the element with respect to time is given as:
y (0,t) = a sin (–ωt), here we have considered the inception of wavefrom x=0
y (0,t) = -a sin (ωt)
As we know, sinusoidal or harmonic motion is periodic in nature, i.e. the nature of the graph of an element of the wave repeats itself at a fixed duration. To mark the duration of periodicity following terms are introduced for sinusoidal waves.
Following is the table explaining other related concepts of waves:
As shown above, the particles move about the mean equilibrium or mean position with time in a sinusoidal wave motion. The particles rise until they reach the highest point, the crest, and then continue to fall until they reach the lowest point, the trough. The cycle repeats itself in a uniform pattern. The time period of oscillation of a wave is defined as the time taken by any string element to complete one such oscillation. For a sine wave represented by the equation:
y (0, t) = -a sin(ωt)
The time period formula is given as:
\(\begin{array}{l}T=\frac{2\pi }{\omega }\end{array} \)
We define the frequency of a sinusoidal wave as the number of complete oscillations made by any wave element per unit of time. By the definition of frequency, we can understand that if a body is in periodic motion, it has undergone one cycle after passing through a series of events or positions and returning to its original state. Thus, frequency is a parameter that describes the rate of oscillation and vibration.
The equation gives the relation between the frequency and the period:
The relation between the frequency and the period is given by the equation:
f=1/T
For a sinusoidal wave represented by the equation:
y (0,t) = -a sin (ωt)
The formula of the frequency with the SI unit is given as:
Formula\(\begin{array}{l}f=\frac{1}{T}=\frac{\omega }{2\pi }\end{array} \)
SI unit HertzOne Hertz is equal to one complete oscillation taking place per second.
For a sinusoidal wave, the angular frequency refers to the angular displacement of any element of the wave per unit of time or the rate of change of the phase of the waveform. It is represented by ω. Angular frequency formula and SI unit are given as:
Formula\(\begin{array}{l}\omega=\frac{2\pi }{T}=2\pi f\end{array} \)
SI unit rads-1Where,
Frequently Asked Questions – FAQs
Q1
A wave is a disturbance that travels through a medium from one location to another location.
Q2
f = 1/T
Q3
We define the frequency of a sinusoidal wave as the number of complete oscillations made by any element of the wave per unit of time.
Q4
The time period of oscillation of a wave is defined as the time taken by any string element to complete an oscillation.
Q5
The angular frequency refers to the angular displacement of any wave element per unit of time or the rate of change of the waveform phase.
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