The periodic nature of SHM and simple pendulum - Simple Harmonic Motion (SHM)

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The periodic nature of SHM and simple pendulum

Lessons

Notes:

In this lesson, we will learn:

  • The nature of the periodic motion
  • The graph analysis of the periodic motion
  • Simple Pendulum

Notes:

  • If each vibration (the back and forth motion) takes the same amount of time, then the motion is Periodic.
  • To discuss the period motion we need to define the following terms;
  • Cycle; a complete to-and-fro movement.
  • Amplitude; maximum displacement, the greatest distance from the equilibrium position.
  • Period; time taken for one complete oscillation.
  • Frequency; number of oscillations in one second.

The Periodic Nature of SHM and Simple Pendulum

  • The following equations represent the mathematical relationship between frequency and period of motion;

f=1T\large f = \frac{1}{T} \quad and T=1f\quad\large T = \frac{1}{f}


Position as a Function of Time
Since the motion is considered as a periodic motion, we would be able to plot position Vs. time graph.

The Periodic Nature of SHM and Simple Pendulum


Looking at the graph we can refer to it as the cosine function, since at t = 0 the position is maximum;

x(t)=Acoswt x \, (t) = Acoswt


w= w = angular velocity 
w=2πf=2π/T w = 2 \pi f = 2 \pi / T

x(t)=Acos(2πft)=Acos x \, (t) = Acos(2 \pi f t) = Acos (2πtT) \large (\frac{2 \pi t}{T})


  • For one complete cycle,


The Periodic Nature of SHM and Simple Pendulum


Velocity as a Function of Time

  • Velocity is defined as the derivative of position with respect to time,

v=v = dxdt\large \frac{dx}{dt} == ddt\large \frac{d}{dt} (Acoswt)=Aw(sinwt) (Acoswt) = - Aw (sinwt)


Since the Velocity is the sine function of time, at t=0,V=0t = 0, V = 0, therefore; the graph starts at zero.

The Periodic Nature of SHM and Simple Pendulum


Acceleration as a Function of Time

  • Acceleration is defined as the derivative of velocity with respect to time,

a=a = dvdt\large \frac{dv}{dt} = = ddt \large \frac{d}{dt} (Awsinwt)=Aw2(coswt) (-Awsinwt) = - Aw^{2} \, (coswt)


The Periodic Nature of SHM and Simple Pendulum


a=a = Fm\large \frac{F}{m} == - kxm\large \frac{kx}{m} = =- (kAm)\large (\frac{kA}{m}) coswt=amaxcoswt coswt = - a_{max} coswt

amax= \Rightarrow a_{max} = kAm \large \frac{kA}{m}

Simple Pendulum

  • A simple pendulum is a small mass attached to the end of a string.
  • The pendulum swings back and forth, ignoring the air resistance, it resembles simple harmonic motion.


The Periodic Nature of SHM and Simple Pendulum

Let’s apply the simple harmonic oscillator to the case of the simple pendulum;

x(t)=Acosx(t) = Acos (2πTt) \large (\frac{2 \pi}{T} \, t)


In the case of simple pendulum;
AA; is the maximum angular displacement, θmax\theta _{max}
xx; is the angle the pendulum is at; the angle is measured from the equilibrium positon (the vertical position),θ \theta .

θ(t)=θmaxcos \theta \, (t) = \theta_{max} cos (2πTt)\large (\frac{2 \pi}{T}\, t)


The Periodic Nature of SHM and Simple Pendulum

  • The restoring force is opposite to the displacement and is equal to the component of the weight;


  • F=mgsinθF = - mgsin \theta

  • In this case, the motion is considered to be simple harmonic motion if the angle is less than 15°, for small angles, sinθθ \sin \theta \approx \theta ;

The Periodic Nature of SHM and Simple Pendulum


Form Hooke’s law;
F=KxF = - Kx
K=\qquad \qquad \qquad \qquad \qquad \qquad \Rightarrow K = mgL\large \frac{mg}{L}

FF \approx - mgxL\large \frac{mgx}{L}


We know from spring- mass system;


T=2πmkT=2πmmg/LT=2πLgT = 2 \pi \, \sqrt{\frac{m}{k}} \, \Rightarrow \, T = 2 \pi \, \sqrt{\frac{m}{mg\, / \, L} } \, \Rightarrow \, T = 2 \pi \, \sqrt{\frac{L}{g}}

and f=f = 12π\large \frac{1}{2 \pi} gLK=mgL \sqrt{\frac{g}{L}} \enspace K = \frac{mg}{L}
  • Intro Lesson
  • 2.
    A 6 kg block is attached to a spring wit a spring constant of 216 N/m. The spring is stretched to a length of 12cm and then released.
  • 3.
    A mass of 2.40 kg is attached to a horizontal spring with a spring constant of 121N/m. It is stretched to a length of 10.0cm and released from test.
  • 4.
    A mass is attached to a horizontal spring, and oscillates with a period of 1.4s and with an amplitude of 12cm. At t=0t=0s, the mass is 12cm to the right of the equilibrium positon.
  • 5.
    A simple Pendulum has a length of 42.0cm and makes 62.0 complete oscillation in 3.0 min.
  • 6.
    The length of a simple pendulum in 0.86m, the pendulum bob has a mass of 265 g and it is released to an angle of 11.0° to the vertical.
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The periodic nature of SHM and simple pendulum

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