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Wize AP Physics C: Mechanics Textbook > Unit 6: Oscillations (6-14%)

Basic Concepts of Oscillation

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Simple Harmonic Motion



Simple Harmonic Motion (SHM) is a type of periodic motion in which:
  • The restoring force is proportional to the displacement from the equilibrium position
  • The Force direction is opposite to that of displacement.

F=−kx\boxed{F=-kx}


Wize Tip
Any physical system with the above type of the force undergoes a Simple Harmonic Motion. Constant kk in the above equations depends on the physical property of the oscillating system.


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The mathematical description of SHM is a sinusoidal function of time:


  x(t)=Acos⁡(ωt)  \boxed{\ \ x\left(t\right)=A\cos\left(\omega t\right)\ \ }




Watch Out!
SHM can be described by either sine or cosine function!

These are the parameters we use to describe a simple harmonic motion:

  • Amplitude (A): The maximum displacement of the object from the equilibrium.
  • Period (T): The time it takes for one complete oscillation (Period is measured in seconds).

T=Total timenumber of cycles\boxed{T=\frac{Total\ time}{number\ of\ cycles}}

  • Frequency (f): The number of oscillations per unit of time (Frequency is measured in Hertz (1 Hz=1/s)).
f=1T\boxed{f=\dfrac{1}{T}}

  • Angular frequency (ω\colorFour{\omega}): The rate at which argument of sinusoidal function describing oscillation changes in time (Angular frequency is measured in rad/srad/s).

ω=2πf=2πT\boxed{\omega=2\pi f=\dfrac{2\pi}{T}}

0:00 / 0:00

Simple Harmonic Motion



Simple Harmonic Motion (SHM) is a type of periodic motion in which:
  • The restoring force is proportional to the displacement from the equilibrium position
  • The Force direction is opposite to that of displacement.

F=−kx\boxed{F=-kx}


Wize Tip
Any physical system with the above type of the force undergoes a Simple Harmonic Motion. Constant kkin the above equations depends on the physical property of the oscillating system.


PAGE BREAK

The mathematical description of SHM is a sinusoidal function of time:


  x(t)=Acos⁡(ωt+φ)  \boxed{\ \ x\left(t\right)=A\cos\left(\omega t+\varphi\right)\ \ }




Watch Out!
SHM can be described by either sine or cosine function!

These are the parameters we use to describe a simple harmonic motion:

  • Amplitude (A): The maximum displacement of the object from the equilibrium.

  • Period (T): The time it takes for one complete oscillation (Period is measured in seconds).

T=Total timenumber of cycles\boxed{T=\frac{Total\ time}{number\ of\ cycles}}

  • Frequency (f): The number of oscillations per unit of time (Frequency is measured in Hertz (1 Hz=1/s)).
f=1T\boxed{f=\dfrac{1}{T}}

  • Angular frequency (ω\colorFour{\omega}): The rate at which argument of sinusoidal function describing oscillation changes in time (Angular frequency is measured in rad/srad/s).

ω=2πf=2πT\boxed{\omega=2\pi f=\dfrac{2\pi}{T}}


  • Phase Constant (φ\colorFour{\varphi}): A term which determines the initial condition of oscillation (Phase constant is measured in radrad).


Phase constant causes a shift in the oscillating function which changes the starting point of oscillation:



Exam Tip
To calculate the phase constant φ\varphi, we need to solve this equation:
x(t=0)=Acos⁡(φ)→φ=cos⁡−1(x(0)A)x\left(t=0\right)=A\cos\left(\varphi\right)\to\varphi=\cos^{-1}\left(\frac{x\left(0\right)}{A}\right)


Watch Out!
It is important to know that solving above equation for the phase constant will give two answers between 00 and 2π2\pi:
φ=θ  or  2π−θ, 0≤θ≪π\varphi=\theta\ \text{ or }\ 2\pi-\theta,\ 0\le\theta\ll\pi
We need to choose between these two values based on the other information of the problem.


Write the equation of motion for a 2.3 kg mass attached to a spring with constant 8.0 N/m, when the mass is released from rest at 0.15 m from equilibrium.

x(t) = A cos (wt)

A = 0.15 m
m = 2.3 kg
k = 8.0 N/m

w = sqrt (k/m) = sqrt (8.0/2.3) = 1.9 rad/s

x(t) = 0.15 cos (1.9t)

A girl on a swing is pushed such that she oscillates with a frequency of 2.3 Hz and an amplitude of 1.7 m. (Mass of the girl is 60kg)

Write an equation (position as a function of time) to describe the motion of the girl on the swing assuming that she starts at amplitude.