High School
SAT
ACT
University
MCAT
LSAT
DAT
Wize University Physics Textbook (Master) > Periodic Motion: Oscillations

Basic Concepts of Oscillation

TS-DiagramsPrevious Section
Velocity and AccelerationNext Section
Popular Courses
Find My Course
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 kk in 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)  \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.
Extra Practice
Practice: Spring and mass system
A block of unknown mass mmis attached to a horizontal spring which has a spring constant k=4.8 Nmk=4.8\ \frac{N}{m}. It is then set oscillating on a frictionless surface with an amplitude A=12 cmA=12\ cm. The position and velocity of the block at t=0t=0are measured to be +6.0 cm+6.0\ cmand +30 cms+30\ \frac{cm}{s}, respectively.
The motion can be modeled as an equation in the form x(t)=Acos⁡(ωt+φ)x\left(t\right)=A\cos\left(\omega t+\varphi\right).
Simple harmonic motion: Pendulum
In two separate experiments a pendulum and a mass-spring are moved from the Earth to the moon surface where gravitational acceleration is 16\frac{1}{6}of the Earth.
a) How does the amplitude of oscillations change for each system?
b) How does the frequency of oscillations change for each system?
The displacement function of a simple harmonic motion over time is shown below. Write the equation of the motion.
Simple Harmonic Motion: Pendulum
You have landed on an unknown planet. You have a simple pendulum with a length of 25.0 cm25.0\ cm. You measure the frequency of oscillation of the pendulum to be 0.5 Hz0.5\ Hz.
a) What is the gravitational acceleration on this planet? (in m/s^2)
b) What would be the period of the oscillation if you used this pendulum on Earth? (in seconds)
A mass is on a spring. The spring is stretched and then released so that the mass bounces back and forth. There are no other forces in the system. At what point is the block moving the fastest?