Wize University Physics Textbook (Master) > Periodic Motion: Oscillations

Basic Concepts of Oscillation

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.

\boxed{F=-kx}

Image illustrating Concepts of Oscillation (Basic Version)

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

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

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).

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

Frequency (f): The number of oscillations per unit of time (Frequency is measured in Hertz (1 Hz=1/s)).

$\boxed{f=\dfrac{1}{T}}$

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

\boxed{\omega=2\pi f=\dfrac{2\pi}{T}}

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.

\boxed{F=-kx}

Image illustrating Concepts of Oscillation (Advanced Version)

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

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

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).

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

Frequency (f): The number of oscillations per unit of time (Frequency is measured in Hertz (1 Hz=1/s)).

$\boxed{f=\dfrac{1}{T}}$

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

\boxed{\omega=2\pi f=\dfrac{2\pi}{T}}

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

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

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.

$x\left(t\right)=1.7\ \cos\ \left(14.5t\right)$

$x\left(t\right)=3.4\ \cos\ \left(14.5t\right)$

$x\left(t\right)=1.7\ \sin\ \left(29t\right)$

$x\left(t\right)=1.7\ \cos\ \left(14.5t^2\right)$

I don't know

Extra Practice

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?

Simple Harmonic Motion: Pendulum

You have landed on an unknown planet. You have a simple pendulum with a length of $25.0\ cm$ . You measure the frequency of oscillation of the pendulum to be $0.5\ Hz$ .

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)

The displacement function of a simple harmonic motion over time is shown below. Write the equation of the motion.

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 $\frac{1}{6}$ 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?

Practice: Spring and mass system

A block of unknown mass $m$ is attached to a horizontal spring which has a spring constant $k=4.8\ \frac{N}{m}$ . It is then set oscillating on a frictionless surface with an amplitude $A=12\ cm$ . The position and velocity of the block at $t=0$ are measured to be $+6.0\ cm$ and $+30\ \frac{cm}{s}$ , respectively.

The motion can be modeled as an equation in the form $x\left(t\right)=A\cos\left(\omega t+\varphi\right)$ .

Basic Concepts of Oscillation

Create a free account to get started

Simple Harmonic Motion

Create a free account to get started

Simple Harmonic Motion

Create a free account to get started

Extra Practice