Work, energy, periodic motion, wave characteristics | Translational motion, forces, work, energy, and equilibrium | Chem/phys

Energy of point object systems

Work and energy are closely related: doing work transfers energy. All forms of energy are measured in Joules (J). A key form is kinetic energy, given by

$K E = \frac{1}{2} m v^{2}$

where $m$ is mass and $v$ is speed.

From this equation, the unit of energy is

$kg \cdot m^{2} / s^{2}$

For a constant speed, kinetic energy is proportional to mass. For example, doubling $m$ doubles $K E$ .

For a fixed mass, speed has a larger effect because it’s squared. Doubling $v$ makes $K E$ four times larger. So changes in speed typically change kinetic energy more dramatically than the same proportional changes in mass.

Potential energy (PE)

Potential energy is energy stored because of an object’s position or configuration.

Near Earth’s surface, gravitational potential energy is

$PE = m g h$ ,

where $m$ is mass, $g$ is the acceleration due to gravity, and $h$ is height above the ground. This expression is local because it assumes Earth’s gravitational field, where $g \approx 9.8 m/ s^{2}$ (and $g$ differs on other planets).

For elastic systems, the energy stored in a spring is

$PE = \frac{1}{2} k x^{2}$ ,

where $k$ is the spring constant and $x$ is the displacement from equilibrium. A larger $k$ means a stiffer spring, so more energy is required to stretch or compress it by the same amount.

More generally, the gravitational potential energy between any two masses is

$PE = - \frac{G m M}{r}$ ,

where $G$ is the universal gravitational constant, $m$ and $M$ are the masses, and $r$ is the distance between their centers.

The negative sign indicates that you must do work to separate the masses against gravity.

Conservation of energy

Conservation of energy means the total energy of an isolated system stays constant: energy can change form, but it isn’t created or destroyed.

For example:

As an object falls, gravitational potential energy is converted into kinetic energy, while the total stays the same.
When a crate slides to a stop on a rough surface, its kinetic energy is converted into heat and sound energy, but the total energy is still conserved.

Power, units

Power is the rate at which energy is transferred or used.

Unit: Watt (W)
Relationship: $1 W = 1 J/s$

Doing the same amount of work in less time requires more power (for example, lifting a crate in one minute instead of one hour).

Periodic motion

Amplitude, frequency, phase

Periodic motion repeats in cycles and is described using amplitude, period, and frequency.

Amplitude ( $A$ ): the maximum displacement from equilibrium (meters). Larger amplitude generally corresponds to more energy in the system.
Period ( $T$ ): the time for one complete cycle (seconds).
Frequency ( $f$ ): the number of cycles per second (Hertz, $Hz$ ).

Period and frequency are inversely related: a shorter period means a higher frequency.

Sometimes frequency is given in revolutions per minute (rpm). To convert from cycles per second to rpm, multiply by $60$ .

Angular frequency ( $ω$ ) is the rate of oscillation in radians per second:

$ω = 2 π f$

It’s also called angular velocity in this context.

Hooke’s Law

$F = - k x$ The restoring force exerted by a spring, where $k$ is the spring constant (a measure of stiffness) and $x$ is the displacement from the equilibrium position. The maximum displacement, or amplitude, is represented by $A$ .

The energy stored in the spring as a result of its deformation is its potential energy ( $PE$ ), calculated as $PE = \frac{1}{2} k x^{2}$
The kinetic energy ( $K E$ ) of a moving mass is given by $K E = \frac{1}{2} m v^{2}$

At the equilibrium position ( $x = 0$ ), the potential energy is zero and the kinetic energy reaches its maximum value.

Conversely, at maximum displacement ( $x = A$ ), the potential energy is at its peak and the kinetic energy is zero. Throughout the motion, the sum of potential and kinetic energy remains constant, equal to the maximum energy ( $\frac{1}{2} k A^{2}$ ), demonstrating the conservation of mechanical energy.

Hooke's law: force vs. elongation with spring illustrations

Simple harmonic motion is periodic motion in which an object’s displacement from equilibrium varies sinusoidally with time.

A common mathematical description is

$x = A \cdot sin (ω t)$

where $A$ is the amplitude (maximum displacement) and $ω$ is the angular frequency.

Common examples include a mass on a spring, a swinging pendulum (for small angles), and the projection of uniform circular motion onto one axis, which produces a sinusoidal pattern over time.

Spring-mass system

For a spring-mass system, the period ( $T$ ) is

$T = 2 π \sqrt (m / k)$

where $m$ is the attached mass and $k$ is the spring constant.

This equation implies:

A heavier mass increases the period
A stiffer spring (larger $k$ ) decreases the period

You can also describe the motion using angular frequency ( $ω$ ):

$ω = \frac{k}{m}$

A higher angular frequency means the system oscillates more rapidly, which happens when the spring is stiffer or the mass is smaller.

Pendulum motion

The period ( $T$ ) of a simple pendulum is

$T = 2 π \frac{L}{g}$

where $L$ is the string length and $g$ is gravitational acceleration (about $9.8 m/ s^{2}$ ).

The angular frequency ( $ω$ ) is

$ω = \frac{g}{L}$

So a pendulum swings faster when gravity is stronger or the string is shorter.

General periodic motion: velocity, amplitude

In periodic motion (without energy losses), the sum of potential energy and kinetic energy stays constant.

At the equilibrium position (displacement $= 0$ ), potential energy is zero and kinetic energy is at its maximum ( $\frac{1}{2} m v^{2}$ ).
At maximum displacement (the amplitude, $A$ ), kinetic energy is zero and potential energy is at its maximum.

For a spring, the maximum potential energy is $\frac{1}{2} k A^{2}$ .

For a pendulum, it is $m g A$ , with $A$ representing the maximum height reached during the swing.

By setting maximum kinetic energy equal to maximum potential energy, you can solve for the amplitude from the equilibrium speed, or solve for the equilibrium speed from the amplitude.

Wave characteristics

Transverse and longitudinal waves: wavelength and propagation speed

A transverse wave has medium displacement perpendicular to the direction of propagation. This is typical of electromagnetic radiation (such as light) and can also be seen in waves on a string. For a string, wave speed depends on the square root of the string tension divided by the mass per unit length, so stiffer, lighter strings produce faster waves.

A longitudinal wave has displacement parallel to the direction of travel. This is characteristic of sound, pressure waves, and seismic vibrations produced during earthquakes.

Wavelength, frequency, and velocity

Wavelength, frequency, and wave speed are related by

$v = f λ$

where $v$ (sometimes written as $c$ ) is the wave speed, $f$ (sometimes written as $ν$ ) is the frequency in Hertz, and $λ$ is the wavelength (meters), the distance between successive wave peaks.

Amplitude, intensity

The amplitude of a wave is its maximum displacement from equilibrium and is directly related to the energy the wave carries: greater amplitude means more energy.

Intensity is the energy transmitted per unit area per unit time (power per area). As a result, increasing amplitude typically increases intensity.

For electromagnetic waves such as light, increasing amplitude and intensity increases the total energy carried by the wave, but it does not change the energy per photon, which depends only on wavelength. Shorter wavelengths (higher frequencies) correspond to higher-energy photons.

Superposition of waves, interference, addition

Superposition means that when waves overlap, their displacements add. This produces interference, which can increase or decrease the resulting amplitude.

When waves are in phase (peaks align with peaks and troughs with troughs), they produce constructive interference, increasing amplitude.
When waves are out of phase (peaks align with troughs), they produce destructive interference, reducing amplitude or canceling the waves.

Resonance

Resonance occurs when an oscillating system reaches maximum amplitude, typically when it is driven at one of its resonance frequencies. For example, a vibrating string or air column can form standing waves with large amplitudes at these frequencies.

Resonance frequencies can be found using

$f = v / λ$

where $f$ is frequency, $v$ is wave speed, and $λ$ is wavelength.

For strings or tubes open at both ends, the condition is $L = n /2 λ$
For tubes with one closed end, the condition is $L = n /4 λ$

Here, $L$ is the length of the medium and $n$ is the harmonic number.

Standing waves formed at resonance do not travel. Instead, they remain fixed in place, with nodes (points of no oscillation) and antinodes (points of maximum oscillation).

Beats

When two waves with slightly different frequencies interfere, beats occur. The beat frequency is the difference between the two frequencies, producing periodic changes in amplitude.

Refraction and diffraction

Refraction is the bending of waves as they pass from one medium into another. It is governed by Snell’s law:

$n_{1} s in θ_{₁} = n_{2} s in θ_{2}$

where $n$ is the refractive index and $θ$ is the angle measured from the normal.

When light enters a denser medium, it bends toward the normal. A specific type of refraction, called dispersion, occurs when light separates into its component colors (for example, in a prism or water droplets), producing a rainbow.

Diffraction is the spreading of waves as they encounter obstacles or pass through narrow openings. This spreading explains why sound can be heard around corners and why light passing through a small aperture forms a diffuse pattern rather than a sharp dot. Diffraction also underlies the interference patterns seen in single- and double-slit experiments.

Energy of point object systems

Work transfers energy; both measured in Joules (J)
Kinetic energy: $K E = \frac{1}{2} m v^{2}$
- Proportional to mass, but more sensitive to speed (squared)
Potential energy forms:
- Near Earth: $PE = m g h$
- Spring: $PE = \frac{1}{2} k x^{2}$
- Universal gravity: $PE = - \frac{G m M}{r}$
Conservation of energy: total energy constant in isolated systems
Power: rate of energy transfer, $1 W = 1 J/s$

Periodic motion

Amplitude ( $A$ ): max displacement from equilibrium
Period ( $T$ ): time for one cycle; Frequency ( $f$ ): cycles per second ( $Hz$ )
- $T = 1/ f$ , $ω = 2 π f$
Hooke’s Law: $F = - k x$
- Spring potential energy: $PE = \frac{1}{2} k x^{2}$
- At $x = 0$ : $PE = 0$ , $K E$ max; at $x = A$ : $K E = 0$ , $PE$ max
- Total mechanical energy conserved: $\frac{1}{2} k A^{2}$
Simple harmonic motion: $x = A sin (ω t)$
Spring-mass system:
- Period: $T = 2 π m / k$
- Angular frequency: $ω = k / m$
Pendulum:
- Period: $T = 2 π L / g$
- Angular frequency: $ω = g / L$
Energy exchange: $K E$ and $PE$ trade off, sum remains constant

Wave characteristics

Transverse waves: displacement perpendicular to propagation (e.g., light, string)
- Wave speed $\propto$ $tension / mass per length$
Longitudinal waves: displacement parallel to propagation (e.g., sound)
Wave equation: $v = f λ$
- $v$ : speed, $f$ : frequency, $λ$ : wavelength
Amplitude: max displacement, relates to wave energy
Intensity: power per area, increases with amplitude
- For light: energy per photon depends on wavelength, not amplitude
Superposition: overlapping waves add displacements
- Constructive interference: in phase, amplitude increases
- Destructive interference: out of phase, amplitude decreases/cancels
Resonance: max amplitude at resonance frequencies
- Standing waves: $f = v / λ$ , nodes and antinodes
- Strings/tubes: $L = n /2 λ$ (open), $L = n /4 λ$ (one end closed)
Beats: interference of close frequencies, beat frequency = frequency difference
Refraction: wave bending at medium boundary, Snell’s law: $n_{1} sin θ_{1} = n_{2} sin θ_{2}$
- Dispersion: separation of colors by wavelength
Diffraction: wave spreading around obstacles/openings
- Explains interference patterns (single/double slit)

Energy of point object systems

Work and energy are closely related: doing work transfers energy. All forms of energy are measured in Joules (J). A key form is kinetic energy, given by

$K E = \frac{1}{2} m v^{2}$

where $m$ is mass and $v$ is speed.

From this equation, the unit of energy is

$kg \cdot m^{2} / s^{2}$

For a constant speed, kinetic energy is proportional to mass. For example, doubling $m$ doubles $K E$ .

Potential energy (PE)

Potential energy is energy stored because of an object’s position or configuration.

Near Earth’s surface, gravitational potential energy is

$PE = m g h$ ,

For elastic systems, the energy stored in a spring is

$PE = \frac{1}{2} k x^{2}$ ,

More generally, the gravitational potential energy between any two masses is

$PE = - \frac{G m M}{r}$ ,

where $G$ is the universal gravitational constant, $m$ and $M$ are the masses, and $r$ is the distance between their centers.

The negative sign indicates that you must do work to separate the masses against gravity.

Conservation of energy

Conservation of energy means the total energy of an isolated system stays constant: energy can change form, but it isn’t created or destroyed.

For example:

As an object falls, gravitational potential energy is converted into kinetic energy, while the total stays the same.
When a crate slides to a stop on a rough surface, its kinetic energy is converted into heat and sound energy, but the total energy is still conserved.

Power, units

Power is the rate at which energy is transferred or used.

Unit: Watt (W)
Relationship: $1 W = 1 J/s$

Doing the same amount of work in less time requires more power (for example, lifting a crate in one minute instead of one hour).

Periodic motion

Amplitude, frequency, phase

Periodic motion repeats in cycles and is described using amplitude, period, and frequency.

Amplitude ( $A$ ): the maximum displacement from equilibrium (meters). Larger amplitude generally corresponds to more energy in the system.
Period ( $T$ ): the time for one complete cycle (seconds).
Frequency ( $f$ ): the number of cycles per second (Hertz, $Hz$ ).

Period and frequency are inversely related: a shorter period means a higher frequency.

Sometimes frequency is given in revolutions per minute (rpm). To convert from cycles per second to rpm, multiply by $60$ .

Angular frequency ( $ω$ ) is the rate of oscillation in radians per second:

$ω = 2 π f$

It’s also called angular velocity in this context.

Hooke’s Law

The energy stored in the spring as a result of its deformation is its potential energy ( $PE$ ), calculated as $PE = \frac{1}{2} k x^{2}$
The kinetic energy ( $K E$ ) of a moving mass is given by $K E = \frac{1}{2} m v^{2}$

At the equilibrium position ( $x = 0$ ), the potential energy is zero and the kinetic energy reaches its maximum value.

Simple harmonic motion is periodic motion in which an object’s displacement from equilibrium varies sinusoidally with time.

A common mathematical description is

$x = A \cdot sin (ω t)$

where $A$ is the amplitude (maximum displacement) and $ω$ is the angular frequency.

Common examples include a mass on a spring, a swinging pendulum (for small angles), and the projection of uniform circular motion onto one axis, which produces a sinusoidal pattern over time.

Spring-mass system

For a spring-mass system, the period ( $T$ ) is

$T = 2 π \sqrt (m / k)$

where $m$ is the attached mass and $k$ is the spring constant.

This equation implies:

A heavier mass increases the period
A stiffer spring (larger $k$ ) decreases the period

You can also describe the motion using angular frequency ( $ω$ ):

$ω = \frac{k}{m}$

A higher angular frequency means the system oscillates more rapidly, which happens when the spring is stiffer or the mass is smaller.

Pendulum motion

The period ( $T$ ) of a simple pendulum is

$T = 2 π \frac{L}{g}$

where $L$ is the string length and $g$ is gravitational acceleration (about $9.8 m/ s^{2}$ ).

The angular frequency ( $ω$ ) is

$ω = \frac{g}{L}$

So a pendulum swings faster when gravity is stronger or the string is shorter.

General periodic motion: velocity, amplitude

In periodic motion (without energy losses), the sum of potential energy and kinetic energy stays constant.

At the equilibrium position (displacement $= 0$ ), potential energy is zero and kinetic energy is at its maximum ( $\frac{1}{2} m v^{2}$ ).
At maximum displacement (the amplitude, $A$ ), kinetic energy is zero and potential energy is at its maximum.

For a spring, the maximum potential energy is $\frac{1}{2} k A^{2}$ .

For a pendulum, it is $m g A$ , with $A$ representing the maximum height reached during the swing.

By setting maximum kinetic energy equal to maximum potential energy, you can solve for the amplitude from the equilibrium speed, or solve for the equilibrium speed from the amplitude.

Wave characteristics

Transverse and longitudinal waves: wavelength and propagation speed

A longitudinal wave has displacement parallel to the direction of travel. This is characteristic of sound, pressure waves, and seismic vibrations produced during earthquakes.

Wavelength, frequency, and velocity

Wavelength, frequency, and wave speed are related by

$v = f λ$

Amplitude, intensity

The amplitude of a wave is its maximum displacement from equilibrium and is directly related to the energy the wave carries: greater amplitude means more energy.

Intensity is the energy transmitted per unit area per unit time (power per area). As a result, increasing amplitude typically increases intensity.

Superposition of waves, interference, addition

Superposition means that when waves overlap, their displacements add. This produces interference, which can increase or decrease the resulting amplitude.

When waves are in phase (peaks align with peaks and troughs with troughs), they produce constructive interference, increasing amplitude.
When waves are out of phase (peaks align with troughs), they produce destructive interference, reducing amplitude or canceling the waves.

Resonance

Resonance frequencies can be found using

$f = v / λ$

where $f$ is frequency, $v$ is wave speed, and $λ$ is wavelength.

For strings or tubes open at both ends, the condition is $L = n /2 λ$
For tubes with one closed end, the condition is $L = n /4 λ$

Here, $L$ is the length of the medium and $n$ is the harmonic number.

Standing waves formed at resonance do not travel. Instead, they remain fixed in place, with nodes (points of no oscillation) and antinodes (points of maximum oscillation).

Beats

When two waves with slightly different frequencies interfere, beats occur. The beat frequency is the difference between the two frequencies, producing periodic changes in amplitude.

Refraction and diffraction

Refraction is the bending of waves as they pass from one medium into another. It is governed by Snell’s law:

$n_{1} s in θ_{₁} = n_{2} s in θ_{2}$

where $n$ is the refractive index and $θ$ is the angle measured from the normal.

Energy of point object systems

Work transfers energy; both measured in Joules (J)
Kinetic energy: $K E = \frac{1}{2} m v^{2}$
- Proportional to mass, but more sensitive to speed (squared)
Potential energy forms:
- Near Earth: $PE = m g h$
- Spring: $PE = \frac{1}{2} k x^{2}$
- Universal gravity: $PE = - \frac{G m M}{r}$
Conservation of energy: total energy constant in isolated systems
Power: rate of energy transfer, $1 W = 1 J/s$

Periodic motion

Amplitude ( $A$ ): max displacement from equilibrium
Period ( $T$ ): time for one cycle; Frequency ( $f$ ): cycles per second ( $Hz$ )
- $T = 1/ f$ , $ω = 2 π f$
Hooke’s Law: $F = - k x$
- Spring potential energy: $PE = \frac{1}{2} k x^{2}$
- At $x = 0$ : $PE = 0$ , $K E$ max; at $x = A$ : $K E = 0$ , $PE$ max
- Total mechanical energy conserved: $\frac{1}{2} k A^{2}$
Simple harmonic motion: $x = A sin (ω t)$
Spring-mass system:
- Period: $T = 2 π m / k$
- Angular frequency: $ω = k / m$
Pendulum:
- Period: $T = 2 π L / g$
- Angular frequency: $ω = g / L$
Energy exchange: $K E$ and $PE$ trade off, sum remains constant

Wave characteristics

Transverse waves: displacement perpendicular to propagation (e.g., light, string)
- Wave speed $\propto$ $tension / mass per length$
Longitudinal waves: displacement parallel to propagation (e.g., sound)
Wave equation: $v = f λ$
- $v$ : speed, $f$ : frequency, $λ$ : wavelength
Amplitude: max displacement, relates to wave energy
Intensity: power per area, increases with amplitude
- For light: energy per photon depends on wavelength, not amplitude
Superposition: overlapping waves add displacements
- Constructive interference: in phase, amplitude increases
- Destructive interference: out of phase, amplitude decreases/cancels
Resonance: max amplitude at resonance frequencies
- Standing waves: $f = v / λ$ , nodes and antinodes
- Strings/tubes: $L = n /2 λ$ (open), $L = n /4 λ$ (one end closed)
Beats: interference of close frequencies, beat frequency = frequency difference
Refraction: wave bending at medium boundary, Snell’s law: $n_{1} sin θ_{1} = n_{2} sin θ_{2}$
- Dispersion: separation of colors by wavelength
Diffraction: wave spreading around obstacles/openings
- Explains interference patterns (single/double slit)

Energy of point object systems

Work and energy are closely related: doing work transfers energy. All forms of energy are measured in Joules (J). A key form is kinetic energy, given by

$K E = \frac{1}{2} m v^{2}$

where $m$ is mass and $v$ is speed.

From this equation, the unit of energy is

$kg \cdot m^{2} / s^{2}$

For a constant speed, kinetic energy is proportional to mass. For example, doubling $m$ doubles $K E$ .

Potential energy (PE)

Potential energy is energy stored because of an object’s position or configuration.

Near Earth’s surface, gravitational potential energy is

$PE = m g h$ ,

For elastic systems, the energy stored in a spring is

$PE = \frac{1}{2} k x^{2}$ ,

More generally, the gravitational potential energy between any two masses is

$PE = - \frac{G m M}{r}$ ,

where $G$ is the universal gravitational constant, $m$ and $M$ are the masses, and $r$ is the distance between their centers.

The negative sign indicates that you must do work to separate the masses against gravity.

Conservation of energy

Conservation of energy means the total energy of an isolated system stays constant: energy can change form, but it isn’t created or destroyed.

For example:

As an object falls, gravitational potential energy is converted into kinetic energy, while the total stays the same.
When a crate slides to a stop on a rough surface, its kinetic energy is converted into heat and sound energy, but the total energy is still conserved.

Power, units

Power is the rate at which energy is transferred or used.

Unit: Watt (W)
Relationship: $1 W = 1 J/s$

Doing the same amount of work in less time requires more power (for example, lifting a crate in one minute instead of one hour).

Periodic motion

Amplitude, frequency, phase

Periodic motion repeats in cycles and is described using amplitude, period, and frequency.

Amplitude ( $A$ ): the maximum displacement from equilibrium (meters). Larger amplitude generally corresponds to more energy in the system.
Period ( $T$ ): the time for one complete cycle (seconds).
Frequency ( $f$ ): the number of cycles per second (Hertz, $Hz$ ).

Period and frequency are inversely related: a shorter period means a higher frequency.

Sometimes frequency is given in revolutions per minute (rpm). To convert from cycles per second to rpm, multiply by $60$ .

Angular frequency ( $ω$ ) is the rate of oscillation in radians per second:

$ω = 2 π f$

It’s also called angular velocity in this context.

Hooke’s Law

The energy stored in the spring as a result of its deformation is its potential energy ( $PE$ ), calculated as $PE = \frac{1}{2} k x^{2}$
The kinetic energy ( $K E$ ) of a moving mass is given by $K E = \frac{1}{2} m v^{2}$

At the equilibrium position ( $x = 0$ ), the potential energy is zero and the kinetic energy reaches its maximum value.

Simple harmonic motion is periodic motion in which an object’s displacement from equilibrium varies sinusoidally with time.

A common mathematical description is

$x = A \cdot sin (ω t)$

where $A$ is the amplitude (maximum displacement) and $ω$ is the angular frequency.

Common examples include a mass on a spring, a swinging pendulum (for small angles), and the projection of uniform circular motion onto one axis, which produces a sinusoidal pattern over time.

Spring-mass system

For a spring-mass system, the period ( $T$ ) is

$T = 2 π \sqrt (m / k)$

where $m$ is the attached mass and $k$ is the spring constant.

This equation implies:

A heavier mass increases the period
A stiffer spring (larger $k$ ) decreases the period

You can also describe the motion using angular frequency ( $ω$ ):

$ω = \frac{k}{m}$

A higher angular frequency means the system oscillates more rapidly, which happens when the spring is stiffer or the mass is smaller.

Pendulum motion

The period ( $T$ ) of a simple pendulum is

$T = 2 π \frac{L}{g}$

where $L$ is the string length and $g$ is gravitational acceleration (about $9.8 m/ s^{2}$ ).

The angular frequency ( $ω$ ) is

$ω = \frac{g}{L}$

So a pendulum swings faster when gravity is stronger or the string is shorter.

General periodic motion: velocity, amplitude

In periodic motion (without energy losses), the sum of potential energy and kinetic energy stays constant.

At the equilibrium position (displacement $= 0$ ), potential energy is zero and kinetic energy is at its maximum ( $\frac{1}{2} m v^{2}$ ).
At maximum displacement (the amplitude, $A$ ), kinetic energy is zero and potential energy is at its maximum.

For a spring, the maximum potential energy is $\frac{1}{2} k A^{2}$ .

For a pendulum, it is $m g A$ , with $A$ representing the maximum height reached during the swing.

By setting maximum kinetic energy equal to maximum potential energy, you can solve for the amplitude from the equilibrium speed, or solve for the equilibrium speed from the amplitude.

Wave characteristics

Transverse and longitudinal waves: wavelength and propagation speed

A longitudinal wave has displacement parallel to the direction of travel. This is characteristic of sound, pressure waves, and seismic vibrations produced during earthquakes.

Wavelength, frequency, and velocity

Wavelength, frequency, and wave speed are related by

$v = f λ$

Amplitude, intensity

The amplitude of a wave is its maximum displacement from equilibrium and is directly related to the energy the wave carries: greater amplitude means more energy.

Intensity is the energy transmitted per unit area per unit time (power per area). As a result, increasing amplitude typically increases intensity.

Superposition of waves, interference, addition

Superposition means that when waves overlap, their displacements add. This produces interference, which can increase or decrease the resulting amplitude.

When waves are in phase (peaks align with peaks and troughs with troughs), they produce constructive interference, increasing amplitude.
When waves are out of phase (peaks align with troughs), they produce destructive interference, reducing amplitude or canceling the waves.

Resonance

Resonance frequencies can be found using

$f = v / λ$

where $f$ is frequency, $v$ is wave speed, and $λ$ is wavelength.

For strings or tubes open at both ends, the condition is $L = n /2 λ$
For tubes with one closed end, the condition is $L = n /4 λ$

Here, $L$ is the length of the medium and $n$ is the harmonic number.

Standing waves formed at resonance do not travel. Instead, they remain fixed in place, with nodes (points of no oscillation) and antinodes (points of maximum oscillation).

Beats

When two waves with slightly different frequencies interfere, beats occur. The beat frequency is the difference between the two frequencies, producing periodic changes in amplitude.

Refraction and diffraction

Refraction is the bending of waves as they pass from one medium into another. It is governed by Snell’s law:

$n_{1} s in θ_{₁} = n_{2} s in θ_{2}$

where $n$ is the refractive index and $θ$ is the angle measured from the normal.

Energy of point object systems

Work transfers energy; both measured in Joules (J)
Kinetic energy: $K E = \frac{1}{2} m v^{2}$
- Proportional to mass, but more sensitive to speed (squared)
Potential energy forms:
- Near Earth: $PE = m g h$
- Spring: $PE = \frac{1}{2} k x^{2}$
- Universal gravity: $PE = - \frac{G m M}{r}$
Conservation of energy: total energy constant in isolated systems
Power: rate of energy transfer, $1 W = 1 J/s$

Periodic motion

Amplitude ( $A$ ): max displacement from equilibrium
Period ( $T$ ): time for one cycle; Frequency ( $f$ ): cycles per second ( $Hz$ )
- $T = 1/ f$ , $ω = 2 π f$
Hooke’s Law: $F = - k x$
- Spring potential energy: $PE = \frac{1}{2} k x^{2}$
- At $x = 0$ : $PE = 0$ , $K E$ max; at $x = A$ : $K E = 0$ , $PE$ max
- Total mechanical energy conserved: $\frac{1}{2} k A^{2}$
Simple harmonic motion: $x = A sin (ω t)$
Spring-mass system:
- Period: $T = 2 π m / k$
- Angular frequency: $ω = k / m$
Pendulum:
- Period: $T = 2 π L / g$
- Angular frequency: $ω = g / L$
Energy exchange: $K E$ and $PE$ trade off, sum remains constant

Wave characteristics

Transverse waves: displacement perpendicular to propagation (e.g., light, string)
- Wave speed $\propto$ $tension / mass per length$
Longitudinal waves: displacement parallel to propagation (e.g., sound)
Wave equation: $v = f λ$
- $v$ : speed, $f$ : frequency, $λ$ : wavelength
Amplitude: max displacement, relates to wave energy
Intensity: power per area, increases with amplitude
- For light: energy per photon depends on wavelength, not amplitude
Superposition: overlapping waves add displacements
- Constructive interference: in phase, amplitude increases
- Destructive interference: out of phase, amplitude decreases/cancels
Resonance: max amplitude at resonance frequencies
- Standing waves: $f = v / λ$ , nodes and antinodes
- Strings/tubes: $L = n /2 λ$ (open), $L = n /4 λ$ (one end closed)
Beats: interference of close frequencies, beat frequency = frequency difference
Refraction: wave bending at medium boundary, Snell’s law: $n_{1} sin θ_{1} = n_{2} sin θ_{2}$
- Dispersion: separation of colors by wavelength
Diffraction: wave spreading around obstacles/openings
- Explains interference patterns (single/double slit)