Physics Study Guide/Print version/Section Two

Rotational Kinematics

Rotational Kinematics concerns with the description of spinning bodies. The orientation of any object can be described by three angular quantities called the Euler angles.

A two dimensional polar co-ordinate system

A three dimentional spherical-polar co-ordinate system

Euler angles which uniquely describe the orientation of an object in space

Torque and Circular Motion

Circular motion is the motion of a particle at a set distance (called radius) from a point. For circular motion, there needs to be a force that makes the particle turn. This force is called the 'centripetal force.' Please note that the centripetal force is not a new type of force-it is just a force causing rotational motion. To make this clearer, let us study the following examples:

1. If Stan ties a piece of thread to a small pebble and rotates it in a horizontal circle above his head, the circular motion of the pebble is caused by the tension force in the thread.
2. In the case of the motion of the planets around the sun (which is roughly circular), the force is provided by the gravitational force exerted by the sun on the planets.

Thus, we see that the centripetal force acting on a body is always provided by some other type of force -- centripetal force, thus, is simply a name to indicate the force that provides this circular motion. This centripetal force is always acting inward toward the center. You will know this if you swing an object in a circular motion. If you notice carefully, you will see that you have to continously pull inward. We know that an opposite force should exist for this centripetal force(by Newton's 3rd Law of Motion). This is the centrifugal force, which exists only if we study the body from a non-inertial frame of reference(an accelerating frame of reference,such as in circular motion). This is a so-called 'pseudo-force', which is used to make the Newton's law applicable to the person who is inside a non-inertial frame . The centrifugal force is equal and opposite to the centripetal force. It is caused due to inertia of a body.

$\omega_{\mathrm{avg} }=\frac{ \omega_1 +\omega_f }{2} =\frac{\theta }{t}$

Average angular velocity is equal to one-half of the sum of initial and final angular velocities assuming constant acceleration, and is also equal to the angle gone through divided by the time taken.

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$\alpha =\frac{ \Delta \omega }{t}$

Angular acceleration is equal to change in angular velocity divided by time taken.

Angular momentum

$\mathbf{l} = \mathbf{r}\times\mathbf{p} = m(\mathbf{r}\times\mathbf{v})$

Angular momentum of an object revolving around an external axis O is equal to the cross-product of the position vector with respect to O and its linear momentum.

$\mathbf{L} = I\boldsymbol{\omega}$

Angular momentum of a rotating object is equal to the moment of inertia times angular velocity.

$L=I\omega$

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$\tau =I\alpha =\frac{\Delta L}{t}$

Torque is equal to moment of inertia times angular acceleration, which is also equal to the change in angular momentum divided by time taken.

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$K_R =\frac{1}{2} I\omega^2$

Rotational Kinetic Energy is equal to one-half of the product of moment of inertia and the angular velocity squared.

IT IS USEFUL TO NOTE THAT

The equations for rotational motion are analogous to those for linear motion-just look at those listed above. When studying rotational dynamics, remember:

• the place of force is taken by torque

• the place of mass is taken by moment of inertia

• the place of displacement is taken by angle

• the place of linear velocity, momentum, acceleration, etc. is taken by their angular counterparts.

Variables

τ: torque, (N·m) I: moment of inertia, (kg·m2) α: angular acceleration, (rad/s2) L: angular momentum, (kg·m2/s) t: time (s) Kr: rotational kinetic energy, (J = kg·m2/s2) ω: angular velocity, (rad/s)

Definition of terms

Torque (τ): Force times distance. A vector. (N·m) Moment of inertia (I): Describes the object's resistance to torque - the rotational analog to inertial mass. (kg·m2) Angular momentum (L): (kg·m2/s) Angular velocity (ω): (rad/s) Angular acceleration (α): (rad/s2) Time (t): (s)

Buoyancy

Buoyancy is the force due to pressure differences on the top and bottom of an object under a fluid (gas or liquid).

Net force = buoyant force - force due to gravity on the object

Bernoulli's Principle

Fluid flow is a complex phenomenon. An ideal fluid may be described as:

• The fluid flow is steady i.e its velocity at each point is constant with time.
• The fluid is incompressible. This condition applies well to liquids and in certain circumstances to gases.
• The fluid flow is non-viscous. Internal friction is neglected. An object moving through this fluid does not experience a retarding force. We relax this condition in the discussion of Stokes' Law.
• The fluid flow is irrotational. There is no angular momentum of the fluid about any point. A very small wheel placed at an arbitary point in the fluid does not rotate about its center. Note that if turbulence is present, the wheel would most likely rotate and its flow is then not irrotational.

As the fluid moves through a pipe of varying cross-section and elevation, the pressure will change along the pipe. The Swiss physicist Daniel Bernoulli (1700-1782) first derived an expression relating the pressure to fluid speed and height. This result is a consequence of conservation of energy and applies to ideal fluids as described above.

Consider an ideal fluid flowing in a pipe of varying cross-section. A fluid in a section of length Δx₁ moves to the section of length Δx₂ in time Δt. The relation given by Bernoulli is:

$P + \frac{1}{2}\rho v^2 + \rho gh = K$

[where: P is pressure at cross-section, K is a constant, h is height of cross-section, ρ is density, and v is velocity of fluid at cross-section.]

In words, the Bernoulli relation may be stated as: As we move along a streamline the sum of the pressure {P), the kinetic energy per unit volume and the potential energy per unit volume remains a constant.

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(To be concluded)

Fields

A field is one of the more difficult concepts to grasp in physics. Simply put, a field is a collection of vectors often representing the force an object would feel if it were placed at any particular point in space. With gravity, the field is measured in newtons, as it depends solely on the mass of an object, but with electricity, it is measured in newtons per coulomb, as the force on an electrical charge depends on the amount of that charge. Typically these fields are calculated based on canceling out the effect of a body in the point in space that the field is desired. As a result, a field is a vector, and as such, it can (and should) be added when calculating the field created by TWO objects at one point in space.

Fields are typically illustrated through the use of what are called field lines or lines of force. Given a source that exerts a force on points around it, sample lines are drawn representing the direction of the field at points in space around the force-exerting source.

There are three major categories of fields:

1. Uniform fields are fields that have the same value at any point in space. As a result, the lines of force are parallel.
2. Spherical fields are fields that have an origin at a particular point in space and vary at varying distances from that point.
3. Complex fields are fields that are difficult to work with mathematically (except under simple cases, such as fields created by two point object), but field lines can still typically be drawn. Dipoles are a specific kind of complex field.

Magnetism also has a field, measured in teslas, and it also has field lines, but its use is more complicated than simple "force" fields. Secondly, it also only appears in a two-pole form, and as such, is difficult to calculate easily.

The particles that form these magnetic fields and lines of force are called electrons and not magnetons. A magneton is a quantity in magnetism.

Definition of terms

Field: A collection of vectors that often represents the force that an object would feel if it were placed in any point in space. Field Lines: A method of diagramming fields by drawing several sample lines showing direction of the field through several points in space.

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Newtonian Gravity

Newtonian Gravity (simplified gravitation) is an apparent force (a.k.a. pseudoforce) that simulates the attraction of one mass to another mass. Unlike the three fundamental (real) forces of electromagnetism and the strong and weak nuclear forces, gravity is purely attractive. As a force it is measured in newtons. The distance between two objects is measured between their centers of mass.

$\|\boldsymbol{F}\| = \frac{Gm_1m_2}{r^2}$

Gravitational force is equal to the product of the universal gravitational constant and the masses of the two objects, divided by the square of the distance between their centers of mass.

$\|\boldsymbol{g}\| = \frac{Gm_1}{r^2}$

The value of the gravitational field which is equivalent to the acceleration due to gravity caused by an object at a point in space is equal to the first equation about gravitational force, with the effect of the second mass taken out.

$U = -\frac{GMm}{r}$

Gravitational potential energy of a body to infinity is equal to the universal gravitational constant times the mass of a body from which the gravitational field is being created times the mass of the body whose potential energy is being measured over the distance between the two centers of mass. Therefore, the difference in potential energy between two points is the difference of the potential energy from the position of the center of mass to infinity at both points. Near the earth's surface, this approximates:

$\Delta U_g = mgh\$

Potential energy due to gravity near the earth's surface is equal to the product of mass, acceleration due to gravity, and height (elevation) of the object.

If the potential energy from the body's center of mass to infinity is known, however, it is possible to calculate the escape velocity, or the velocity necessary to escape the gravitational field of an object. This can be calculated as follows:

$\|\boldsymbol{v}_{esc}\| = \sqrt{\frac{2GM}{R}}$

This is derived based on utilizing the law of conservation of energy and the equation to calculate kinetic energy. The proof of this will be left to the reader.

Variables

F: force (N) G: universal constant of gravitation, (6.67x10-11 N•m2/kg2) m1: mass of the first body m2: mass of the second body r: the distance between the point at which the force or field is being taken, and the center of mass of the first body g: acceleration due to gravity (on the earth’s surface, this is 9.8 m/s2) U: potential energy from the location of the center of mass to infinity (J) ΔUg: Change in potential energy (J) m and M: mass (kg) h: height of elevation (m) vesc: escape velocity (m/s)

Definition of terms

Universal constant of gravitation (G): This is a constant that is the same everywhere in the known universe and can be used to calculate gravitational attraction and acceleration due to gravity. 6.67x10-11 N·m2/kg2 Mass one (m1): One of two masses that are experiencing a mutual gravitational attraction. We can use this for the mass of the Earth (1023 kg). Mass two (m2): One of two masses that are experiencing a mutual gravitational attraction. This symbol can represent the mass of an object on or close to earth. Units: kilograms (kg) Acceleration due to gravity (g): This is nearly constant near the earth's surface because the mass and radius of the earth are essentially constant. At extreme altitudes the value can vary slightly, but it varies more significantly with latitude. This is also equal to the value of the gravitational field caused by a body at a particular point in space (9.8 m/s2) Escape velocity (vesc): The velocity necessary to completely escape the gravitational effects of a body.

Waves

$v = f \lambda\$

Wave speed is equal to the frequency times the wavelength. It can be understood as how frequently a certain distance (the wavelength in this case) is traversed.

$f = \frac{v}{\lambda}$

Frequency is equal to speed divided by wavelength.

$T = \frac{1}{f}$

Period is equal to the inverse of frequency.

Variables

λ: wavelength (m) v: wave speed (m/s) f: frequency (1/s), (Hz) T: period (s)

Definition of terms

Wavelength (λ): The length of one wave, or the distance from a point on one wave to the same point on the next wave. Units: meters (m). In light, λ tells us the color. Wave speed (v): the speed at which the wave pattern moves across the surface. Units: meters per second, (m/s) Frequency of oscillation (f) (or just frequency): the number of times the wave pattern repeats itself in one second. Units: seconds-1 = (1/s) = hertz (Hz) In sound, f tells us the pitch. The inverse of frequecy is the period of oscillation. Period of oscillation (T) (or just period): duration of time betweeen one wave and the next one passing the same spot. Units: seconds (s). The inverse of the period is frequency. Use a capital, italic T and not a lowercase one, which is used for time. Amplitude (A): the maximum height of the wave measured from the average height of the wave (the wave’s center). Unit: meters (m)

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The wave’s extremes, its peaks and valleys, are called antinodes. At the middle of the wave are points that do not move, called nodes.

Examples of waves: Water waves, sound waves, light waves, seismic waves, shock waves, ultrasonic waves …

Oscillation

A wave is said to oscillate, which means to move back and forth in a regular, repeating way. This fluctuation can be between extremes of position, force, or quantity.

Different types of waves have different types of oscillations.

Longitudinal waves: Oscillation is parallel to the direction of the wave. Examples: sound waves, waves in a spring.

Transverse waves: Oscillation is perpendicular to direction of the wave. Example: light

Interference

When waves overlap each other it is called interference. This is divided into constructive and destructive interference.

Constructive interference: the waves line up perfectly and add to each others’ strength.

Destructive interference: the two waves cancel each other out, resulting in no wave.

Resonance

In real life, waves usually give a mishmash of constructive and destructive interference and quickly die out. However, at certain wavelengths standing waves form, resulting in resonance. These are waves that bounce back into themselves in a strengthening way, reaching maximum amplitude.

Resonance is a special case of forced vibration when the frequency of the impressed periodic force is equal to the natural frequency of the body so that it vibrates with increased amplitude, spontaneously.

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Wave overtones

For resonance in a taut string, the first harmonic is determined for a wave form with either one antinode and two nodes. That is, the two ends of the string are nodes because they do not vibrate while the middle of the string is an antinode because it experiences the greatest change in amplitude. This means that one half of a full wavelength is represented by the length of the resonating structure.

The frequency of the first harmonic is equal to wave speed divided by twice the length of the string. (Recall that wave speed is equal to wavelength times frequency.)

F1 = v/2L

The wavelength of the first harmonic is equal to double the length of the string.

λ1 = 2L

The "nth" wavelength is equal to the fundamental wavelength divided by n.

λn = λ1/n

Harmonics for a taut string*

	Harmonic number	Overtone number	F =	λ =
F1	First harmonic	---	F1 = v/2L	λ1 = 2L
F2	Second harmonic	First overtone	F2 = 2F1	λ2 =λ1/2
F3	Third harmonic	Second overtone	F3 = 3F1	λ3 = λ1/3
Fn	Nth harmonic	(Nth - 1) overtone	Fn = nF1	λn = λ1/n

* or any wave system with two identical ends, such as a pipe with two open or closed ends. In the case of a pipe with two open ends, there are two antinodes at the ends of the pipe and a single node in the middle of the pipe, but the mathematics work out identically.

Definition of terms

Frequency (F): Units: (1/s), hertz (Hz) Fundamental frequency, first harmonic (F)1: The lowest frequency (longest wavelength) allowed for the system. Length of string (L): (or pipe, etc.) Units: meters (m). Wavelength (λ): Units: meters (m).

The first overtone is the first allowed harmonic above the fundamental frequency (F₁).

In the case of a system with two different ends (as in the case of a tube open at one end), the closed end is a node and the open end is an antinode. The first resonant frequency has only a quarter of a wave in the tube. This means that the first harmonic is characterized by a wavelength four times the length of the tube.

F1 = v/4L

The wavelength of the first harmonic is equal to four times thelength of the string.

λ1 = 4L

The "nth" wavelength is equal to the fundamental wavelength divided by n.

λn = λ1/n

Note that "n" must be odd in this case as only odd harmonics will resonate in this situation.

Harmonics for a system with two different ends*

	Harmonic number	Overtone number	F =	λ =
F1	First harmonic	---	F1 = v/4L	λ1 = 4L
F2	Third harmonic	First overtone	F2 = 3F1	λ2 =2λ1/3
F3	Fifth harmonic	Second overtone	F3 = 5F1	λ3 = 2λ1/5
Fn	Nth harmonic†	(Nth - 1)/2 overtone	F(n-1)/2 = nF1	λn = 2λ1/n

* such as a pipe with one end open and one end closed
†In this case only the odd harmonics resonate, so n is an odd integer.

V_s: velocity of sound

• dependant on qualities of the medium transmitting the sound, (the air) such as its density, temperature, and “springiness.” A complicated equation, we concentrate only on temperature.
• increases as temperature increases (molecules move faster.)
• is higher for liquids and solids than for gasses (molecules are closer together.)
• for “room air” is 340 meters per second (m/s).
• Speed of sound is 343 meters per second at 20 degrees C. Based on the material sound is passing through and the temperature, the speed of sound changes.

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Standing waves

$\|\vec{v}\|=\sqrt{\frac{\|\vec{T}\|}{\mu}}$

Wave speed is equal to the square root of tension divided by the linear density of the string.

μ = m/L

Linear density of the string is equal to the mass divided by the length of the string.

λmax = 2L

The fundamental wavelength is equal to two times the length of the string.

Variables

λ: wavelength (m) λmax: fundamental wavelength (m) μ: linear density (g/m) v: wave speed (m/s) F: force (N) m: mass (g) L: length of the string (m) l: meters (m)

Definition of terms

Tension (F): (not frequency) in the string (t is used for time in these equations). Units: newtons (N) Linear density (μ): of the string, Greek mu. Units: grams per meter (g/m) Velocity (v) of the wave (m/s) Mass (m): Units: grams (g). (We would use kilograms but they are too big for most strings). Length of the string (L): Units: meters (m)

Fundamental frequency: the frequency when the wavelength is the longest allowed, this gives us the lowest sound that we can get from the system.

In a string, the length of the string is half of the largest wavelength that can create a standing wave, called its fundamental wavelength.

Sound

style="background-color: #F3F3FF; border: solid 1px #D6D6FF; padding: 1em;" valign=top | $decibel(\mbox{dB}) = 10\cdot \log\left(\frac{I_1}{I_0}\right)$

Fig. 1: The Fletcher-Munson equal-loudness contours. Phons are labelled in blue.

The amplitude is the magnitude of sound pressure change within a sound wave. Sound amplitude can be measured in pascals (Pa), though its more common to refer to the sound (pressure) level as Sound intensity(dB,dBSPL,dB(SPL)), and the percieved sound level as Loudness(dBA, dB(A)). Sound intensity is flow of sound energy per unit time through a fixed area. It has units of watts per square meter. The reference Intensity is defined as the minimum Intensity that is audible to the human ear, it is equal to 10^-12 W/m², or one picowatt per square meter. When the intensity is quoted in decibels this reference value is used. Loudness is sound intensity altered according to the frequency response of the human ear and is measured in a unit called the A-weighted decibel (dB(A), also used to be called phon).

The Decibel

The decibel is not, as is commonly believed, the unit of sound. Sound is measured in terms of pressure. However, the decibel is used to express the pressure as very large variations of pressure are commonly encountered. The decibel is a dimensionless quantity and is used to express the ratio of one power quantity to another. The definition of the decibel is $10\cdot \log_{10}\left(\frac{x}{x_0}\right)$, where x is a squared quantity, ie pressure squared, volts squared etc. The decibel is useful to define relative changes. For instance, the required sound decrease for new cars might be 3 dB, this means, compared to the old car the new car must be 3 dB quieter. The absolute level of the car, in this case, does not matter.

$I_0 = 10^{-12} \mbox{ W}/\mbox{m}^2$

Definition of terms

Intensity (I): the amount of energy transferred through 1 m2 each second. Units: watts per square meter

Lowest audible sound: I = 0 dB = 10-12 W/m2 (A sound with dB < 0 is inaudible to a human.) Threshold of pain: I = 140 dB = 10 W/m2

Sample equation: Change in sound intensity
Δβ = β₂ - β₁
= 10 log(I₂/I₀) - 10 log(I₁/I₀)
= 10 [log(I₂/I₀) - log(I₁/I₀)]
= 10 log[(I₂/I₀)/(I₁/I₀)]
= 10 log(I₂/I₁)
where log is the base-10 logarithm.

Doppler effect

$f' = f( \frac{v \pm v_0}{v \mp v_s} )$

f' is the observed frequency, f is the actual frequency, v is the speed of sound ($v=336+0.6T$), T is temperature in degrees Celsius $v_0$ is the speed of the observer, and $v_s$ is the speed of the source. If the observer is approaching the source, use the top operator (the +) in the numerator, and if the source is approaching the observer, use the top operator (the -) in the denominator. If the observer is moving away from the source, use the bottom operator (the -) in the numerator, and if the source is moving away from the observer, use the bottom operator (the +) in the denominator.

Example problems

A. An ambulance, which is emitting a 40 Hz siren, is moving at a speed of 30 m/s towards a stationary observer. The speed of sound in this case is 339 m/s.

$f' = 40( \frac{339 + 0}{339 - 30} )$

B. An M551 Sheridan, moving at 10 m/s is following a Renault FT-17 which is moving in the same direction at 5 m/s and emitting a 30 Hz tone. The speed of sound in this case is 342 m/s.

$f' = 30( \frac{342 + 10}{342 + 5} )$