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Force

This section is just to post a couple of reminders. Equations are hard to post correctly on Wikidot, do your best and we can work collectively to make the equations look good. Example, if you enter F_x = F cos(theta) for an equation you would like entered, someone (or I) will change it to

*[[math label1]]
F_x = F cos (\theta).
[[/math]]


Chapter 1: Intro and Vectors

  • Scalar: quantity specified by a number (speed)
  • Vector: quantity specifies by direction and magnitude (velocity)

Powers of 10

  • 10-15 = femto ( f )
  • 10-12 = pico ( p )
  • 10-9 = nano ( n )
  • 10-6 = micro ( u )
  • 10-3 = milli ( m )
  • 10-2 = centi ( c )
  • 103 = kilo ( k )
  • 106 = mega ( M )
  • 109 = giga ( G )
  • 1012 = tera ( T )

*SI System Units: Meters (m) Kilograms (kg) and Seconds (s)
*Density: mass per unit volume. p=m/V
*Vectors: quantities that have both magnitude and direction and obey the vector law of addition.
*Scalars: Quantities that add algebraically. When two or more vectors are added together they must all have the same units.
*The magnitude of a vector is always larger than the absolute value of each component, unless there is only one nonzero component, in which case the magnitude of the vector is equal to the absolute value of that component.
*Two vectors can be added using the triangle method. (Figure 1.9)

Chapter 2: Motion in One Dimension

  • Kinematic Equations for Motion in a Straight Line Under Constant Acceleration
Equation Information given by equation
vxf=vxi+axt Velocity as a function of time
xf=xi+1/2(vxf+vxi)t Position as a function of velocity and time
xf=xi+vxit+1/2axt2 Position as a function of time
vxf2=vxi2+2ax(xf-xi) Velocity as a function of position
  • Acceleration : change in velocity over time (vector), with units of m/s2 or ft/s2

- When velocity and acceleration are in same direction = object is speeding up
- When velocity and acceleration are in opposite directions = object is slowing down
- A free falling object at earth's surface has acceleration of - 9.8 m/s2 (toward earth)

*Average Speed: The average speed of a particle during some time interval is equal to the ratio of the distance d traveled by the particle and the time interval delta t.
*Average Velocity: The average velocity of a particle moving in one dimension during some time interval is equal to the ratio of the displacement delta x and the time interval delta t.
*Instantaneous Speed: This is defined as the limit if the ration delta x/delta t as delta t approaches zero.
*Acceleration is a measure of how rapidly the velocity is changing.

Chapter 3: Motion in Two Dimensions

Projectile Motion

  1. The free Fall acceleration g is constant over the range of motion and is directed downward
  2. The effect of air resistance is negligible

*Choose reference frame such that the y direction is vertical and positive upward, ay=-g, and ax=0.

Centripetal Acceleration:

  • ax = v2 / r

*Centripetal acceleration vector is not constant as it always points toward the center of the circle, but it continuously changes direction as the partcile moves around the circular path.

Free Fall Motion

  • trajectory is always a parabola
  • nowhere are velocity and acceleration vectors parallel, but at peak they are perpindicular
  • distance maximum when angle is 45 degrees

Chapter 4: The Laws of Motion

http://www.youtube.com/watch?v=7_fbH-muvlw

Newton's First Law:

  • An object in motion stays in motion unless acted upon by an outside force

Newton's Second Law:

  • F = ma, where F is force, m is mass, a is acceleration

Newton's Third Law:

  • Every force has an equal and opposite force (action - reaction pairs act on different objects)

Chapter 5: More Applications of Newton's Laws

  • Fundamental Forces

1) Gravitational (weakest)
2) Electromagnetics (2nd weakest, responsible for electricity, magnetism, contact forces)
3) Weak Nuclear Force (Radioactive decay)
4) Strong Nuclear Force (holds protons and neutrons together)

  • Force of Friction: resistance to motion because an object interacts with its surroundings
  • Force of Static Friction: counteracts friction
  • Force of Kinetic Friction: friction force for an object in motion

Fk= μkn
μk is a given constant and n is the normal force

Chapter 6: Energy and Energy Transfer

The scalar or dot product of any two vectors A and B:

  • A * B= AB cos θ

Work: A force on an object applied over a distance

  • W=F Δr cosθ

Work Kinetic-Energy System: If you do work on something its kinetic energy changes

  • W=ΔKE=0.5mV2 (final) - 0.5mV2 (initial)

Work-Kinetic Energy Theorem: If you do work on a system and the only change in the system is its speed, the total work done on the system by outside forces equals the change in the system's kinetic energy.

  • Wtotal=Kf-Ki=ΔK

Average power is the time rate of energy transfer:

  • Pavg=W/Δt
  • When work is the form of energy transfer.

The general equation for power is:

  • P=dE/dt

SI unit of power is J/s, also called a watt:

  • 1 W = 1 J/s = 1 kg(m2/s3)

US customary system unit of power is the horsepower:

  • 1 hp = 550 ft(lb/s) = 746 W

Chapter 7: Potential Energy

Gravitational Potential Energy:

  • Ug = mgy or Ug = mgh
  • m is mass, g is the force of gravity and y or h is the height an object is raised.

Elastic Potential Energy:

  • Us = (1/2)kx2
  • k is the spring coefficient, x is the distance the spring is pushed OR pulled from the point it is at equilibrium

Conservation of Energy in an isolated system:

  • U_total,i + K_total,i = U_total,f + K_total,f
  • U is the total potential energy, K is the total kinetic energy, W is the work due to non conservative forces

Conservation of Energy in a non-isolated system:

  • U_total,i + K_total,i - W_nc = U_total,f + K_total,f
  • U is the total potential energy, K is the total kinetic energy, W is the work due to non conservative forces

Nonconservative Forces:

  • A force between members of a system that causes transformation from mechanical energy to internal energy
  • Ex: the force of friction

Relationship between force and energy

  • Fx = -du/dx
  • The conservative force in a system equals the negative change of potential energy with respect to distance

Chapter 8: Momentum and Collisions

Momentum is a term used to describe objects in motion. The linear momentum of an object with mass m moving with velocity v is defined as:

  • p=mv

Collisions:

  • In any type of collision, the total momentum of an isolated system before the collision equals the total momentum just after the collision.
  • In an elastic collision, both momentum and kinetic energy of an isolated system are conserved.
  • In an inelastic collision, the total momentum of an isolated system is conserved, but the total kinetic energy is not conserved.
  • In a perfectly (or totally) inelastic collision, the two colliding objects stick together following the collision.

When the external force acting on a system of particles is zero, the total linear momentum of the system is conserved. This is known as the law of conservation of momentum.

  • P1i+P2i=P1f+P2f
  • Impulse-momentum theorem:

The total impulse of the net force on a particle equals the change in the momentum of the particle
I= ∆P

  • Rocket Propulsion:

The operation of a rocket depends on the law of conservation of momentum as applied to a system, where the system is the rocket plus its ejected fuel.

Chapter 9: Relativity

  • Twin paradox One ages slower in space than on earth

Time Dilation: Δtp = 2d/c
c= speed of light
Δt= Δtp/ [1-(v2/c2)]1/2

Chapter 10 : Rotational Kinematics

A comparison of Equations for Rotational and Translational

Rotational Motion Translational Motion
ωfi+αt vf=vi+at
θfi+1/2(ωfi)t xf=xi+1/2(vf+vi)t
θfiit+1/2αt2 xf=xi+vit+1/2at2
ωf2i2+2α(θfi) vf2=vi2+2a(xf-xi)

A x B = ABsin θ

Chapter 12 : Oscillatory Motion

Hooke's Law:
Force exerted by a spring is:
F_s = -kx where k is force constant or spring constant of the spring
Mathematical representation of Simple harmonic motion:
x(t) = A cost(wt + phi)

  • Simple Pendulum: exhibits periodic motion; consists of an object of mass with a light string of length where the upper end of the string is fixed.
  • Physical Pendulum: when a hanging object that cannot be modeled as a particle oscillates about a fixed axis and does not pass through the center of its mass.
  • Damped Oscillations: Any oscillation in which the amplitude of the oscillating quantity decreases with time. Damped_spring.gif

Chapter 13: Mechanical Waves

Mechanical waves: waves that disturb and propagate through a medium. ex: a pebble falls into a river and there is a ripple in the water that was caused by the pebble.
Electromagnetic waves: waves that don't need a medium to propagate. ex: light waves and radio waves.

* TRANSVERSE WAVE- elements of the disturbed medium move perpendicularly to the direction of propagation.
Onde_cisaillement_impulsion_1d_30_petit.gif
* LONGITUDINAL WAVE- elements of the medium undergo displacements parallel to the direction of propagation. an example is- sound waves in air. sound waves need to travel through a material medium and their speed depends on the specific properties of that medium. sound waves cannot travel through space or in a vacuum, however.
Onde_compression_impulsion_1d_30_petit.gif
Wave definitions/formulas—

* Period: the period (T) is the time interval required for an element of medium to undergo one oscillation.
T=1/f

* Frequency: (f) of a sinusoidal wave is the same as the frequency of simple harmonic motion of an element of the medium.

* Wavelength: (lambda) is the smallest distance between two identical points on a wave.
lambda= vT

* Angular Wave Number: (k) or wave number
k=2π/lambda

*Angular frequency: (w)
w=2π/T=2πf

*Wave function for a sinuisoidal wave——> y=Asin(kx-wt)

v=w/k

v=lambda times f

v= square root of T/µ

*DOPPLER EFFECT—> the doppler effect is seen when there is motion between a source creating sound and an observer. when the source and the observer move toward each other- the observer hears a frequence that is higher than the true frequency that the source is emitting. when the source and the observer move away from each other- the observer hears a frequency that is lower than the true frequency that the source is emitting.
f'=f((v+vo)/(v-vs))

*Seismic Waves-
epicenter: point on the earth's surface radially above hypocenter.
hypocenter: when an earthquake happens, a release of energy takes place at this point.
seismic waves: the released energy will propagate away from the hypocenter using seismic waves.
P waves: longitudinal waves that arrive first at the seismograph.
S waves: transverse waves that arrive second at the seismograph.

Chapter 14 : Superposition and Standing Waves

Principle of Superposition
*When two waves traveling through a medium combine at a certain point, a resulting wave will occur at that point that is the sum of each wave's position.
*Symmetrical waves are not required for superposition
*The combination of these two waves in the same region is called interference
*After the waves combine at that instant, they will continue in their original direction and remain unaltered

Interference

  • Interfere constructively
          • The difference between the phase of two interacting waves is 0 degrees or any multiple of 2pi
          • The waves completely add on to each other, only if both of their amplitudes are equal
  • Interfere destructively
          • The difference between the phase of two interacting waves is 180 degrees or any multiple of pi
          • The waves completely cancel each other out, only if both of their amplitudes are equal
  • If the difference between the phase of the two waves are between 0 and 180 degrees, the resulting wave will be between 0 and 2A (where A is the amplitude)

Standing Wave

  • An oscillation pattern that results from two waves traveling in opposite directions
  • The function of this wave is:
    • y=(2A sin kx) cos ωt

Harmonics

  • higher natural freuencies together with the fundamental frequency form a harmonic series.
  • the various frequencies are called harmonics

Beats
*Combination varies between large and small amplitudes and this results in beats.
fb= absolute value of (f1-f2)

Buoyancy
Buoyancy.jpg

B=pfgV

Continuity Equation
A1V1=A2V2

Bernouli's Equation
P1 + (1/2)pv1^2 + pgy1 = P2 + (1/2)pv2^2 + pgy2

The speed of fluid flow is due to pressure differences. If the fluid moves faster, the pressure must decrease.

Reynold's Number
If it reaches or exceeds 1000, the the flow of a liquid turns from being laminar to turbulent.

Viscosity: A fluid's 'thickness' or resistance to flow.

  • Archimedes's Principle: any object completely or partialy submerged in a fluid experiences an upward buoyant force whose magnitude is equal to the weight of the fluid displaced by the object.
  • Laminar: if each particle of the fluid follows a smooth path so that the paths of different particles never cross each other.
  • Turbulent: irregular flow characterized by small, whirlpool-like regions.

Chapter 16: Gases & Temperature

Ideal Gas Law:
PV=NKBT

  • P= pressure
  • V= volume
  • N= # of particles
  • KB = Boltzmann's Constant
  • T= Temperature in kelvin (K)

KE = 1/2mv2 = 3/2KBT
Kelvin = Degree C + 273.15
K=0 is absolute zero

Pressure: Measures particles bouncing off wall barriers
Temperature: Measures average kinetic energy

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