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Potential Energy
A form of stored energy that depends on the spatial position or configuration of objects. The capacity of an object to do work in the future based on work completed in the past. ______ accumulates in mechanical system because of the work performed by a force applied over some distance.
Gravitational potential energy accumulates when:
The height of an object increases because a force must act against gravity to raise the object.
Stretching or compressing a spring away from its equilibrium point:
Increases its elastic potential energy. The potential energy in stretched or compressed springs explains why harmonic oscillations
Elastic Potential Energy Equation
U(el) = ½ kx² (k = spring constant, x = spring displacement)
Elastic Potential Energy
Refers to the potential energy of springs and other elastic objects.
Elastic Potential Energy Increases:
Following the application of a stretching or compressing force that displaces a spring from its equilibrium position.
Power Equation
Power = Work/Time (Watt = Joules/Second)
Power (Translational Motion) and Power (Mechanical Work)
Power = (Force x Distance) / Time = Force x Velocity
Mechanical Work Equation
W = F x d x cos Theta
Cardiac Output Equation
CO = A x v (A = cross sectional area of aorta) (v = velocity of the blood)
for cm³ / L there is:
1000 cm³ / L
Kinetic Energy Equation
KE = 1/2mv² (Joules kg(m²/s²)
Exercise (Blood Flow)
During exercise, blood flow to the skin increases to maintain normal body temperature. Excess body heat generated during exercise can be transferred to the surface to the surface through convection and dissipated into the environment through radiation.
The molar volume for all gases at standard temperature and pressure
22 L / mol
Displacement (Delta X)
Delta X = Xf - Xi = (v)(t). The area under the curve of a. velocity (y axis) vs time (x axis) graph is equal to displacement.
Displacement (graph equation)
Displacement = Rectangle Area + Triangle Area
Displacement = R base length x hR eight length + ½ T base length x T height length
Conservation of Energy
E = KE + PE = constant
Potential Energy Equation: PE = mgh
Kinetic Energy Equation: KE = 1/2 mv²
Torque
The tendency of an applied force to cause the rotation of an object around a pivot point located at some radial distance from where the force is applied.
Torque Equation
r(f)(sin)(theta)
Rotational Equilibrium
When the sum of all torques acting on an object is equal to zero.
Left Torque = Right Torque
F r sin theta (right side values) = F r si5n theta (left side values)
F = mass x gravity
Add torques if there are multiple torques on one side
Frictional Forces
Attractive intermolecular forces between the atoms of two surfaces contribute frictional forces that oppose motion along the interface between two objects that are in direct contact with another.
Static Equilibrium
Motionless, a frictional force that resists the initiation of motion is called static friction.
Static Friction Equation
F(s) = (Normal Force)(Static Coefficient)
If the force applied to an object does not exceed Fs, the object will not move. If Fs is exceeded, the object will accelerate and static friction becomes kinetic friction.
Tension
T = Weight = mg
Horizontal Component of Tension (Tx = T cos Theta)
Vertical Component of Tension (Ty = T sin Theta)
Tension and Force for a block in static equilibrium
F(s) must be equal to T(x) and F(n) + T(y) = weight of other block.
So coeffecient (static friction)(vertical block - T(y) = T(x) or F(s) (same thing)
Gravitational Potential Energy
Measures the potential of gravity to do work on a mass. Conventionally, gravitational potential energy is expressed using negative numbers and is inversely proportional to the radial distance between masses. U = - (G)(M)(m)/r)
Periodic Motion
Occurs when an object moves in a cyclical fashion, returning to the same location after a certain period of time. Is modeled in the terms of wave motion where the cycle of motion is defined as a period (T).
Frequency (f) Equation
f = 1/T
(cycle/time in Hz)
Work-Energy Principle
The total work (W) done by a force is equal to the charge in kinetic energy (Delta KE) of the object. W = Delta KE
Conservation of Energy in Mechanical Systems
Delta KE = - Delta PE
Delta KE + Delta PE = 0
KEf - KEi + PEf - PEi = 0
KEi + PEi = KEf + PEf
The kinetic energy of an object launched upward in gravity is converted:
into gravitational potential energy
When R and f is perpendicular Theta is equal to:
90 degrees and the torque is maximal
When R and f is parallel Theta is equal to:
0 degrees and the torque is zero.
Clockwise or Counterclockwise Direction (Torque)
The direction the force is applied
Electromagnetic Waves (EM)
EM waves are arranged by frequency or wavelength in the EM spectrum, ranges from very low frequency/long wavelength to high frequency/small wavelength.
EM Spectrum ,
Big Wavelength/Low Energy to Small Wavelength/High Energy
Radio
Microwaves
Infrared
Ultraviolet
X Rays
Gamma
Raging Martians Invaded Venus Using X-ray Guns
Photon Energy
When EM waves are emitted or absorbed they behave as photons.
Photon Energy = (h)(f)
h = Planck’s constant
f = frequency
Radiation Energy (Temperature)
Radiation is related to the temperature of the object that emits radiation. Hotter objects radiate waves at the higher frequency region in the spectrum and colder objects radiate waves at the lower frequency region.
Power Equation
Power = Energy / Time
average veloctiy
v(x)f -v(x)i / t
v(y)f -v(y)i / t
Hooke’s Law
Elastic Force = -kx
k = spring constant
x = displacement from equilibrium point
Models the elastic forces generated by sprnigs
Newton’s Second Law of Motion
An object will accelerate if the net force acting along each axis of a given coordinate system is a nonzero quantity
Dynamic Equilibrium
Describes an object moving at velocity without acceleration. Free body diagrams can be used to assess the forces.
Parallel Angle of Incline is:
Sine
Perpendicular Angle of Incline is:
Cos
Static Equilibrium of Torque on Fulcrum Equation
Torque = CCW (T) - CW(T) = 0
Pressure Equation
Pressure = Force / Area
If the torque on one side of the fulcrum decreases:
The torque on the other side of the fulcrum must also decrease for the system to remain equilibrium.
Mechanical Work Equation
Work = (Force)(Distance) = - (Pressure)(Volume)
In a graph of force on object vs distance on object
The work performed on the object is equal to the area under the curve.
Hooke’s Law (Derived)
F = k/x
k = F/x = slope = (F2 - F1) / (x2 - x1)
Power Equation
Power = Work/Time or Power = Energy/Time (Watts J/s)
Elastic Potential Energy (Spring)
Elastic Potential Energy = ½ kx²
Work Energy Theorem
Work = Delta PE = mg (delta)h
P = W / t = mg (delta)h / t
Work (from area under the curve)
Work = triangle area + rectangle area
triangle area = ½ (triangle height)(triangle base)
rectangle area = (rectangle height)(rectangle base)
Mechanical Advantage
The force amplification that occurs through the use of a device/mechanical system. The mechanical advantage is related to the number of load sharing pulleys but not fixed pulleys.
Heat of Fusion
The amount of heat energy required to transform a substance between its liquid phase and solid phase. The material remains at its freezing temperature during this phase change until it is in its solid form.
Snell’s Law
The refraction of light is given by snell’s law. n1 sin theta1 = n2 sin theta2
(n1 = refractive index of air, theta1 = incident angle of light, n2 = refractive index of glass, theta2 = incident angle of air)
Refractive Indexes
Light travels between materials and undergoes refraction, it bends as it passes to a new material. The refractive index depends on the speed of light with agreater refractive index indicating a slower speed of light.
Work Equation (Pull object with constant force along horizontal surface)
Work = (Force)(Distance)
Newton’s First Law of Motion
Net force of an object is zero when it is in constant motion
Reflection of light from mirror equation
Angle of incident light = Angle of reflected light
Block Sliding on an Incline (Steps)
F parallel = 0 = F(w)sin theta - Fk (Fk = F(n) and k constant)
F perpendicular = 0 = F(N) - F(w) cos theta
get k constant alone so k constant = F(N) / F(w)
plug in Fw sin theta / Fw cos theta and cross out and solve
Doppler Effect
Occurs when a periodic signal (sound or light) is perceived by an observer to have a different wavelength due to the relative motion between the source and the observer. The magnitude of the change in wavelength is proportional to the relative speed between source and observer.
If the sound/light source is moving toward the observer:
the perceived wavelength of the signal decreases.
If the sound/light source is moving away from the observer:
the perceived wavelength of the signal increases.
Y Forces and X Forces (Static Equilibrium)
Y forces = (+ y forces) + (- y forces) = 0 (- y force = gravity)
X forces = (+ x forces) + (- x forces) = 0
Displacement Equation
Delta x = x(f) - x(i) = v(x) multiplied by t
Delta y = y(f) - y(i) = v(y) multiplied by t
Velocity
A vector quantity characterized by magnitude and direction
Conservation of Energy Equation
1/2mv² + mgh = 1/2mv^2 + mgh
Law of Conservation of Energy
Energy cannot be created or destroyed, only transformed from one form to another.
Electric Force F(E)
F(E) on a charged object equals the product of the object’s charge (q) and electric field (E)
F(E) = (q)(E)
Electric Field
The electric field exists when there is a difference in the electric potential (Voltage) between two locations. The component of the electric field in the y direction is:
E(y) = Delta V / Delta y
change in voltage over change in distance
Newton’s Second Law
F = ma
Period Equation
Period = Distance / Speed
Distance needed to complete one cycle in a circle
Is equal to the circumference so d = 2 pi r
The area under the curve in a graph of force versus distance equa;s:
the work done on an object
Power Equation
Power = Energy / Delta Time
Distance and Kinetic Energy
Distance is proportional to Kinetic Energy
Instantaneous Velocity (Graph)
The line with the steepest and most positive slope has the greatest instantaneous velocity.
Scalar Quantities
Physical properties with only size or magnitude. (time and mass)
Vector Quantities
properties with only direction and magnitude. (velocity)
Vector Addition
Displacement is a vector and can be added.
d = d(1) + d(2) or d = (x1 + x2) , (y1 + y2)
The magnitude of any vector is equal to the vector squared so:
d = sqrt( x2 + y2 )
d total = d (of one given) + d (of second given)
Determining Velocity from plot of position and time
The slope of the position vs time is equal to the velocity. The velocity of the subject is greatest where the slope of the plot is greatest.
Work Equation
W = Fd = mgd
When walking towards laser Doppler sensor:
Higher frequency and shorter wavelength
f = (c + v) F(t) / c
When walking away from laser Doppler:
Lower frequency and higher wavelength
f = (c - v) F(t) / c
Law of Conservation Energy
E = PE + KE = constant
E = PE initial + KE initial = PE final + KE final
Cardiac Output Equation
Cardiac Output = Heart Volume + Stroke Volume
Stroke Volume = End Diastolic Volume - End Systolic Volume
Static Equilibrium and Pivot Point
For an object in static equilibrium, the sum of all torques exerted on the object equals zero. The torque on an object is equal to the product of the distance from the pivot point that the force is applied, magnitude, and sine angle.
Translational Motion Equation (Displacement/Delta X)
v = v(0) + at
Translational Motion Equation (Velocity)
v² = v(0)² + 2a (delta x)
For a projectile traveling straight upward: delta x = height and a = gravity, and v@ = 0 so the new equation is: 0 = v(0)²
Power Equation
Power = (Force)(Velocity)
Magnification of Two Lens System (Equation)
M = (M1)(M2)
M = Magnification
M1 = Lens 1 of Magnification
M2 = Lens 2 of Magnification
Magnification Equation
M = i / o
i = image, o = object
Distance Ratio
The positions of two objects circling a central point can be represented by two vectors. The objects are nearest to each other when the position vectors align in the same direction and farthest away from each other when position vectors in opposite direction.
Distance Ratio = Distance Short : Distance Far
Distance Ratio = Distance 1 - Distance 2 : Distance 1 + DIstance 2
Vector Components
V(x) = v cos theta
V(y) = v sin theta
Work and Tension Equations
Work = (F)(d)(cos theta) or (T)(d)(cos theta)
Tension can be swapped with force
Torque Equation
Torque = r f
Torque 1 = Torque 2 + Torque 3 (r = distance/cm and F = newtons/force)
r1 F1 = r2 F2 + r3 F3 (force could be mg)