Chapter 4-6 Flashcards
Conservative Force
any force (f) for which a potential energy function (V) can be defined such that: F(x) = -dV/dx
Work (W) Done By a Conservative Force
W = integral from x sub 0 to x of F dot dx = - delta V
- W is independent of the path taken (only depends on the initial and final positions)
- W = 0 for a closed path
- If F violates W = 0, then it violates W = -delta V (W depends on the path taken!) and F is called a “nonconservative” force and V cannot be defined (ex: friction)
Test for a Conservative Force
The curl of F (del cross F) = 0
Del Operator
Del = ihat (d/dx) + jhat(d/dy) + khat(d/dz)
where d is a partial derivative
Gradient of a Scalar
Del V = dV/dx ihat + dV/dy jhat + dV/dz khat
(where d is a partial derivative)
produces a vector
Divergence of a Vector
Del dot F = d(F sub x)/dx + d(F sub y)/dy + d(F sub z)/dz
(where d is a partial derivative)
produces a scalar
Curl of a Vector
(on equation sheet)
produces a vector
Conservation of Energy (E)
The total energy of an isolated system must remain constant
- If only conservative forces are present: Ta + Va = Tb + Vb (where T + V = E = Total energy)
- If both conservative forces and nonconservative forces are present: Ta + Va + integral from a to b of Fnc dot dr = Tb + Vb (work done by c forces are included in V with Fc = -dV/dx and work done by nc forces must be calculated directly since V doesn’t exist for them with Fnc != -dV/dx)
Separable Force
a force where each component only contains that particular coordinate (ex: x-comp only contains “x”)
F = Fx(x) ihat + Fy(y) jhat + Fz(z) khat
all separable forces are conservative
Steps to solving Projectile Motion in 3D
trajectory: (g/2v0^2cos^2(alpha))x^2 -(tan(alpha))x + z = 0
zmax = v0^2sin^2(alpha)/2g
range = v0^2sin(2alpha)/g
Isotropic Harmonic Oscillator
restoring force is the same in all directions
Nonisotropic Harmonic Oscillator
restoring force is not the same in all directions
Constrained Motion of a Particle
motion in which a particle is constrained to move along a definite surface or curve
F + R = ma
Reference Frame
a coordinate system used to describe an object’s position and motion
Inertial Reference Frame
reference frame in which Newton’s 1st Law is obeyed (If the net force on an object is zero, then a = 0)
any reference frame that moves at a constant velocity with respect to an inertial reference frame is also an inertial reference frame
laws of physics look the same in all inertial reference frames
Noninertial Reference Frame
reference frames in which Newton’s 1st Law is not obeyed (object accelerates with Fnet = 0)
all accelerating reference frames are noninertial reference frames
laws of physics do not look the same in all inertial reference frames
Inertial Forces
“fictitious forces” that are not due to interactions with real objects. They are always present when motion is described from a noninertial reference frame (to force the laws of physics to have the same mathematical form)
Transverse Force
created when O’ has an angular acceleration
directed perpendicular to r
Ftrans = -m(omega_dot cross r)
Coriolis Force
created when particle moves in O’ (unless v’ is ll to omega)
directed perpendicular to v and omega
Fcor = -2m(omega cross v)
Centrifugal Force
created when r is not ll to omega (otherwise omega cross r = 0)
directed outward from rotational axis
Fcent = -m omega cross (omega cross r)
Centripetal Acceleration
inward acceleration that an object experiences when it moves in a circle of radius r at a constant speed v
a = v^2/r
Centripetal Force
Real inward force that keeps an object moving in a circle of radius r at constant speed v
F = mv^2/r
The Foucault Pendulum
a spherical pendulum free to swing in any direction that is affected by the Earth’s rotation
Kepler’s 1st law
the orbit of each planet is an ellipse with sun at a focus point
Kepler’s 2nd Law
a line drawn between sun and planet sweeps equal areas in equal times
Kepler’s 3rd Law
the square of a planet’s sidereal period (tal) is directly proportional to the cube of semimajor axis (a) of orbit
tal^2 = ka^3
Sidereal Period
(tal) time for a planet to orbit once about the sun relative to stars
Newton’s Law of Universal Gravitation
every object in the universe attracts ever other object with a force that is directly proportional to the product of their masses, inversely proportional to the square of the distance between them, and points along the line connecting them
Point Mass
an object whose mass is concentrated at a single point in space and has no volume
Central Force
a force that is directed toward or away from a single point and magnitude only depends on the distance from this point
F = f(r) rhat
Angular Momentum
(L) cross product of a particle;s position vector (r) and linear momentum (p)
L = r cross p
Torque
(Tal) any influence that can cause an object to rotate (accelerate angularly)
Tal = r cross F
Conservation of Angular Momentum
if tal external = 0 then dL/dt = 0 and L is constant
- L is constant for all central forces
- if a particle has L = constant, it is confined to the plane formed by r cross p
Area of a Parallelogram
Area = absolute value of c cross d
Ellipse
set of all points whose total distance from 2 foci (f and f’ is constant)
Pericenter of an Orbit
distance at which objects are closest together
rmin = a(1-epsilon) or rmin = alpha/1+epsilon
perihelion for solar orbits and perigee for earth orbits
Apocenter of an Orbit
distance at which objects are farthest apart
rmax = alpha/1-epsilon or rmax = a(1+epsilon)
aphelion for solar orbits and apogee for earth orbits
Conic Section
figure formed by the intersection of plane and cone
- Circle epsilon = 0
- Ellipse 0 1
Escape Velocity
the value of v that makes rmax = infinity
Rutherford Scattering
describes the motion of a particle that experience an inverse square repulsive electrical force
Impact Parameter
(b) perpendicular distance from the origin (scattering center = nucleus) to the initial line of motion of the projectile (alpha particle)
Solid Angle
2D angle in 3D space that an object subtends at a point (dimensionless unit = steradians)
