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