Chapter 4-6

Question 1

Conservative Force

Answer 1

any force (f) for which a potential energy function (V) can be defined such that: F(x) = -dV/dx

Question 2

Work (W) Done By a Conservative Force

Answer 2

W = integral from x sub 0 to x of F dot dx = - delta V

1. W is independent of the path taken (only depends on the initial and final positions)
2. W = 0 for a closed path
3. 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)

Question 3

Test for a Conservative Force

Answer 3

The curl of F (del cross F) = 0

Question 4

Del Operator

Answer 4

Del = ihat (d/dx) + jhat(d/dy) + khat(d/dz)

where d is a partial derivative

Question 5

Gradient of a Scalar

Answer 5

Del V = dV/dx ihat + dV/dy jhat + dV/dz khat

(where d is a partial derivative)

produces a vector

Question 6

Divergence of a Vector

Answer 6

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

Question 7

Curl of a Vector

Answer 7

(on equation sheet)

produces a vector

Question 8

Conservation of Energy (E)

Answer 8

The total energy of an isolated system must remain constant

1. If only conservative forces are present: Ta + Va = Tb + Vb (where T + V = E = Total energy)
2. 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)

Question 9

Separable Force

Answer 9

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

Question 10

Steps to solving Projectile Motion in 3D

Answer 10

trajectory: (g/2v0^2cos^2(alpha))x^2 -(tan(alpha))x + z = 0

zmax = v0^2sin^2(alpha)/2g

range = v0^2sin(2alpha)/g

Question 11

Isotropic Harmonic Oscillator

Answer 11

restoring force is the same in all directions

Question 12

Nonisotropic Harmonic Oscillator

Answer 12

restoring force is not the same in all directions

Question 13

Constrained Motion of a Particle

Answer 13

motion in which a particle is constrained to move along a definite surface or curve

F + R = ma

Question 14

Reference Frame

Answer 14

a coordinate system used to describe an object’s position and motion

Question 15

Inertial Reference Frame

Answer 15

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

Question 16

Noninertial Reference Frame

Answer 16

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

Question 17

Inertial Forces

Answer 17

“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)

Question 18

Transverse Force

Answer 18

created when O’ has an angular acceleration

directed perpendicular to r

Ftrans = -m(omega_dot cross r)

Question 19

Coriolis Force

Answer 19

created when particle moves in O’ (unless v’ is ll to omega)

directed perpendicular to v and omega

Fcor = -2m(omega cross v)

Question 20

Centrifugal Force

Answer 20

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)

Question 21

Centripetal Acceleration

Answer 21

inward acceleration that an object experiences when it moves in a circle of radius r at a constant speed v

a = v^2/r

Question 22

Centripetal Force

Answer 22

Real inward force that keeps an object moving in a circle of radius r at constant speed v

F = mv^2/r

Question 23

The Foucault Pendulum

Answer 23

a spherical pendulum free to swing in any direction that is affected by the Earth’s rotation

Question 24

Kepler’s 1st law

Answer 24

the orbit of each planet is an ellipse with sun at a focus point

Question 25

Kepler’s 2nd Law

Answer 25

a line drawn between sun and planet sweeps equal areas in equal times

Question 26

Kepler’s 3rd Law

Answer 26

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

Question 27

Sidereal Period

Answer 27

(tal) time for a planet to orbit once about the sun relative to stars

Question 28

Newton’s Law of Universal Gravitation

Answer 28

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

Question 29

Point Mass

Answer 29

an object whose mass is concentrated at a single point in space and has no volume

Question 30

Central Force

Answer 30

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

Question 31

Angular Momentum

Answer 31

(L) cross product of a particle;s position vector (r) and linear momentum (p)

L = r cross p

Question 32

Torque

Answer 32

(Tal) any influence that can cause an object to rotate (accelerate angularly)

Tal = r cross F

Question 33

Conservation of Angular Momentum

Answer 33

if tal external = 0 then dL/dt = 0 and L is constant

1. L is constant for all central forces
2. if a particle has L = constant, it is confined to the plane formed by r cross p

Question 34

Area of a Parallelogram

Answer 34

Area = absolute value of c cross d

Question 35

Ellipse

Answer 35

set of all points whose total distance from 2 foci (f and f’ is constant)

Question 36

Pericenter of an Orbit

Answer 36

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

Question 37

Apocenter of an Orbit

Answer 37

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

Question 38

Conic Section

Answer 38

figure formed by the intersection of plane and cone

1. Circle epsilon = 0
2. Ellipse 0 1

Question 39

Escape Velocity

Answer 39

the value of v that makes rmax = infinity

Question 40

Rutherford Scattering

Answer 40

describes the motion of a particle that experience an inverse square repulsive electrical force

Question 41

Impact Parameter

Answer 41

(b) perpendicular distance from the origin (scattering center = nucleus) to the initial line of motion of the projectile (alpha particle)

Question 42

Solid Angle

Answer 42

2D angle in 3D space that an object subtends at a point (dimensionless unit = steradians)