PHYS 380: STUDY GUIDE FOR PART 2.
TWO AND THREE DIMENSIONAL PARTICLE MOTION
Outline:
1. Review of Vectors, Vector Algebra, and Vector Calculus
a) need for vectors (3-D universe)
b) vector addition
c) scalar multiplication
d) dot product – Work and line integrals
e) cross product - Torque
f) differentiation and integration
g) unit vectors and kinematics:
(1) rectangular (2-D, 3-D)
(2) polar (2-D)
(3) cylindrical (3-D)
(4) spherical (3-D)
h) gradient, divergence, curl
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15,16
17
2. General Physical Quantities (2-D, 3-D)
a) momentum
b) work
c) angular momentum
d) Newton's 2nd law
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3. Potential Energy - Independence of Path
a) Harmonic Oscillator in 2-D
19
4. Projectiles
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5. Central Force:
a) orbits
b) scattering
21,22,23
6. Motion in E&M Fields
Homework:
Problem #14: A particle of mass, m, in the x-y plane is attracted to the origin (direction is
negative radial) by a force that is inversely proportional to the distance from the x-axis, F(y) = k/y.
Calculate the work done by the force from an initial point (x=0, y=a) to a final point (x=2a, y=0)
along two different paths:
a) along the sides of a rectangle: (x=0, y=a) to (x=2a, y=a) and then from (x=2a, y=a) to (x=2a,
y=0); and
b) along part of an ellipse: x=2a*sin() and y=a*cos() for  from 0o to 90o. Note that the angle,
, is only a parameter used to describe the path, and it is different from the angle the particle
makes with the origin (which we usually label as ).
HINT:  du / [1+u²] = ln[u + SQR(1+u²)]
PHYS 380
Study Guide for Part 2
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Problem #15: Find the jerk, da/dt, in plane polar coordinates.
Problem #16: Given that A is a vector function, find d2A/dt2 in cylindrical coordinates (a 3-D
vector problem).
Problem #17: Calculate the curl of A, A, in cylindrical coordinates.
Problem #18: A particle of mass, m, moves according to the equations:
x(t) = xo + at2 (where a is a constant)
y(t) = bt3 (where b is a constant)
z(t) = ct (where c is a constant).
a) Find an expression for the angular momentum, L(t).
b) Find the force, F(t).
c) Find the torque, (t).
d) Verify that (t) = dL(t)/dt .
Problem #19: Determine if the force is conservative, and determine the potential energy function
if it is:
a) Fx = 18abyz3 - 20bx3y2 ; Fy = 18abxz3 – 10bx4y, Fz = 6abxyz2 (where a and b are constants).
b) F = Fx(x)x + Fy(y)y + Fz(z)z.
HINT: if F is conservative, then the line integral of the force is Independent Of Path
Problem #20: Extend the procedure in the Projectile Motion section to get the second order
correction (term with b2 ) for either the range (section 2f ) or the sideways wind drift (section 3e).
Problem #21: For an inverse square law force with effective potential energy,
V’(r) = K/r + L2/2mr2 ,
a) find the frequency of small radial oscillations about the equilibrium (circular) radius of orbit;
b) show that this frequency of small radial oscillations is equal to the frequency of the circular
orbit. (This makes the circular orbit into an elliptical one).
Problem #22: Explorer I satellite had a perigee of 360 km and an apogee of 2,549 km above the
earth’s surface. Find its distance above the earth’s surface when it passed over a point 90 o
around the earth from its perigee.
Problem #23: The earth moves in a fairly circular (assume perfectly circular) orbit with radius of
1.49 x 1011 m. Mars moves in an elliptical orbit with a perihelion distance of 2.06 x 1011 meters
and aphelion distance of 2.485 x 1011 meters.
a) Find the speed of the Earth in its circular orbit about the sun.
b) Find the speed of Mars at perihelion.
c) Find an orbit for a spaceship that has its perihelion at the earth distance, and its aphelion at
the perihelion distance of Mars.
d) Find the speed the spaceship should have at its perihelion (at the Earth’s distance) and
compare to the Earth’s speed. (This difference is the additional speed we would need to supply
to the spaceship after we lift it off the earth’s surface.)
e) Find the speed the spaceship should have at its aphelion (at the Mar’s perihelion) and
compare to Mar’s speed at that location. (This is the speed we would need to add to the
spaceship to have it match Mars’ speed.)
PHYS 380
Study Guide for Part 2
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Study Questions:
1. Be able to express the 3-D position vector:
a) in rectangular coordinates (x,y,z) with rectangular unit vectors (x,y,z)
b) in cylindrical coordinates (,,z) with cylindrical unit vectors (,,z)
(here  = the greek letter phi,  = rho)
c) in cylindrical coordinates with rectangular unit vectors;
d) in spherical coordinates (r,,) with spherical unit vectors (r,,) ;
(here  = the greek letter theta)
e) in spherical coordinates with rectangular unit vectors.
2. Be able to derive an expression for the 3-D acceleration vector for cases a) and d) of #1 in
terms of the coordinates and their time derivatives.
3. Be able to derive an expression for dA/dt where A is located at a moving point for the
cases a) and d) of #1.
4. From u  dr = du, be able to derive an expression for  in
a) rectangular coordinates;
b) cylindrical coordinates;
c) spherical coordinates.
5. Be able to set up the 3-D equations of motion for a projectile near the earth's surface with air
resistance and with a horizontal wind; be able to indicate how to proceed to solve for the
location where the objects lands using the method of successive approximations.
6. Be able to set up the 3-D equations of motion for a projectile far from the earth's surface with
air resistance that decreases exponentially with height and with a constant horizontal wind.
7. Given: F = -K r-n r with n>2]
a) be able to determine whether the force is conservative;
b) if it is, be able to determine the potential energy function, V(r);
c) given the angular momentum, L, be able to determine for what initial radius, ro, will circular
motion occur and what angular frequency, c, it will have;
d) if the energy, E, is slightly above this value (from c above), be able to determine at what
angular frequency, r, the radius will oscillate about ro .
8. For a central force problem with F = Arp r
a) be able to derive a differential equation from which r() can be obtained;
b) for an inverse square law force, write down the general solution for r();
c) be able to describe the four possible types of orbits for part b above, and show how e
and E determine which of these orbits applies.