PHYS 380: STUDY GUIDE FOR PART 2.

TWO AND THREE DIMENSIONAL PARTICLE MOTION

Outline:

 

 

 

Description

Homework Problems

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   

14

 

e

 

Cross product - Torque

 

 

f

 

Differentiation and integration

 

 

g

 

unit vectors:

15,16

 

 

1

Rectangular (2-D, 3-D)

 

 

 

2

polar (2-D)

 

 

 

3

Cylindrical (3-D)

 

 

 

4

Spherical (3-D)

 

 

h

 

Gradient, divergence, curl

17

2

 

 

General Physical Quantities (2-D, 3-D)     

18

 

a

 

Momentum

 

 

b

 

Work

 

 

c

 

Angular momentum

 

 

d

 

Newton's 2nd law

 

3

 

 

Harmonic Oscillator - Lissajous Figures

 

4

 

 

Projectiles     

19

5

 

 

Potential Energy - Independence of Path

20

6

 

 

Central Force:

21,22,23

 

a

 

Orbits

 

 

b

 

Scattering

 

7

 

 

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(f) and y=a*cos(f) for f from 0o to 90o.  Note that the angle, f, 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 q).

HINT: ò du / Ö [1+u²] = ln[u + SQR(1+u²)]

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 curl of A, Ñ ´ 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(t).

d)  Verify that  t(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 line integral 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 90o 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.)

 

Study Questions:

1. Be able to express the 3-D position vector:

  1. in rectangualr coordinates (x,y,z) with rectangular unit vectors (x,y,z)
  2. in cylindrical coordinates (r ,j ,z) with cylindrical unit vectors (r  ,j  ,z)
    (here     j = the greek letter phi, r = rho)
  3. in cylindrical coordinates with rectangular unit vectors;
  4. in spherical coordinates (r,q ,j ) with spherical unit vectors (r,q ,j ) ;
    (here     q = the greek letter theta)
  5. 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 Ñu · dr = du, be able to derive an expression for Ñ in

  1. rectangular coordinates;
  2. cylindrical coordinates;
  3. 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]

  1. be able to determine whether the force is conservative;
  2. if it is, be able to determine the potential energy function, V(r);
  3. given the angular momentum, L, be able to determine for what initial radius, ro, will circular motion occur and what angular frequency, wc, it will have;
  4. if the energy, E, is slightly above this value (from c above), be able to determine at what angular frequency, wr, the radius will oscillate about ro .

8. For a central force problem with F = Arp r

  1. be able to derive a differential equation from which r(q) can be obtained;
  2. for an inverse square law force, write down the general solution for r(q);
  3. be able to describe the four possible types of orbits for part b above, and show how e (eccentricity) and E (energy) determine which of these orbits applies.

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