#### Transcript hp1f2013_class04_3d

```Honors Physics 1
Class 04 Fall 2013
Vectors
Non-Cartesian coordinate systems
Motion in multiple dimensions
Uniform circular motion
Applications
1
Activity – Motion with constraints
Frictionless pulley and table.
Only net external force=gravity on mass 2.
Rope massless and fixed length.
Find acceleration of mass 1.
3rd Law: Fs 2   F2 s ; Fs1   F1s ; F1s  F2 s  T
Constraint: The velocity and acceleration of M1 is the same as M2
Do "Free Body", dude! Mass 1, Mass 2
T  M1a
M 2 a  M 2 g  T  M 2 g  M1a
2
Vectors
r ( x, y, z )  rxiˆ  ry ˆj  rz kˆ  rxiˆ  ry ˆj  rz kˆ
dx ˆ dy ˆ dz ˆ dr
ˆ
ˆ
ˆ
v  vxi  v y j  vz k  i 
j k 
dt
dt
dt
dt
Note: differentiating vectors requires care.
ˆ
dx
di
The derivative of xiˆ is really:
iˆ  x .
dt
dt
2
2
2
2
d
x
d
y
d
z
d
r
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
a  axi  a y j  az k  2 i  2 j  2 k  2
dt
dt
dt
dt
v   adt ( v0  at for constant a )
Examples: Projectile motion; Uniform circular motion.
3
Cartesian coordinates
iˆ, ˆj , kˆ form a right-handed
coordinate system
iˆ  ˆj  kˆ; ˆj  kˆ  iˆ; kˆ  iˆ  ˆj
r  xiˆ  yjˆ  zkˆ
dr  dxiˆ  dyjˆ  dzkˆ
dV  dxdydz
4
Cylindrical coordinates

dV   d  d dz
ds  d ˆ   dˆ  dzzˆ
x   cos  ,
y   sin  ,
zz
  ( x 2  y 2 )1/2 , tan   y / x
uˆ   uˆ  0; uˆ   uˆ z  0; uˆ  uˆ z  0
uˆ   uˆ  uˆ z ; uˆ  uˆ z  uˆ  ; uˆ z  uˆ   uˆ
  uˆ 

1 

 uˆ
 uˆ z

 
z
5
5
Spherical coordinates
x  r sin  cos  ; y  r sin  sin  ; z  r cos
r 
x  y  z ; tan   y / x; tan 
2
2
2
x2  y 2
z
dV  r 2 sin drd d
ds  drrˆ  rdˆ  r sin dˆ

1 
1

 uˆ
 uˆ
r
r 
r sin 
uˆr  uˆ  uˆ ;...
  uˆr
6
6
Exponentials and complex numbers
Complex numbers and the Euler relation are
sometimes useful tools for solving problems.
And the representation of a complex number,
z  a  ib on a plane where a is the value on the x-axis

and b is the y-axis so z = a  b
2

2 1/2
.
By comparing series representations of
sin , cos , and ei we know that:
ei  cos   i sin 
ei  e i
ei  e i
and cos  
;sin  
2
2i
Discuss powers, square roots, derivatives.
7
Newton’s Laws in 3d form
F   Fi  ma
Fx  ma x ; Fy  ma y
Because x and y components are
at right angles, the equations can be solved
independently.
Choice of coordinate system is a powerful tool.
8
Motion Example:
Projectile trajectory
A tennis ball is hit at 50 mps at an angle of 5 degrees
above the horizontal. The initial height is 2 m.
Neglecting air drag, how far does the ball go
before hitting the ground?
Choose +y to be up and x to be horizontal.
y0  2; v0 x  v0 cos  ; v0 y  v0 sin  ; a x  0; a y  9.8
1
y  y0  v0 y t  a y t 2
2
Solve for time at which y=0.
Use x  x0  v0 xt to find distance.
9
Motion Example:
Choosing coordinates well is important
A tennis ball is hit at 50 mps at an angle of 5 degrees
above the horizontal. The initial height is 2 m.
Neglecting air drag, how far does the ball go
before hitting the ground?
Choose +x to be in the direction the ball starts at.
Choose +y to be at right angles to that.
Choose the origin to be the starting point.
y0  0; x0  0; y f  2 / cos 
v0 x  v0 ; v0 y  0;
a0 x  9.8sin  ; a0 y  9.8cos 
(The x that you first solve for is not the landing point.)
10
Activity: Uniform circular motion
Let's describe the position, velocity, and
acceleration of an object moving in uniform circular
motion. (Constant angular speed . Constant radius  .)
x   cos   cos t ; y   sin t
r  xiˆ  yjˆ  iˆ cos t  ˆj  sin t
v ?
a ?
a ?
Which way does a point?
Which way does v point?
v ?
11
Activity: Application
The Spinning Terror ride
The spinning terror is a large vertical drum which spins
so fast that everyone stays pinned to the wall when
the floor drops out. For a typical ride the radius of the
drum is 2 m.
What is the minimum angular velocity if the coefficient
of friction between the patron and the wall is 0.3?
12
Application example:
Falling through a viscous fluid
Assume that the density of the fluid is very
small compared to the density of the falling
object. (e.g – a human body in air)
Assume that the body falls under the action of
constant gravity and drag force only.
Assume that the drag force is linear in speed:
FD  Cv
Is there a terminal velocity?
If there is, find the terminal velocity.
13
Application example:
Mass on a spring
Equilibrium position x  0; Starting point=x0
F  k  x 
Write the F  ma equation.
Assume a solution of the form: x(t )  Ae t .
See what conditions have to be met by A and 
to solve the relation F  ma and satisfy initial
conditions.
14
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