Transcript AP Physics C - Heritage High School

AP Physics C
Math Review
Vector Components
• All vectors can be broken into their individual x & y
components.
• Typically -
x  r cos
y  r sin 
• However, if you have an inclined plane this will not be
the case. Make sure you determine the x & y
components with the angles in your diagrams.
Vector Addition
• Break vectors into x & y components
• Add the components together
• Report answer using magnitude and direction or
components.
• î & ĵ are used to represent x and y
• You will use vector addition in kinematics, with
forces, momentum, electric and magnetic field, and
anything else that is a vector.
• Energy and electric potential are not vectors
Vector Multiplication
Dot Product
• If we want the parallel components of two vectors we
use the dot product.
• We use the cosine of the angle between the two
vectors. Work is shown below…
W  F  d  Fd cos 
• Dot product is used with work, power (P=Fv), electric
and magnetic flux, and Ampere’s Law.
Vector Multiplication
Cross Product
• If we want the perpendicular components of two vectors we
use the dot product.
• We use the sine of the angle between the two vectors. Torque
is shown below…
  r  F  rF sin 
• Cross product is used with torque, angular momentum,
magnetic force (for both a charge and a current)
• Direction is given by right hand rule. Index finger points in the
direction of the first variable, middle finger with the second
variable and the thumb is the direction of the result.
Differential Equations
• Differential equations are used to solve for
objects that experience air resistance,
harmonic motion, RL circuits, RC circuits, LC
circuits
Differential Equations
Retarding Force (air resistance)
Moving Forward
FD  kv
dv
m a  m  kv
dt
Falling Object
F   m g  kv
dv
ma m
  m g  kv
dt
• Use Newton’s Second Law and identify the forces
on the object.
• Separate variables and integrate to solve. Be sure
to include limits of integration
• You will end with something like e-kx or A(1 - e-kx )
Harmonic Motion
• The motion is periodic – it repeats itself and
can be modeled as a sine function
• Springs and LC circuits all have the same form.
• Pendulums are also harmonic
Spring
F  m a   kx
d 2x k
 x0
2
dt
m
k

m
LC Circuit
dU
di q dq
 Li 
0
dt
dt C dt
d 2q
1

q0
2
dt
LC
1

LC
RC Circuits
Charging
• Use Kirchoff’s Loop rule to find voltage across
each component. Set the current to dq/dt.
• Separate the variables and solve for q.
ε
C
R
q
   iR  0
C
dq q
R
 
dt C
q  C (1  e t / RC )
dq
   t / RC
 i (t )   e
dt
R
RL Circuits
Rise of Current
• Use the same approach as the RC circuit – the
difference is that we look at current instead of
charge.
di
  iR  L  0
dt
di
L  iR  
dt
i

R
(1  e  Rt / L )
Integration for AP Physics
• There are many cases where you will be asked
to integrate on the AP Physics exam or when
you are asked to find the area under a curve.
• The next few slides will deal with the more
difficult integration that you may encounter.
• There are some common themes among:
Center of Mass, Rotational Inertia, Electric
Field and Electric Potential
Integration for AP Physics
For the situations listed in the last slide there are
some common strategies to approach the
problems.
1. Identify symmetry and choose your axes so
that you integrate along a line of symmetry
2. Use the mass/charge density and break your
object into pieces that fit the shape.
Integration for AP Physics
Linear
• Break the long, thin rod into small pieces dx
• Use the mass/charge density to find dm or dq
in terms of dx.
• Substitute in order to get a single variable in
your integral
• Plug values in, set limits of integration and go
Integration for AP Physics
Two Dimensional
• Break the area into shapes – either rings or
rectangles
• Use the mass/charge density to solve for dm
or dq in your ring
• Substitute in order to get a single variable in
your integral
• Plug values in, set limits of integration and go
Integration for AP Physics
Three Dimensional
• Break the area into shapes – a sphere breaks
into spherical shells. A cylinder breaks into
cylindrical shells.
• Use the mass/charge density to solve for dm
or dq in your shape
• Substitute in order to get a single variable in
your integral
• Plug values in, set limits of integration and go
Integration
Other Notes
• E-fields are vectors and you need to keep the
direction in mind as you solve
• Center of Mass, Rotational Inertia and
Potential are scalar quantities