Download AP Physics C Review Mechanics

AP Physics C Review
Mechanics
CHSN Review Project
This is a review guide designed as preparatory information for the AP1 Physics C
Mechanics Exam on May 11, 2009. It may still, however, be useful for other purposes
as well. Use at your own risk. I hope you find this resource helpful. Enjoy!
This review guide was written by Dara Adib based on inspiration from Shelun Tsai’s
review packet.
This is a development version of the text that should be considered a work-inprogress.
This review guide and other review material are developed by the CHSN Review
Project.
Copyright © 2009 Dara Adib. This is a freely licensed work, as explained in the Definition of Free Cultural Works (freedomdefined.org). Except as noted under “Graphic
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1
“Why do we love ideal worlds? . . . I’ve been doing this for 38 years and school is an
ideal world.” — Steven Henning
Contents
Kinematic Equations
3
Free Body Diagrams
3
Projectile Motion
4
Circular Motion
4
Friction
5
Momentum-Impulse
5
Center of Mass
5
Energy
5
Rotational Motion
7
Simple Harmonic Motion
8
Gravity
9
Graphic Credits
• Figure 1 on page 3 is based off a public domain graphic by Concordia College and vectorized
by Stannered: http://en.wikipedia.org/wiki/File:Incline.svg.
• Figure 2 on page 3 is based off a public domain graphic by Mpfiz: http://en.wikipedia.
org/wiki/File:AtwoodMachine.svg.
• Figure 5 on page 7 is a public domain graphic by Rsfontenot: http://en.wikipedia.org/
wiki/File:Reference_line.PNG.
• Figure 6 on page 7 was drawn by Enoch Lau and vectorized by Stannered: http://en.
wikipedia.org/wiki/File:Angularvelocity.svg. It is licensed under the Creative Commons Attribution-Share Alike 2.5 license: http://creativecommons.org/licenses/by-sa/
2.5/.
• Figure 7 on page 8 is based off a public domain graphic by Mazemaster: http://en.wikipedia.
org/wiki/File:Simple_Harmonic_Motion_Orbit.gif.
• Figure 8 on page 9 is a public domain graphic by Chetvorno: http://en.wikipedia.org/
wiki/File:Simple_gravity_pendulum.svg.
2
Figure 1: Normal Force
Kinematic Equations
1
∆x = at2 + v0 t
2
Figure 2: Atwood’s Machine
∆v = at
(v)2 − (v0 )2 = 2a(∆x)
Figure 3: Draw a banked curve diagram
v0 + v
∆x =
×t
2
Pulled Weights
Free Body Diagrams
a=
F−f
Σm
N Normal Force
f Frictional Force
T = ma
T Tension
mg Weight
Elevator
Normal force acts upward, weight acts downward.
F = ma
• Accelerating upward: N = |ma| + |mg|
In a particular direction:
• Constant velocity: N = |mg|
ΣF = (Σm)a
• Accelerating downward: N = |mg| − |ma|
Atwood’s Machine2
a=
2 Pulley
Banked Curve
|(m2 − m1 )|g
m1 + m2
Friction can act up the ramp (minimum velocity
when friction is maximum) or down the ramp
(maximum velocity when friction is maximum).
videal =
and string are assumed to be massless.
3
p
rg tan θ
Range
s
vmin =
rg(tan θ − µ)
µ tan θ + 1
θ represents the smaller angle from the x-axis to
the direction of the projectile’s initial motion.
Starting from a height of x = 0:
s
vmax =
rg(tan θ + µ)
1 − µ tan θ
xmax =
Projectile Motion
Circular Motion
Position
Centripetal (radial)
(v0 )2 sin 2θ
g
Centripetal acceleration and force is directed towards the center. It refers to a change in direction.
∆x = vx t
1
∆y = − gt2 + (vy )0 t
2
ac =
v2
r
Velocity
Fc = mac =
θ represents the smaller angle from the x-axis to
the direction of the projectile’s initial motion.
mv2
r
Tangential
(vx )0 = v0 cos θ
Tangential acceleration is tangent to the object’s
motion. It refers to a change in speed.
(vy )0 = v0 sin θ
at =
d|v|
dt
∆vx = 0
Combined
∆vy = −gt
atotal =
q
(ac )2 + (at )2
Height
θ represents the smaller angle from the x-axis to Vertical loop
the direction of the projectile’s initial motion.
In a vertical loop, the centripetal acceleration is
Starting from a height of x = 0:
caused by a normal force and gravity (weight).
ymax =
(v0 sin θ)2
2g
4
Top
Elastic
Kinetic energy is conserved.
F = ma
N + mg = m ×
N =
v2
r
m1 v1 + m2 v2 = m1 v10 + m2 v20
mv2
− mg
r
−(v20 − v10 ) = v2 − v1
Bottom
Inelastic
F = ma
Kinetic energy is not conserved.
v2
N − mg = m ×
r
2
mv
+ mg
N =
r
m1 v1 + m2 v2 = (m1 + m2 )v0
Center of Mass
Friction
Friction converts mechanical energy into heat.
Static friction (at rest) is generally greater than
kinematic friction (in motion).
rcm =
λ=
fmax = µN
Σm =
p = mv
Z
dm =
λdx
(Σm)vCM = Σmv = Σp
dp
dt
Fnet = (Σm)aCM
Z
I=
dm
M
=
dx
L
Z
Momentum-Impulse
F=
Z
Z
1
1
Σmr
rdm =
xλdx
=
Σm
Σm
Σm
Energy
Fdt = F∆t = ∆p = m∆v
Work
Collisions
Z
Total momentum is always conserved when there
are no external forces (F = dp
dt = 0).
5
W=
Fdx = ∆K
Power
Pavg =
Fx
W
=
t
t
Pinstant =
dW
= Fv
dt
Kinetic Energy
Linear
1
K = mv2
2
Potential Energy
dU
F=−
dx
Z xf
∆U = −
FC dx = −WC
Z
UHooke
v=
dx
dt
=
∆x
∆t
ω=
dθ
dt
=
∆θ
∆t
a=
dv
dt
=
∆v
∆t
α=
dω
dt
=
∆ω
∆t
∆x = 12 at2 + v0 t
∆θ = 12 αt2 + ω0 t
∆v = at
∆ω = αt
(v)2 − (v0 )2 = 2a(∆x)
(ω)2 − (ω0 )2 = 2α(∆ω)
∆x =
xi
Z
1
= − FHooke dx = − −kxdx = kx2
2
Ug = mgh
equilibrium point F = − du
dx = 0 (extrema)
stable equilibrium U is a minimum
v0 + v
×t
2
∆θ =
ω0 + ω
×t
2
F = ma
Rx
W = x0 Fdx
τ = Iα
Rθ
Wrot = θ0 τdθ
W = 12 mv2 − 12 m(v0 )2
Wrot = 21 mω2 − 12 m(ω0 )2
P = Fv
Prot = τω
p = mv
L = Iω
F=
unstable equilibrium U is a maximum
Angular
dp
dt
τ=
Figure 4: Rotational Motion
Total
E = K+U
Ei + WNC = Ef
WNC represents non-conservative work that converts mechanical energy into other forms of energy. For example, friction converts mechanical
energy into heat.
6
dL
dt
Torque
τ = r × F = rF sin θ
τ = Iα
Moment of Inertia
Z
2
I = Σmr =
r2 dm
Figure 5: Arc Length
I = Icm + Mh2
(h represents the distance from the center)
Values
1
2
12 ml
1
2
3 ml
rod (center)
Figure 6: Angular Velocity
rod (end)
hollow hoop/cylinder mr2
Rotational Motion
1
2
2 mr
The same equations for linear motion can be mod- hollow sphere 2 mr2
3
ified for use with rotational motion (Figure 4 on
2
solid sphere 5 mr2
the previous page).
solid disk/cylinder
Atwood’s Machine
Angular Motion
θ=
s
r
ω=
v
r
α=
at
r
at = r
p
a=
|(m2 − m1 )|g
m1 + m2 + 12 M
Angular Momentum
L = Iω
L = r × p = rp sin θ = rmv sin θ
α2 + ω4
ac = ω2 r
τ=
1
1
Krolling = Iω2 + mv2
2
2
dL
dt
Total angular momentum is always conserved
when there are no external torques (τ = dL
dt = 0).
7
ω = 2πf
r
A=
(x0 )2 +
φ = arctan
v 2
0
ω
−v0
ωx0
1
E = kA2
2
Spring
Figure 7: Simple Harmonic Motion
Fs = −kx
Simple Harmonic Motion
r
Simple harmonic motion is the projection of uniform circular notion on to a diameter. Likewise,
uniform circular motion is the combination of
simple harmonic motions along the x-axis and
y-axis that differ by a phase of 90◦ .
Ts = 2π
r
ωs =
m
k
k
m
amplitude (A) maximum magnitude of displacePendulum
ment from equilibrium
cycle one complete vibration
Simple
period (T ) time for one cycle
s
frequency (f) cycles per time
T = 2π
angular frequency (ω) radians per time
r
x = Acos(ωt + φ)
ω=
A cable with a moment of inertia swings back
and forth. d represents the distance from the
pendulum’s pivot to its center of mass.
s
I
T = 2π
mgd
a = −ω2 A cos(ωt + φ) = −ω2 x
2π
1
=
ω
f
r
f=
g
L
Compound
v = −ωA sin(ωt + φ)
T=
L
g
1
ω
=
T
2π
ω=
8
mgd
I
frictionless pivot
Energy
amplitude θ
U=
massless rod
E=
bob's
trajectory
massive bob
equilibrium
position
−Gm1 m2
R
−GMm
2r
v=
Figure 8: Simple Pendulum
2πR
T
r
vescape =
Torsional
2GM
re
For orbits around the earth, re represents the raA horizontal mass with a moment of inertia is dius of the earth.
suspended from a cable and swings back and
forth.
r
T = 2π
ω=
I
k
k
I
Gravity
F=
−Gm1 m2
R2
G ≈ 6.67 × 10−11
Nm2
kg2
Kepler’s Laws
1. All orbits are elliptical.
2. Law of Equal Areas.
2
4π
3. T 2 = GM
R3 = Ks R3 , where Ks is a uniform
constant for all satellites/planets orbiting
a specific body
9