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=
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|
⌃F = (⌃m)a
• Accelerating downward: N = |mg| - |ma|
Atwood’s Machine2
a=
2 Pulley
F-f
⌃m
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).
and string are assumed to be massless.
videal =
3
p
rg tan ✓
Range
vmin =
s
rg(tan ✓ - µ)
µ tan ✓ + 1
✓ represents the smaller angle from the x-axis to
the direction of the projectile’s initial motion.
Starting from a height of x = 0:
vmax =
s
rg(tan ✓ + µ)
1 - µ tan ✓
xmax =
Projectile Motion
Circular Motion
Position
Centripetal (radial)
(v0 )2 sin 2✓
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 =
✓ represents the smaller angle from the x-axis to
the direction of the projectile’s initial motion.
mv2
r
Tangential
(vx )0 = v0 cos ✓
Tangential acceleration is tangent to the object’s
motion. It refers to a change in speed.
(vy )0 = v0 sin ✓
at =
vx = 0
d|v|
dt
Combined
vy = -gt
atotal =
Height
q
(ac )2 + (at )2
✓ 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 ✓)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
mv2
N =
+ mg
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
Z
Z
⌃mr
1
1
=
=
rdm =
x dx
⌃m
⌃m
⌃m
=
fmax = µN
Momentum-Impulse
⌃m =
p = mv
F=
I=
Z
dm
M
=
dx
L
Z
dm =
Z
dx
(⌃m)vCM = ⌃mv = ⌃p
dp
dt
Fnet = (⌃m)aCM
Energy
Fdt = F t = p = m v
Work
Collisions
Total momentum is always conserved when there
are no external forces (F = dp
dt = 0).
5
W=
Z
Fdx = K
Power
Pavg =
W
Fx
=
t
t
Pinstant =
dW
= Fv
dt
Kinetic Energy
Linear
1
K = mv2
2
Potential Energy
F=-
U=-
Z xf
dU
dx
Angular
v=
dx
dt
=
x
t
!=
d✓
dt
=
✓
t
a=
dv
dt
=
v
t
↵=
d!
dt
=
!
t
x = 12 at2 + v0 t
✓ = 12 ↵t2 + !0 t
v = at
! = ↵t
(v)2 - (v0 )2 = 2a( x)
FC dx = -WC
x=
xi
Z
v0 + v
⇥t
2
F = ma
Rx
W = x0 Fdx
Z
1
UHooke = - FHooke dx = - -kxdx = kx2
2
Ug = mgh
equilibrium point F = - du
dx = 0 (extrema)
stable equilibrium U is a minimum
✓=
!0 + !
⇥t
2
⌧ = I↵
R✓
Wrot = ✓0 ⌧d✓
W = 12 mv2 - 12 m(v0 )2
Wrot = 12 m!2 - 12 m(!0 )2
P = Fv
Prot = ⌧!
p = mv
L = I!
F=
unstable equilibrium U is a maximum
(!)2 - (!0 )2 = 2↵( !)
dp
dt
⌧=
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
⌧ = r ⇥ F = rF sin ✓
⌧ = I↵
Moment of Inertia
I = ⌃mr2 =
Figure 5: Arc Length
Z
r2 dm
I = Icm + Mh2
(h represents the distance from the center)
Values
rod (center)
Figure 6: Angular Velocity
rod (end)
1
2
12 ml
1
2
3 ml
Rotational Motion
hollow hoop/cylinder mr2
Angular Motion
Atwood’s Machine
solid disk/cylinder
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).
✓=
s
r
!=
v
r
↵=
at
r
at = r
p
a=
|(m2 - m1 )|g
m1 + m2 + 12 M
Angular Momentum
L = I!
L = r ⇥ p = rp sin ✓ = rmv sin ✓
↵ 2 + !4
ac = !2 r
⌧=
dL
dt
Total angular momentum is always conserved
when there are no external torques (⌧ = dL
dt = 0).
1
1
Krolling = I!2 + mv2
2
2
7
! = 2⇡f
A=
r
(x0 )2 +
✓
= arctan
⇣ v ⌘2
0
!
-v0
!x0
◆
1
E = kA2
2
Spring
Figure 7: Simple Harmonic Motion
Simple Harmonic Motion
Fs = -kx
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 = 2⇡
!s =
r
r
m
k
k
m
amplitude (A) maximum magnitude of displacePendulum
ment from equilibrium
cycle one complete vibration
Simple
period (T ) time for one cycle
frequency (f) cycles per time
T = 2⇡
angular frequency (!) radians per time
x = Acos(!t + )
!=
g
L
2⇡
1
=
!
f
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 = 2⇡
mgd
1
!
=
T
2⇡
r
a = -!2 A cos(!t + ) = -!2 x
f=
r
L
g
Compound
v = -!A sin(!t + )
T=
s
!=
8
mgd
I
Energy
U=
E=
-Gm1 m2
R
-GMm
2r
v=
Figure 8: Simple Pendulum
vescape =
Torsional
2⇡R
T
r
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.
T = 2⇡
r
!=
k
I
I
k
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
4⇡
3. T 2 = GM
R3 = Ks R3 , where Ks is a uniform
constant for all satellites/planets orbiting
a specific body
9