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-in- progress. 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 Defi- nition of Free Cultural Works (freedomdefined.org). Except as noted under “Graphic Credits” on the following page, the work is licensed under the Creative Commons Attribution-Share Alike 3.0 United States License. To view a copy of this license, visit http://creativecommons.org/licenses/by-sa/3.0/us/ or send a letter to Creative Commons, 171 Second Street, Suite 300, San Francisco, California, 94105, USA. This review guide is provided “as is” without warranty of any kind, either expressed or implied. You should not assume that this review guide is error-free or that it will be suitable for the particular purpose which you have in mind when using it. In no event shall the CHSN Review Project be liable for any special, incidental, indirect or consequential damages of any kind, or any damages whatsoever, including, without limitation, those resulting from loss of use, data or profits, whether or not advised of the possibility of damage, and on any theory of liability, arising out of or in connec- tion with the use or performance of this review guide or other documents which are referenced by or linked to in this review guide.
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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 Com- mons 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 2 ∆x = at + v0t 2 Figure 2: Atwood’s Machine
∆v = at
2 2 (v) − (v0) = 2a(∆x)
Figure 3: Draw a banked curve diagram v + v ∆x = 0 × t 2 Pulled Weights
Free Body Diagrams F − f a = Σm N Normal Force f Frictional Force T = ma T Tension mg Weight Elevator
F = ma Normal force acts upward, weight acts down- ward.
In a particular direction: • Accelerating upward: N = |ma| + |mg| • Constant velocity: N = |mg| ΣF = (Σm)a • Accelerating downward: N = |mg| − |ma|
Atwood’s Machine2 Banked Curve
|(m − m )|g a = 2 1 Friction can act up the ramp (minimum velocity m1 + m2 when friction is maximum) or down the ramp (maximum velocity when friction is maximum).
2 p Pulley and string are assumed to be massless. videal = rg tan θ
3 Range s rg(tan θ − µ) θ represents the smaller angle from the x-axis to vmin = µ tan θ + 1 the direction of the projectile’s initial motion. Starting from a height of x = 0: s rg( θ + µ) tan (v )2 sin 2θ vmax = x = 0 1 − µ tan θ max g
Projectile Motion Circular Motion
Position Centripetal (radial)
Centripetal acceleration and force is directed to- ∆x = v t x wards the center. It refers to a change in direc- tion.
1 2 ∆y = − gt + (vy)0t v2 2 a = c r Velocity mv2 F = ma = θ represents the smaller angle from the x-axis to c c r the direction of the projectile’s initial motion. 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 d|v| a = t dt ∆vx = 0 Combined
∆vy = −gt q 2 2 atotal = (ac) + (at) Height Vertical loop θ represents the smaller angle from the x-axis to 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).
(v sin θ)2 y = 0 max 2g
4 Top Elastic
Kinetic energy is conserved. F = ma v2 N + mg = m × m v + m v = m v0 + m v0 r 1 1 2 2 1 1 2 2 mv2 N = − mg r 0 0 −(v2 − v1) = v2 − v1 Bottom Inelastic
F = ma Kinetic energy is not conserved. v2 N − mg = m × r 0 mv2 m1v1 + m2v2 = (m1 + m2)v N = + mg r Center of Mass Friction
Σmr 1 1 Friction converts mechanical energy into heat. r = = rdm = xλdx cm Σm Σm Σm Static friction (at rest) is generally greater than Z Z kinematic friction (in motion). dm M λ = = fmax = µN dx L
Momentum-Impulse Σm = dm = λdx Z Z
p = mv (Σm)vCM = Σmv = Σp
dp F = dt Fnet = (Σm)aCM
I = Fdt = F∆t = ∆p = m∆v Energy Z Work Collisions
Total momentum is always conserved when there W = Fdx = ∆K dp are no external forces (F = dt = 0). Z
5 Power
W Fx P = = avg t t
dW P = = Fv instant dt
Kinetic Energy
1 Linear Angular K = mv2 2 dx ∆x dθ ∆θ v = dt = ∆t ω = dt = ∆t dv ∆v dω ∆ω Potential Energy a = dt = ∆t α = dt = ∆t 1 2 1 2 ∆x = 2 at + v0t ∆θ = 2 αt + ω0t dU F = − dx ∆v = at ∆ω = αt 2 2 2 2 (v) − (v0) = 2a(∆x)(ω) − (ω0) = 2α(∆ω) xf v + v ω + ω ∆U = − FCdx = −WC ∆x = 0 × t ∆θ = 0 × t Zxi 2 2 F = ma τ = Iα 1 2 UHooke = − FHookedx = − −kxdx = kx x θ 2 W = x Fdx Wrot = θ τdθ Z Z 0 0 1 2R 1 2 1 R2 1 2 W = 2 mv − 2 m(v0) Wrot = 2 mω − 2 m(ω0) U = mgh g P = Fv Prot = τω du equilibrium point F = − dx = 0 (extrema) p = mv L = Iω stable equilibrium U is a minimum dp dL F = dt τ = dt unstable equilibrium U is a maximum Figure 4: Rotational Motion Total
E = K + U
Ei + WNC = Ef
WNC represents non-conservative work that con- verts mechanical energy into other forms of en- ergy. For example, friction converts mechanical energy into heat.
6 Torque
τ = r × F = rF sin θ
τ = Iα
Moment of Inertia
I = Σmr2 = r2dm Figure 5: Arc Length Z
2 I = Icm + Mh
(h represents the distance from the center)
Values
1 2 rod (center) 12 ml Figure 6: Angular Velocity 1 2 rod (end) 3 ml 2 Rotational Motion hollow hoop/cylinder mr 1 2 solid disk/cylinder 2 mr The same equations for linear motion can be mod- 2 2 hollow sphere 3 mr ified for use with rotational motion (Figure 4 on solid sphere 2 mr2 the previous page). 5
Angular Motion Atwood’s Machine
s |(m2 − m1)|g θ = a = 1 r m1 + m2 + 2 M v ω = r Angular Momentum
a α = t L = Iω r
p 2 4 L = r × p = rp sin θ = rmv sin θ at = r α + ω
dL a = ω2r τ = c dt
1 2 1 2 Total angular momentum is always conserved Krolling = Iω + mv dL 2 2 when there are no external torques (τ = dt = 0).
7 ω = 2πf
r v 2 A = (x )2 + 0 0 ω
−v0 φ = arctan ωx0
1 E = kA2 2
Figure 7: Simple Harmonic Motion Spring
Simple Harmonic Motion Fs = −kx
rm Simple harmonic motion is the projection of uni- Ts = 2π form circular notion on to a diameter. Likewise, k uniform circular motion is the combination of r simple harmonic motions along the x-axis and k ω = y-axis that differ by a phase of 90◦. s m amplitude (A) maximum magnitude of displace- ment from equilibrium Pendulum cycle one complete vibration Simple period (T) time for one cycle s frequency (f) cycles per time L T = 2π angular frequency (ω) radians per time g
rg x = Acos(ωt + φ) ω = L
v = −ωA sin(ωt + φ) Compound
A cable with a moment of inertia swings back and forth. d represents the distance from the a = −ω2A cos(ωt + φ) = −ω2x pendulum’s pivot to its center of mass. s 2π 1 I T = = T = 2π ω f mgd
r mgd 1 ω ω = f = = I T 2π
8 frictionless pivot Energy
amplitude −Gm m θ U = 1 2 massless rod R
−GMm E = 2r bob's trajectory massive bob 2πR equilibrium v = position T
Figure 8: Simple Pendulum r2GM vescape = re Torsional
For orbits around the earth, re represents the ra- A horizontal mass with a moment of inertia is dius of the earth. suspended from a cable and swings back and forth. r I T = 2π k
k ω = I
Gravity
−Gm m F = 1 2 R2
Nm2 G ≈ 6.67 × 10−11 kg2
Kepler’s Laws
1. All orbits are elliptical. 2. Law of Equal Areas. 2 4π2 3 3 3. T = GM R = KsR , where Ks is a uniform constant for all satellites/planets orbiting a specific body
9