PH2100 Formula Sheet - Physics
PH2100 Formula Sheet - Physics
PH2100 Formula Sheet - Physics
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<strong>PH2100</strong> <strong>Formula</strong> <strong>Sheet</strong> Knight<br />
Concepts of Motion<br />
distance traveled<br />
average speed = time interval spent traveling<br />
<br />
∆ r = rf<br />
−ri<br />
<br />
∆r<br />
vavg<br />
= ∆ t<br />
<br />
∆v<br />
aavg<br />
= ∆ t<br />
Kinematics: The Mathematics of Motion<br />
For Motion in a Straight Line:<br />
ds<br />
vs<br />
= = slope of position-time graph<br />
dt<br />
dvs<br />
as<br />
= = slope of velocity-time graph<br />
dt<br />
tf<br />
⎧area under velocity curve<br />
sf = si + ∫ vsdt = si<br />
+⎨<br />
ti<br />
⎩ from ti<br />
to tf<br />
tf<br />
⎧area under acceleration curve<br />
vfs = vi s<br />
+ ∫ asdt = vi<br />
s<br />
+ ⎨<br />
ti<br />
⎩from<br />
ti<br />
to tf<br />
Uniform Motion:<br />
s = s + v ∆t<br />
f<br />
i<br />
Uniformly Accelerated Motion:<br />
v = v + a ∆t<br />
fs is s<br />
1<br />
f i is<br />
2<br />
v = v + 2a ∆s<br />
2 2<br />
fs is s<br />
s<br />
s<br />
( )<br />
s = s + v ∆ t+ a ∆t<br />
Motion on an Inclined Plane : a =± gsinθ<br />
Vectors and Coordinate Systems<br />
2<br />
s<br />
Force and Motion<br />
1 <br />
a = Fnet<br />
m<br />
<br />
where F = F + F + F + ...<br />
net 1 2 3<br />
Dynamics I: Motion Along a Line<br />
<br />
F = F = ma<br />
( µ<br />
sn, direction as necessary to prevent motion)<br />
( µ<br />
kn, direction opposite the motion)<br />
( µ n, direction opposite the motion)<br />
Need additional help? Then visit the <strong>Physics</strong> Learning Center located in 228 Fisher. Walk-in hours are Sunday<br />
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net<br />
∑<br />
i<br />
i<br />
A Model for Friction :<br />
<br />
Static : fs<br />
≤<br />
<br />
Kinetic : fk<br />
=<br />
<br />
Rolling : fr<br />
=<br />
r<br />
<br />
1 2<br />
Drag : D ≈<br />
4<br />
Av , direction opposite the motion<br />
( )<br />
Dynamics II: Motion in a Plane<br />
<br />
r = xiˆ+<br />
yj ˆ<br />
<br />
dr dx<br />
ˆ<br />
dy<br />
v = = i + ˆj = v ˆ ˆ<br />
xi + vy<br />
j<br />
dt dt dt<br />
<br />
dv dv dv<br />
x y<br />
a = = iˆ+ ˆj = a ˆ ˆ<br />
xi + ay<br />
j<br />
dt dt dt<br />
F = F = ma<br />
( net )<br />
( )<br />
net<br />
x<br />
F = F = ma<br />
y<br />
∑<br />
∑<br />
Constant Acceleration :<br />
( ) ( )<br />
1<br />
2<br />
1<br />
2<br />
f<br />
=<br />
i<br />
+<br />
ix∆ +<br />
2 x<br />
∆<br />
f<br />
=<br />
i<br />
+<br />
iy∆ +<br />
2 y<br />
∆<br />
v = v + a ∆t<br />
fx ix x<br />
x<br />
y<br />
x<br />
y<br />
x x v t a t y y v t a t<br />
v = v + a ∆t<br />
fy iy y<br />
y<br />
Projectile Motion :<br />
1<br />
2<br />
A <br />
<br />
xf = xi + vi x∆ t yf = yi + viy∆t− g( ∆t<br />
2 )<br />
Ay<br />
= Ay<br />
ĵ<br />
vf x<br />
= vix = constant vf y<br />
= vi<br />
y<br />
−g∆t<br />
θ 2 2<br />
A ˆ<br />
vf y<br />
= viy<br />
−2g( yf<br />
− y )<br />
x<br />
= Axi<br />
i<br />
x<br />
y y′<br />
<br />
A= A ˆ ˆ<br />
x<br />
+ Ay = Axi + Ay<br />
j<br />
Relative Motion:<br />
<br />
In the figure above:<br />
rPS = rPS '<br />
+ vS ' St<br />
• P<br />
<br />
Ax<br />
= Acos<br />
θ Ay<br />
= Asinθ<br />
vPS = vPS '<br />
+ vS ' S<br />
S<br />
A<br />
2 2 −1⎛<br />
y ⎞<br />
′<br />
x′<br />
A= Ax<br />
+ Ay<br />
θ = tan<br />
<br />
⎜ ⎟<br />
⎝ A<br />
aPS<br />
= aPS<br />
'<br />
S<br />
x ⎠<br />
x
Dynamics III: Motion in a Circle<br />
s<br />
θ =<br />
r<br />
∆θ<br />
average angular velocity = ∆ t<br />
dθ<br />
ω =<br />
dt<br />
v = ωr<br />
t<br />
Uniform Circular Motion :<br />
2πr<br />
2 π rad<br />
v = ω =<br />
T T<br />
θ = θ + ω ∆t<br />
f<br />
i<br />
2<br />
v 2<br />
ar<br />
= = ω r<br />
r<br />
Nonuniform Circular Motion (constant a ):<br />
dv<br />
at<br />
=<br />
dt<br />
at<br />
2<br />
θf = θi + ωi∆ t+ ( ∆t)<br />
2r<br />
at<br />
ωf<br />
= ωi<br />
+ ∆t<br />
r<br />
2<br />
mv<br />
2<br />
Fn<br />
et<br />
=<br />
r ∑ Fr<br />
= mar<br />
= = mω<br />
r<br />
r<br />
⎧0 uniform motion<br />
Fnet<br />
= F<br />
t ∑ t<br />
=⎨<br />
⎩ mat<br />
nonuniform motion<br />
( )<br />
( )<br />
( F )<br />
net<br />
z<br />
∑<br />
= F = 0<br />
z<br />
t<br />
Energy<br />
K =<br />
U<br />
g<br />
mech<br />
1<br />
2<br />
2<br />
f f i i<br />
1<br />
( Fsp ) =−k∆ s Us<br />
=<br />
2<br />
k( ∆s)<br />
s<br />
mv<br />
= mgy<br />
E = K + U<br />
K + U = K + U<br />
(conservation of mechanical energy)<br />
Perfectly Elastic 1-D Collisions ( m initially at rest):<br />
m − m 2m<br />
+ +<br />
1<br />
1 2 1<br />
( v ) = ( v ) ( v ) = ( v )<br />
Work<br />
fx 1 ix 1 fx 2<br />
ix<br />
m1 m2 m1 m2<br />
sf<br />
⎧<br />
⎪ Fds<br />
s<br />
=<br />
si<br />
area under the force-position curve<br />
W =<br />
∫<br />
⎨ ⎪<br />
<br />
<br />
⎩F⋅∆ r = F∆rcos θ if F is a constant force<br />
∆ K = W = W + W + W (work-kinetic energy theorem)<br />
net c diss ext<br />
f i c<br />
th micro micro<br />
th<br />
diss<br />
( i f)<br />
∆ U = U − U =−W<br />
→<br />
dU<br />
Fs<br />
=−<br />
ds<br />
E = K + U<br />
∆ E<br />
E<br />
sys<br />
= −W<br />
= K + U + E<br />
th<br />
Kf + Uf +∆ Eth = Ki + Ui + Wext<br />
dE<br />
dW <br />
= = = ⋅ =<br />
dt<br />
dt<br />
sys<br />
P P F v Fv<br />
2<br />
2<br />
cosθ<br />
Newton’s Third Law<br />
<br />
F<br />
A on B<br />
<br />
=−F<br />
B on A<br />
Impulse and Momentum<br />
<br />
p = mv<br />
f<br />
tf<br />
⎧<br />
⎪∫<br />
Fx<br />
t dt =<br />
ti<br />
( ) area under force curve<br />
J<br />
x<br />
= ⎨<br />
⎪⎩ Favg∆t<br />
∆ px<br />
= Jx<br />
(impulse-momentum theorem)<br />
<br />
P = p1+ p2 + p3+<br />
...<br />
<br />
P = P (conservation of momentum)<br />
i<br />
Newton’s Theory of Gravity<br />
GMm<br />
FM on m<br />
= Fm on M<br />
=<br />
2<br />
r<br />
GM<br />
gsurface =<br />
2<br />
R<br />
GM ⎛ 4π<br />
⎞<br />
Circular Orbit: v= T =⎜ ⎟r<br />
r ⎝GM<br />
⎠<br />
Physical Constants<br />
g = 9.80 m/s<br />
earth<br />
earth<br />
2<br />
G = 6.67 × 10 N ⋅m / kg<br />
M<br />
R<br />
= ×<br />
−11 2 2<br />
24<br />
5.98 10 kg<br />
= ×<br />
6<br />
6.37 10 m<br />
2<br />
2 3<br />
Need additional help? Then visit the <strong>Physics</strong> Learning Center located in 228 Fisher. Walk-in hours are Sunday<br />
7:00 – 9:00 p.m. and Monday through Thursday 3:00 - 9:00 p.m.