Exam 1 Review
Ch 1 Units and Measurements
• Fundamental physical quantities: Length [m], Mass [Kg],
Time [s]
• Scienti c notation: 5.2 × 10
• Signi cant Figures
• Convert Units
• Dimensional Analysis
PHY2170/75-Exam 1 Review
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•
•
•
•
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A signi cant gure is a reliably known digit in the measurement
All non-zero digits are signi cant
Leading zeros are not signi cant; Only tailing zeros count
If you are give an integer number, you can specify it has in nite precision
Perform operations with full accuracy and round o digit in the nal result
When multiplying or dividing two or more quantities, the number of signi ant
gures in the nal results is the same of the least number of signi ant gures
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When adding or subtracting, round the result to the smallest number of decimal
places of any term in the sum (or di erence)
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PHY2170/75-Exam 1 Review
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Signi icant Figures
Measurement Uncertainty
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•
The uncertainty in a measurement, A, is often denoted as δA
The measurement result would be recorded as
A ± δA
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•
δA
× 100 %
Precent uncertainty =
A
Uncertainty Propagation Formula
δ(A ± B) =
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δA + δB
δA
δB
δ(AB) = AB
+
(A) (B)
2
A
A
δ
=
(B) B
2
2
2
PHY2170/75-Exam 1 Review
δA
δB
+
(A) (B)
2
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Ch 3. 1D Motion
•
Instantaneous velocity is the slope of the position curve x(t) and acceleration is the
•
Given a particle’s position as a function of time, calculate the instantaneous velocity
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•
slope of the velocity curve v(t)
2
dx
d x
for any given time v(t) =
and acceleration a(t) =
dt
dt 2
Given a graph of a particle’s position versus time, determine the instantaneous
velocity for any time
Master the kinematic equations for constant acceleration to solve various problems
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PHY2170/75-Exam 1 Review
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Ch 3. 1D Motion
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•
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Displacement Δx ⃗ = x2 − x1
dx⃗
Velocity v ⃗ =
dt
dv⃗ d x⃗
Acceleration a ⃗ =
=
dt
dt 2
2
Velocity
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PHY2170/75-Exam 1 Review
Position
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Summary of Kinematic Equations (constant a)
v(t) = v0 + at
v0 + v(t)
v̄(t) =
2
x(t) = x0 + v̄t
1 2
x(t) = x0 + v0t + at
2
2
v (t) =
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v0
+ 2a(x(t) − x0)
PHY2170/75-Exam 1 Review
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Ch 2. Vector
•
A vector is a mathematical object with magnitude and direction
⃗
s⃗ = a ⃗ + b
Cartesian
Coordinates
⃗
b
a⃗
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x
ϕ
vx
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v ⃗ + w ⃗ = (vx + wx, vy + wy)
y
= vx i ̂ + vy j ̂
v
s⃗
The head-to-tail method
vy
⃗
V = (v , v )
y
2D
x
vx = v cos(ϕ)
vy = v sin(ϕ)
v ⃗ − w ⃗ = (vx − wx, vy − wy)
PHY2170/75-Exam 1 Review
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Scalar (Dot) Product
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A vector is a mathematical object with magnitude and direction
Vector multiplication: Scalar (Inner/dot) Product a ⃗ ⋅ b ⃗ = | a ⃗ | | b ⃗ | cos θab
= axbx + ayby + azbz
π o
⃗
⃗
θab < (90 ) : a ⋅ b > 0
2
π
θab = : a ⃗ ⋅ b ⃗ = 0
2
π
θab > : a ⃗ ⋅ b ⃗ < 0
2
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PHY2170/75-Exam 1 Review
a⃗
θab
⃗
b
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Vector (Cross) Product
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•
⃗
⃗
⃗
⃗
⃗
⃗
|
|
|
|
|
b | sin(θab)
Type 3: Cross (vector) product c = a × b ; c = a
With vector components,
⃗
̂
̂
̂
̂
̂
̂
⃗
a × b = (ax i + ay j + azk) × (bx i + by j + bzk) c ⃗
̂i × j ̂ = k̂
̂i × k̂ = − j ̂
̂i × i ̂ = 0
̂
̂
̂
ax i × by j = axbyk
⃗
⃗
⃗
a × b =− b × a⃗
⃗
b
θab
a⃗
i ̂ j ̂ k̂
⃗
a
a
a
̂
̂
̂
=
x
y
z
⃗
a × b = (aybz − byaz) i + (azbx − bzax) j + (axby − bxay)k
bx by bz
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PHY2170/75-Exam 1 Review
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Displacement, Velocity, and Acceleration
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•
•
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The position vector from the origin of the coordinate system to
⃗
point P is r(t)
̂
̂
̂
r(t)
⃗ = x(t) i + y(t) j + z(t)k
Δ r ⃗ = r(t
⃗ 2) − r(t
⃗ 1)
r(t
⃗ + Δt) − r(t)
⃗
dr ⃗
⃗ = lim
=
Velocity: v (t)
t→0
Δt
dt
⃗ + Δt) − v (t)
⃗
v (t
dv⃗
⃗ = lim
Acceleration: a (t)
=
t→0
Δt
dt
The displacement vector Δ r ⃗ is de ned
PHY2170/75-Exam 1 Review
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Independence of Motion
In the kinematic description of motion, we are able to treat the horizontal and vertical
components of motion separately.
r(t)
⃗ = x(t) i ̂ + y(t) j ̂ + z(t)k̂
dx(t)
dy(t)
dz(t)
vx(t) =
, vy(t) =
, vz(t) =
⃗ = vx(t) i ̂ + vy(t) j ̂ + vzk̂
v (t)
dt
dt
dt
2
2
2
dv
(t)
dvz(t)
dvx(t)
d
x(t)
d
y(t)
d
z(t)
y
̂
̂
̂
̂
̂
̂
⃗
a (t) =
i+
j+
k =
i
+
j
+
k
dt
dt
dt
dt 2
dt 2
dt 2
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PHY2170/75-Exam 1 Review
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Independence of Motion: Constant Acceleration
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A particle moves in two-dimension with a constant acceleration
⃗ = a0x i ̂ + a0y j ̂
a (t)
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r(t)
⃗ = x(t) i ̂ + y(t) j ̂
Independence of motion in
x and y directions
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⃗ = vx(t) i ̂ + vy(t) j ̂
v (t)
PHY2170/75-Exam 1 Review
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Kinematics of projectile motion
Horizontal Motion
Vertical Motion
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PHY2170/75-Exam 1 Review
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Uniform Circular Motion
•
Centripetal acceleration
Δv⃗
a ⃗ = lim
Δt→0 ( Δt )
Δr
| Δ v ⃗ | ≃ vΔθ = v
r
v
Δr
v
a = lim
=
r Δt→0 ( Δt )
r
2
Points to the center of the circle
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