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Chapter 1 Unit(s) / Mechanics / Sig-Figs / Vectors
Speed = (d/t) || (m/s) |
d = distance : m = meters |
| |
t = time : s = seconds |
1 km = 1000 m |
1 kg = 1000 g |
mass = (kg) |
1 hour = 3600 seconds |
time = (seconds) |
1 mile = 1.609 km |
length = (meter) |
Volume = 1 cm3 |
Sig Figs |
π = 3.14 (3 sigfig) |
π = 3.14159 (6 sigfig) |
Density = (mass / volume) || (kg / m3) || (g / cm3) |
√ = square root |
Vector (Displacement) = √(x)2+(y)2 |
Total distance = x + y |
Vector A = Vector B if |Vector A| = |Vector B| |
Magnitude: √(x)2+(y)2 = (Answer in Units) : 1 Direction |
Components of Vector |
Vector A = Ax + Ay |
Ax = A ⋅ cos(Θ)
Ay = A ⋅ sin(Θ) |
A = √(A ⋅ cos(Θ))2 + (A ⋅ sin(Θ))2 |
Θ = Angle |
x = cos(Θ)
y = sin(Θ) |
cos(Θ) = Ax / A
sin(Θ) = Ay / A |
tan(Θ) = (y / x) or (Ay / Ax) or (By / Bx) |
x = Î
y = ĵ
z = k̂ |
Vector A = AxÎ + Ayĵ
Vector B = BxÎ + Byĵ |
Vector R = Vector A + Vector B
Vector R = (Ax + Bx)Î + (Ay + By)ĵ
Vector R (direction) = (x)Î + (y)ĵ
Vector R (magnitude) = √(x)Î2 + (y)ĵ2 |
Quadratic Formula
x = (-b +/- √b2 - 4 ⋅ a ⋅ c ) / (2 ⋅ a) |
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Chapter 2: Motion along A Straight Line
One Dimensional Motion |
Average Speed
= (total distance) / (time) |
Displacement
= Final Point - Initial Point |
Not Constant Velocity |
Average Velocity (V)
= (displacement / time) |
Average Velocity (V)
= (∆x / ∆t) |
Instantaneous Velocity = derivative of the given equation
Instantaneous Velocity = ( (a-final) - (a-initial) ) / ( (t-final)-(t-initial) ) |
∆t = (t-final) - (t-initial) |
∆x = (x-final) - (x-initial) |
Acceleration |
∆V = (V-final) - (V-initial)
∆t = (t-final) - (t-initial) |
Acceleration (a) = (∆V) / (∆t)
[a is constant] |
if a > 0 (positive)
if a < 0 (negative) |
Instantaneous Acceleration = derivative of the given equation |
Constant Acceleration
= constant acceleration motion in 1D |
V-final
= (a ⋅ t) + V-initial V-final
2 = (v-initial) 2 + 2 ⋅ a ( (t-final) - (t-initial) ) |
∆x = (x-final) - (x-initial)
∆x = (v-average) ⋅ (seconds)
∆x = (1/2 ⋅ (V-final) + (V-initial) ) ⋅ t (seconds) |
x-final
= 1/2 ( (V-initial) + (V-final) ) ⋅ t + (x-initial) x-final
= x-initial + (V-initial) ⋅ t(seconds) + 1/2 ⋅ a ⋅ t 2 |
Gravity (g) = -9.8 m/s2 |
V-final = (V-initial) + g * t (seconds) |
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Chapter 3: 2D or 3D Motion
The Acceleration Vector |
a = ∆V / ∆t |
(v-final) = (v-initial) + ∆V
∆V = (v-final) - (v-initial)
∆V = (v-final) + (-(v-initial)) |
Constant Speed Changing Direction |
a = ∆V / ∆t |
(v-final) = (v-initial) + ∆V
∆V = (v-final) - (v-initial) |
Projectile Motion
two assumptions: |
1. The freefall acceleration (g) is constant
2. Air resistance is negligible |
y-direction = constant acceleration motion
x-direction = constant velocity motion
Acceleration is only negative (y-direction)
g = -9.8 m/s2 |
Constant Velocity Motion |
x = (x-initial) + (v [x-direction] ) ⋅ t |
V (y-direction) = (v-initial) [y-direction] + g ⋅ t |
(y-final) = (y-initial + (v-initial) [y-direction] ⋅ t + 1/2 ⋅ g ⋅ t2 |
V (y-direction)2 = (v-initial) [y-direction]2 + 2 ⋅ g ( (y-final) - (y-initial) ) |
V (y-direction) = (v-initial) [y-direction] + g ⋅ t |
Trig Identity |
sin(ΘΘ) = sinΘcosΘ + cosΘsinΘ |
Constant Speed Motion
velocity is always changing |
r = radius |
V = (2πr)2 : 4π2r |
T = time-period |
a = ∆V / ∆t : never zero
∆V = (V / r) · ∆r |
Centripetal Acceleration |
Ac = (V2) / r
Ac = (2πr)2 / r
Ac = 4π2r / T2 |
Tangential and Radial Acceleration |
Ac = a-rad |
Vector A-total = Vector A-tangential + Vector A-radical
A-total = √(A-tan)2 + (A-rad)2 |
Relative Motion |
r ' = ( (v-initial) ⋅ t ) - (vector-r) |
Vector-r = √( (v-initial) ⋅ t)2 + (r ')2 |
Vector-V ' = (v-final) - (v-initial) |
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Chapter 4: Newtons Laws
Superposition of Forces |
Vector-R = Vector-F1 + Vector-F2 |
N = Net Force |
Fx = N · cos(ϴ)
Fy = N · sin(ϴ) |
Rx = ∑Fx
Ry = ∑Fy |
R = √(Rx)2 + Ry2 |
Newton's 1st Law
No Force; No Acceleration; No Motion |
Inertia:
the tendency of an object to resist any attempt to change its velocity |
Newton's 2nd Law |
Net Force = m · g |
a (x-direction) = (Fx total) / mass
a (y-direction) = (Fy total) / mass |
tan(ϴ) = y / x |
Newton's 3rd Law |
Fn = Normal Force |
Fy = Fn - m · g · cos(ϴ) |
Fx = m · g · sin(ϴ) |
Chapter 5: Applying Newton's Laws
vector-F = m · a |
Fx = m · ax |
T = tension : friction |
Fy= m · ay |
y = T - m · g |
Fr = Fn : Normal Force (Fn) |
No Friction |
α = Coefficient |
Fn = m · g |
Fx = T1· cos(ϴ) + T2· cos(ϴ) |
| |
Fy = T1· sin(ϴ) + T2· sin(ϴ) |
Friction |
Static Friction (fs): Object not in motion
Kinetic Friction (fK): Object is in motion |
| |
Empirical Formula |
μk: Coefficient of Kinetic Friction
μs: Coefficient of Static Friction
Static: fs ≤ μs · Fn
Static: fk = μk · Fn |
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Terminal Speed |
Fr α v |
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Fr α v2 |
Uniform Circular Motion |
Fc = m · ac : m · V2 / r |
Vertical Circle |
Top: Fy = -m · (V2 / r)
Bottom: Fy = μs * m · (g + V2 / r)
maxV = √(fs · r) / m
maxV = √ μs · g · r |
Top View |
T · sin(ϴ) = m · ac
ac = tan(ϴ) · g |
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