F=ma Formula Sheet
LATEX
January 2022
1
1.1
Error Propagation
Addition and Subtraction
If x and y have independent random errors δx and δy ,
the error in z = x ± y is given by
q
δz = δx2 + δy2 .
1.2
Multiplication
If x and y have independent random errors δx and δy ,
the error in z = x × y is given by
s 2
2
δy
δz
δx
=
+
(Percent Uncertainty).
z
x
y
1.3
Function Uncertainty
If z = f (x) for some function f (x), then
δz = |f 0 (x)| δx .
2
2.1
Kinematics
Big 5 Equations
When acceleration is constant,
∆x
∆x
vf
vf2
∆x
1
= vi t + at2
2
1
= vf t − at2
2
= vi + at
= vi2 + 2a∆x
vi + vf
=
t
2
1
F=ma Formula Sheet
2.2
Changing Acceleration
Z
x(t)
dx
dt
d2 x
dx2
2.3
=
t
vdt
x0 +
0
Z t
=
adt
v0 +
0
=
a(t)
Projectile Motion
Constant acceleration in the y direction and no acceleration in the x direction.
ax
=
0
ay
=
−9.81m/s2
The time of flight, t, can be given by
t=
2v0 sin θ
g
The maximum height reached, H, can be given by
H=
v02 sin2 θ
2g
The horizontal range, R, can be given by
R=
2.4
v02 sin2 θ
g
(Where ∆y = 0)
Projectile Motion on an Incline
The acceleration and velocity in the x and y axis are given by
ax
=
−g sin(φ)
ay
=
−g cos(φ)
vx
= vi cos(θ − φ)
vy
= vi sin(θ − φ)
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3
Forces and Newton’s Laws
3.1
Newton’s First Law
A body will remain at rest or at constant velocity unless it is acted upon by a
net external force.
3.2
Newton’s Second Law
F~ = m~a
3.3
Newton’s Third Law
If two bodies exert a force on one another, the forces are equl in magnitude but
opposite in direction.
3.4
Gravitational force
Fg = mg
Where g = 9.81m/s2
3.5
Spring Force
F = −kx
Where k is the spring constant with units of N/m
3.6
Centripetal Force
Fc =
mv 2
r
Where
v = 2πωr
ω in rev/s
Summing the above,
Fc = 4π 2 ω 2 mr
3.7
Friction
Friction is the force that resists the sliding or rolling of one solid object over
another.
3.7.1
Static Friction
Fs ≤ µs Fn
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F=ma Formula Sheet
3.7.2
Kinetic Friction
Fk = µk Fn
If an object is moving down an incline at constant velocity, the coefficient of
kinetic friction can be given by
µk = tan θ
4
Work and Energy
4.1
Work
The work a force applies on an object is given by
W = F × s × cos θ
Where
F = Force
s = Displacement
θ = Angle between the force and direction of motion
The above equation can also be written as
W = F~ · ~s
4.2
Work Energy Theorem
Work can be expressed as
W = ∆K
4.3
Kinetic Energy
1
mv 2
2
K=
4.4
Gravitational Potential Energy
Ug = mgh
4.5
Spring Potential Energy
Us =
4.6
1 2
kx
2
Force Energy Relationships
Z
x
U (x) = −
F (x)dx
x0
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4.7
4.7.1
Conservative and Non-conservative Forces
Conservative Forces
Conservative forces are forces that only depends on its initial and final positions.
Examples: gravity, force in elastic spring, electrostatic force, etc.
4.7.2
Non-conservative forces
Non-conservative forces are forces that not only depends on its initial and final
positions, but also on the path between them.
Examples: friction,tension, normal force, etc.
4.8
Potential Energy Graphs
dU = −F dx
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5
Momentum
Momentum is the vector denoting the quantity of motion of a moving body.
5.1
Conservation of Momentum
In a system without any net external force, momentum is always conserved.
5.2
Equation for Momentum
p~ = m~v
5.3
Center of Mass
The distance to the center of mass in a two body system, xcm , is given by
(m1 + m2 )xcm = m1 x1 + m2 x2
Where m1 , m2 are the masses of the two individual bodies and x1 , x2 are the
distance to the center of mass of the two individual bodies.
5.4
Momentum in Collisions
Total momentum before collision = Total momentum after collision
m1 vi1 + m2 vi2 = m1 vf 1 + m2 vf 2
5.5
Collisions
A collision takes place when bodies move toward each other and come near
enough to interact and exert a mutual influence.
5.5.1
Elastic Collisions
In perfect elastic collisions, objects do not stick together.
Kinetic energy is conserved.
For 1 dimention elastic collisions involving 2 objects:
mA − mB
2mB
VA =
VA0 +
V B0
mA + mB
mA + mB
2mB
mA − mB
VA =
VA0 +
V B0
mA + mB
mA + mB
5.5.2
Inelastic Collisions
In perfect inelastic collisions, objects stick together and looses energy.
Kinetic energy is not conserved.
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5.6
Force-Momentum Relationships
d~
p
F~ =
dt
5.7
Force vs. Time Graphs
The area under a Force vs. Time graph is equal to the impulse, J,or ∆P .
Z
J
t2
=
F dt
t1
Z t2
=
t1
p2
dp
dt
dt
Z
=
dp
p1
6
=
p2 − p1
=
∆p
Torque and Angular Momentum
Torque and angular momentum are the force and momentum for rotating objects.
6.1
Terms
s = Translational Displacement
v
= Translational Speed
a
= Translational Acceleration
t
= Time
θ
= Angular Displacement
ω
= Angular Speed
α
= Angular Acceleration
t
= Time
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F=ma Formula Sheet
6.2
Conversion
Displacement
s = rθ
θ=
s
r
v = rω
ω=
v
r
a = rα
α=
a
r
Velocity
Acceleration
6.3
Big 5 Equations
When angular acceleration is constant,
θ
= ωt
ω
= ω0 + αt
1
= ω0 t + αt2
2
= ω02 + αt2
ω0 + ω
=
2
θ
ω2
ω̄
6.4
Torque
torque = τ = ~r × F~ = rF sinθ
Also,
τ = Iα
Where I is the moment of inertia.
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6.5
Moment of Inertia
Rotational inertia is a scalar value which tells how difficult it is to change the
rotational velocity of the object around a given rotational axis.
Moment of inertia has SI units of kg · m2
6.6
Angular Momentum
angular momentum = L = I × ω
In the absence of net external torques, angular momentum is conserved.
6.7
Rolling Without Slipping
When a circular object is rolling without slipping,
v = Rω
a = Rα
6.8
Rotational Energy
When a circular object is rolling without slipping,
K=
6.9
1 2
Iω
2
Net Torque
The net torque in a system is given by
X
~τ = I α
~ = ~τ1 + ~τ2 + ~τ3 + · · · + ~τn
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F=ma Formula Sheet
7
7.1
Gravitation
Newton’s law of universal gravitation
Gm1 m2
F~g =
r2
Where G = 6.67 × 10−11 kg −2 m2 .
Gravitational force is a vector that always points at the other mass.
The gravitational field, g, is given by
GM
r2
g=
Where
M = Mass of the earth, 5.972 × 1024 kg
r = Radius of the earth, 6.371 × 106 m
7.2
Gravitational Potential Energy
Z
7.3
Ug
=
Ug
=
Ug
=
Ug
=
Ug
=
r
Gm1 m2
dr
r2
∞
Z r
1
Gm1 m2
dr
2
r
∞
r
−1
Gm1 m2
r
∞ 1
1
−Gm1 m2
−
r
∞
−Gm1 m2
r
Escape Velocity
r
vesc =
7.4
2GM
R
Kepler’s Laws
Kepler’s Laws are three theorems describing orbital motion.
7.4.1
Kepler’s First Law
Planets move around a star in an elliptical orbit.
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F=ma Formula Sheet
7.4.2
Kepler’s Second Law
The radius vector from the star to a planet sweeps equal areas in equal times.
7.4.3
Kepler’s Third Law
The orbital period squared is proportional to the semi-major axis cubed.
T2 =
7.5
4π 2 3
a
GM
Total energy of an orbit
Etot = −
7.6
GM m
2a
Properties of Inverse Square Law
The inverse square law of gravitation shows that objects have the following
properties:
7.6.1
Shell Theorem
• A spherically symmetric body affects external objects gravitationally as
though all of its mass were concentrated at a point at its center.
• If the body is a spherically symmetric shell (i.e., a hollow ball), no net
gravitational force is exerted by the shell on any object inside, regardless
of the object’s location within the shell.
7.6.2
Elliptical Orbit
Planets orbit around a star in an elliptical orbit.
7.6.3
Square Cube Law
The squares of the orbital periods of the planets are directly proportional to the
cubes of the semi-major axes of their orbits.
The period of a planet orbiting a star increases rapidly with the radius of its
orbit.
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8
Fluids
8.1
Pressure
Pressure (P ) =
8.2
Force (Fn )
Area (A)
Ideal Fluid
• Incompressible: density is constant.
• Irrotational: flow is smooth, no turbulence.
• Nonviscous: fluid has no internal friction.
8.3
Pressure at Depth
The pressure at depth h in a fluid is given by
P = ρgh
Where ρ is the density of the fluid.
8.4
Buoyant Force
FB = ρf l vg
Where
FB is the buoyant force.
ρf l is the fluid density.
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F=ma Formula Sheet
v is the volume of fluid displaced.
g is the gravitational constant.
The equation can also be expressed as
FB = m f l g = Fg
Where
mf l is the mass of the displaced fluid.
Fg is the weight of the displaced fluid.
8.5
Continuity Equation
With ideal fluid,
A1 v 1 = A2 v 2
Where
A1 , A2 are the area of the fluid pathway at two points.
v1 , v2 are the velocity of the fluid at those two points.
8.6
Bernoulli’s Principle
For ideal fluid,
1
1
P1 + ρv12 + ρgh1 = P2 + ρv22 + ρgh2
2
2
9
Oscillations
9.1
Simple Harmonic Motion
An object will have simple harmonic motion when the restoring force is proportional to displacement (F = −kx).
9.2
Common Equations
Acceleration
a = −(2πf )2 x
Displacement
x = Acos(2πf t)
Speed
v = ±2πf
p
A2 − x 2
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Maximum Speed
vmax = 2πf A
Maximum Acceleration
amax = (2πf )2 A
for a mass-spring system
r
T = 2π
m
k
for a simple pendulum
s
T = 2π
10
10.1
l
g
Waves
Transverse Waves
Waves that oscillate perpendicular to the direction of motion.
λ=
v
f
Where
λ = Wavelength
v = Velocity of propagation
f = Frequency of signal
s
v0 =
T
=
µ
r
TL
M
Where
T = Tension
L = Length
µ= M
L = The linear mass of the substance that the wave is traveling through.
10.2
Longitudinal Waves
Waves that oscillate parallel to the direction of motion.
10.3
Fundamental Frequency
When the ends of a string are kept in place, the frequency of the waves in the
string is equal to multiples of the fundamental frequency, where the fundamental
frequency is the frequency of the wave with a wavelength of twice the length of
the string.
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