Download Physics – Mechanics

Physics – Mechanics
Rule #1: DRAW A PICTURE (Pictorial Representation, Motion Diagrams, Free-Body Diagrams)
Draw a picture of what the question is describing. Use motion diagrams and free-body diagrams to assist you in seeing
what is going on in the question.
→ +x
Motion Diagrams:
a=0
0
1
2
3
Ball rolling
Constant speed
Free-Body Diagrams:
a>0
a<0
01 2
3
4
Car speeding up
0
1
2 34
Box sliding to a stop
FDrag
FN
Ffriction
FN
FEngine
Fg
Fg
Rule #2: LABEL/LIST YOUR KNOWN VALUES AND THOSE DESIRED (Include units!!)
Label or list all values that were given to you in the problem. Also, include any values that were not explicitly stated, but

can be inferred from the problem (gravity occurs in most problems, but goes unsaid, so list ag = −9.8 m
s2
). Lastly,
label those values that you are seeking as well.
Motion in One Dimension (Translational Kinematics)
Equations for one-dimensional motion follow, but it is important to note that not all motion is one-dimensional. In order
to still use these equations, motion that occurs in multiple dimensions must be broken apart into one-dimensional
components. The one-dimensional equations may then be used for each component separately.
Equations of One-Dimensional Motion ( x has been used to denote
the position in any general direction; s also commonly used)



∆x = area under v -t graph = ∫ v ⋅ dt



∆v = area under a -t graph = ∫ a ⋅ dt
  
∆x x f − x0


vavg
= slope of x -t graph
= =
∆t t f − t0
  
∆v v f − v0


aavg
= slope of v -t graph
= =
∆t t f − t0


dx
dv


vinst =
ainst =
dt
dt
Equation for Constant Velocity (zero acceleration)



x f= v0 ⋅ ∆t + x0
Equations for Constant Acceleration
1



x=
a ⋅ (∆t ) 2 + v0 ⋅ ∆t + x0
f
2



v f = a ⋅ ∆t + v0
 2 2
 
v f=
v0 + 2a ⋅ ∆x
Variables
∆ denotes "change in"
t = Time

x = Position in any general direction

x0 = Initial Position

x f = Final Position

v = Velocity

vavg = Average Velocity

vinst = Instantaneous Velocity

v0 = Initial Velocity

v f = Final Velocity

a = Acceleration

aavg = Average Acceleration

ainst = Instantaneous Acceleration
Projectile Motion
Projectile motion can be simplified into two separate one-dimensional motions: one motion of the object going up & down and a
separate motion of the object going left or right. These motions can be considered independently. Once the motions are separated
we are free to use the equations for one-dimensional motion on each component.
y

v y0

v0
θ

vx0
x
Figure 2
Figure 1
Variables
Equations of Projectile Motion

θ Launch Angle off + x axis
=

a
x = 0 (ignoring air resistance)
vx0= v0 ⋅ cos θ




v y0= v0 ⋅ sin θ
a y = ag
v0 = Initial Velocity

vx0 = Initial Velocity in x direction

Horizontal Range Equation
v y0 = Initial Velocity in y direction
(distance traveled when an object launches and lands at same height)
This equation can be derived using equations for one-dimensional motion, equations

ax = Acceleration in x direction
for projectile motion, and the fact that y f = y0 .

a y = Acceleration in y direction
2
v ⋅ sin ( 2θ )

x f − x0 =∆x = 0
ag = Acceleration due to gravity
ag

Vertical Range Equation (maximum height of object)
This equation can be derived using equations for one-dimensional motion, equations

for projectile motion, and the fact that at the maximum height v y = 0 .
2

v02 ⋅ ( sin θ )
y f − y0 =∆y =
2a g
Motion down a Frictionless Inclined Plane

ag y θ 
ag

ag x
y
θ
Variables

FN

ax = ± ag sin θ
x

Fg
NOTE! ag has magnitude and direction since
it is the vector for gravity (direction would be a
positive or negative sign). ag has only the
magnitude since it is the scalar for gravity
(meaning ag is always positive).
θ = Angle of Incline
θ

ax = Acceleration down plane

ag = Acceleration due to Gravity

ag x = Gravity in the x direction

ag y = Gravity in the y direction
Circular Motion (Rotational Kinematics)
Circular motion seems like a fairly complicated two-dimensional motion, but when broken down it can be seen that many of the
equations and ways we approach circular motion are nearly identical to the one-dimensional equations. When looking at the
equations below you may notice that they seem to be one-dimensional equations that just have different variables, and that is
exactly what they are. θ replaces s or x or y , ω replaces v , and α replaces a . The difference with circular motion is we have
additional equations to solve the additional pieces that exist when motion goes from linear to circular.
where at ≠ 0, α ≠ 0

vt1
where
 at = 0
vt
ω
ω

vt ω

vt
ω
ω1

vt2 ω2

vt
α
ω3
∆θ
area=
under ω -t graph
∆ω
area=
under α -t graph
ar2 + at2
∫ ω dt
∫ α dt
slope of θ -t graph
ω=
=
avg
∆θ θ f − θ 0
=
t f − t0
∆t
slope of ω -t graph
α=
=
avg
∆ω ω f − ω0
=
t f − t0
∆t
ωinst =
dθ
dt
α inst =
Variables
r = Radius
T = Period
vt = Tangential Velocity
=
at Tangential Acceleration (∆ in speed)
=
ar Radial Acceleration (∆ in direction)
dω
dt
Equation for Constant Angular Velocity/
Uniform Circular Motion (zero angular acceleration)
θ f = ω ⋅ ∆t + θ 0
Equations for Constant Angular Acceleration/
Non-Uniform Circular Motion (constant angular acceleration)
1
2
=
θf
α ( ∆t ) + ω0 ⋅ ∆t + θ 0
2
ω f = α ⋅ ∆t + ω0
2
ω=
ω0 2 + 2α ⋅ ∆θ
f

at


anet ar

vt3
General Equations for Circular Motion
vt2
a=
= ω 2r
∆s = r ⋅ ∆θ
r
r
∆vt
2π r
vt = ω ⋅ r =
at=
= α ⋅r
T
∆t
a=
net

 
anet= at + ar
θ = Angular Position
θ 0 = Initial Angular Position
θ f = Final Angular Position
ω = Angular Velocity
ωavg = Average Angular Velocity
ωinst = Instantaneous Angular Velocity
ω0 = Initial Angular Velocity
ω f = Final Angular Velocity
α = Angular Acceleration
α avg = Average Angular Acceleration
α inst = Instantaneous Angular Acceleration
α 0 = Initial Angular Acceleration
α f = Final Angular Acceleration
Forces
The basic idea behind forces is that a force is a push or pull exerted on an object. We have used equations to show an object’s
motion, and now we use forces to show why an object may be changing its motion. When looking at forces acting on an object we
will tend to separate forces into one-dimensional components just as we did with motion but we can also sum all the forces in those
dimensions to see what we call the resultant force or net force. The idea is that when numerous forces act on an object you can add
them all together to see what the net force is, and this net force determines what kind of change in motion the object experiences.
We use the acceleration from this net force to link forces to our motion equations.
Variables

F = Force
m = Mass

a = Acceleration

Fnet = Net Force

Fx = Forces in x-direction

Fsp = Spring Force
General Force Equations


F= m ⋅ a



Fnetx =
∑ Fx =⋅
m ax






∑ Fx = F1x + F2x + F3x + ... + Fnx = m ⋅ ax
Specific Force Equations
Friction:


Fsp = −k ⋅ ∆s


Fg= m ⋅ g
f s ≤ µs ⋅ n
f=
µk ⋅ n
k
f=
µr ⋅ n
r
D = CD ⋅ A ⋅ v
Fg =
2
k = Spring Constant
∆s =Distance Stretched/Compressed

n = Normal Force

f s = Static Friction
Gm1m2
r2
µ s = Coefficient of Static Friction
Note on Frictional Forces
When picking which frictional force to use it is important to note when each one

f k = Kinetic Friction
should be used. Static friction,
µk = Coefficient of Kinetic Friction

f s , should be used when the object we are
looking at is not in motion or is being powered in its roll or braking.

f k , should be used when an object is moving/sliding across a

surface. Rolling friction, f r , should be used when an unpowered object is rolling
Kinetic friction,

f r = Rolling Friction
µr = Coefficient of Rolling Friction

D = Drag Force (opposite to motion)
=
A Cross-Sectional Area ( ⊥ to motion)
CD = Coefficient of Drag
across a surface. Frictional forces always point in the direction opposite to the
motion, or in the case of static friction, in the direction to prevent motion.
Momentum & Impulse
Momentum can be thought of as a quantity that represents how difficult it is to stop an object in motion or change an object’s
direction of motion. Our main use for momentum comes from the fact that in a closed system the total momentum is constant
(Conservation of Momentum). This fact allows us to have a better understanding of the interaction of objects, particularly in collisions
and explosions. What seem like chaotic interactions in collisions and explosions can be broken into parts, and so long as our system is
closed, the sum of the momenta before the event is equal to the sum of the momenta after the event. If the system is not closed,
then we have to take into consideration any outside forces taking or giving momentum to our system. Impulse is the change in
momentum for an object and is equal to the product of a force and the duration of time that it is applied.
Momentum & Impulse Equations


p= m ⋅ v



J x = ∆px ≈ Fxavg ⋅ ∆t

=
Jx
tf

dt
∫ F (t ) ⋅=
x

area under F -t graph
t0
Variables

p = Momentum

J = Impulse

Fxavg = Average Force in x-direction
∆t =Change in Time
Conservation of Momentum










∑ px0 =
∑ px f
px10 + px 20 + px 30 + ... + pxn0= px1 f + px 2 f + px 3 f + ... + pxn f
Remember: You can only use the conservation of momentum if the system is “isolated” or “closed”. This means that you can only
use the conservation of momentum if there are no outside forces that are adding or taking momentum from the system.
Energy, Work & Power
We can think of systems as having energy, and if there are no outside forces on these systems, then the energy is conserved much in
the way momentum is conserved. Just as an outside force can change the momentum of the system, an outside force can change
the energy of a system through what we call work. Power is the rate at which energy is transferred of transformed.
Energy & Work Equations
Variables
1
m ⋅ v2
2
1
2
U sp=
k ⋅ ( ∆s )
2
U g = m ⋅ g ⋅ h or m ⋅ g ⋅ "y"
K = Kinetic Energy
U sp = Elastic Potential Energy (Spring)
k = Spring Constant
∆r =Change in position in any general direction
U g = Gravitational Potential Energy
=
h Height=
 ∆y Change in Vertical Position
∆Eth =
Change in Thermal Energy
W = Work

∆r =
Distance Traveled in Same Direction as Force
Wext = Work External
=
K
∆Eth=
W=
∆U g= m ⋅ g ⋅ ∆y
f ⋅ ∆r
∫
xf
Fx ⋅ dx
= Area under F -x curve
 
W= F ⋅ ∆r , if F is constant and straight-line motion

W= F ⋅ ∆r cos θ
x0
Conservation of Energy
K 0 + U 0 + Wext
= K f + U f + ∆Eth
The conservation of energy is useful in many situations because, unlike the conservation of momentum, we can still use the
conservation of energy if there are outside forces. Outside forces are taken into account by work done on the system.
Variables
Power
P = Power
Esys = Energy of the System
t = Time
F = Force
v = Velocity
∆Esys dEsys
=
dt
∆t
P =F ⋅ vinst =F ⋅ v cos θ
=
P
Rotation of a Rigid Body
The following rotational motion equations can be used when you have a rigid body that is being revolved around a fixed point.
Rotational Motion Equations
M = ∑ mi = m1 + m2 + m3 + ...
i




m1 x1 + m2 x2 + m3 x3 + ...
1

=
X cm
∑ ( m=
i ⋅ xi )
M i
m1 + m2 + m3 + ...

1 
=
X cm
x ⋅ dm
M∫
1
I ⋅ω 2
2
I = ∫ r 2 ⋅ dm =
K=
rot
I = I cm + M ⋅ d
∑m r
i i
2
(for point masses)
Variables
M = Total Mass
X cm = Center of Mass in any general direction
K rot = Rotational Kinetic Energy
ω = Angular Velocity
r = Radius
I cm = Inertia at Center of Mass
d = Distance between Axis of Rotation and Center of Mass
I = Inertia at distance d from Center of Mass
about parallel axis
2
Because the integral to find the inertia about a center of mass can be very difficult to solve, most classes do not require the
calculation. General equations for the inertia of different objects will be provided to you or can be found in your text.
Torque
Torque can be thought of as the rotational equivalent of force. So for an example when you push a door open you are applying a
force to the door, this force exerts a torque around the pivot point (in this case is the hinges) which will cause the door to open.
Positive torque provides counterclockwise rotation. Negative torque provides clockwise rotation.
Torque Equations


τ = r ⋅ F⊥
τ = r ⋅ F ⋅ sin ϕ
τ= r × F
τ net = I ⋅ α

F
Pivot Point
ϕ
Variables
τ = Torque
r = Distance from Pivot Point

F⊥ = Force perpendicular to r


ϕ = Angle between vectors r and F

F⊥

F → Causes No Rotation

r
(Extended to meet one another)
I = Moment of Inertia
α = Angular Acceleration
Useful Trigonometric Equations
opp
hyp
opp
tan θ =
adj
sin θ =
cos θ =
c
hypotenuse ( hyp )
adj
hyp
2
c=
a 2 + b2
b
opposite side ( opp )
θ
a
adjacent ( adj )
Constants
Coefficients
M
=
= 5.98 ×10 kg
Mass of Earth
e
24
=
Re Radius of Earth
= 6.37 ×106 m
M moon
= Mass of Moon
= 7.36 × 1022 kg
Rmoon
= Radius of Moon
= 1.74 ×106 m
R=
= 1.50 ×1011 m
Radius of Earth Orbit
EO
−9.81 m 2 =
−32 ft 2
g=
ag =
s
s
−11 Nm 2
=
G 6.67 ×10
kg 2
=
of sound in air 343 m
vsound Speed
s
=
m p Mass of a proton or neutron
= 1.67 ×10−27 kg
=
me Mass of an electron
= 9.11×10−31 kg
=
ε 0 Permittivity constant
= 8.85 ×10−12 C
2
Nm 2
2
=
K Coulomb's law constant 1 = 8.99 ×109 Nm 2
4πε 0
C
=
= 1.26 ×10−6 Tm
µ0 Permeability constant
A
=
e Fundamental unit of charge
= 1.60 ×10 C
=
c Speed of light in a vacuum
= 3.00 ×108 m
s
−19
Material
Rubber on concrete
Steel on steel (dry)
Steel on steel (lubricated)
Wood on wood
Wood on snow
Ice on ice
Metal
aluminum
copper
gold
iron
silver
tungsten
Static
µs
1.00
0.80
0.10
0.50
0.12
0.10
Resistivity(Ωm)
2.8 ×10−8
1.7 ×10−8
2.4 ×10−8
9.7 ×10−8
1.6 ×10−8
5.6 ×10−8
Conversions
Kinetic Rolling
µk
µr
0.80
0.60
0.05
0.20
0.06
0.03
0.02
0.002
Conductivity(1/Ωm)
3.5 ×107
6.0 ×107
4.1×107
1.0 ×107
6.2 ×107
1.8 ×107
=
1 mile 5280
=
feet 1609
=
meters 1.609 kilometers
1 inch = 2.54 centimeters
=
1 hour 60
=
minutes 3600 seconds
1 revolution
= 360=
° 2π radians
mi
=
=
1m
2.24
3.28 ft
s
hr
s
−19
1 eV
= 1.60 ×10 J
1=
u 1.66 ×10−27 kg
1 kg ≈ 2.2lbs on Earth