Download Forces - faculty at Chemeketa

The following is a tip sheet that should help highlight some useful
strategies and common pitfalls encountered by students studying
physics. It should not be used as a substitute for the textbook, your
notes, or anything else.
General physics problem solving tips
If you do not know when a formula is useful (correct and pertinent), then that formula is
useless. Becoming a good physicist does not come from memorizing a bunch of formulas
and practicing math. Strive to understand which formulas are universally valid (ex:
Newton’s laws) or definitions and which formulas only apply in special cases (ex: N =
mg). Also strive to understand when certain problem solving methods are more useful
than others.
Solve all problems symbolically before plugging in numbers. This will provide insight
into which variables affect your answer. It will also help avoid excessive rounding error.
Kinematics
The most abused formula in kinematics is the definition vaverage = Δx/Δt. The first formula
can be used correctly, though vaverage doesn’t appear anywhere in the other kinematics
formulas and is therefore not particularly useful. It is useful when acceleration is zero.
Here are some more generally applicable (any fixed acceleration) kinematics formulas:
Δx = vot + ½at2
vf = vo + at
Δx = displacement
vo = initial velocity
t = time
a = acceleration
vf = final velocity
The symbol “g” is used for the magnitude of the free fall acceleration due to gravity. It is
always positive since it is a magnitude, so don’t ever write, “g = -9.8 m/s2.” It varies from
place to place, but the typical textbook uses 9.81 m/s2. We use 9.80 m/s2 in labs in Salem,
Oregon. The acceleration of an object under the influence of gravity alone using a
conventional coordinate system will be -g.
A common misunderstanding is that positive acceleration is speeding up and negative
acceleration is slowing down. This is not necessarily true. You need to look at the
relationship between the velocity and acceleration vectors to determine this. If the
velocity and acceleration vectors are in the same direction, then the object is speeding up.
A dropped ball is an example of negative acceleration and increasing speed. If the
velocity and acceleration vectors are in the opposite direction, then the object is slowing
down. A car driving to the left and applying the brakes has a negative velocity and a
positive acceleration, yet is slowing down. If the velocity and acceleration are
consistently perpendicular, then there is no change in speed. This occurs in uniform
circular motion.
Forces
Generally, start your problem with a free body diagram for each object. It is best practice
to identify the type, direction, source, and receiver for each force. The source of the force
should be the agent that directly acts upon the object. For weight forces, this is almost
always the earth. All mathematical relationships should be consistent with your free body
diagram and a choice of Newton’s first or second law.
Newton’s first law states if ΣF = 0 then v will be constant and a = 0. It also works in
reverse: the observation of v being constant leads to the conclusion that ΣF = 0.
Newton’s second law gives a mathematical relationship between the net force on an
object, the mass of an object, and the acceleration of an object as follows: ΣF = ma.
Newton’s third law states: Fab = -Fba. The forces are of the same type with the source and
receiver reversed. Do not make the mistake of calling a normal force and a weight force
acting on the same object an action-reaction pair. Though they may be equal and opposite
forces, they are not the same type of force and they do not act on different objects.
The weight force on an object near the surface of the earth has a magnitude of m*g and a
direction of towards the center of the earth. See the notes on “g” in the kinematics
section.
Do not write N = mg or T = mg unless you have a good reason. A good reason is if those
are the only two forces in the y direction and there is no acceleration in that direction.
You typically use Newton’s first or second law to calculate N or T. There are no general
formulas for these forces. The direction of the normal force is always perpendicular to the
surface and repulsive. The direction of the tension force is always attractive.
Frictional forces always acts parallel to the contact surface and perpendicular to the
normal force. The static frictional force will take the magnitude (up to a maximum) and
direction necessary to hold the object in place relative to the surface. The kinetic
frictional force will always have a fixed magnitude and a direction opposite the relative
motion.
The buoyant force exerted by a fluid (liquid or gas) is equal to the weight of fluid
displaced. The most useful formulation of this statement is the following:
Fb = ρfVdg
Fb = magnitude of buoyant force
ρf = density of fluid (1.29 kg/m3 for typical air, 1000 kg/m3 for water)
Vd = volume of fluid displaced
g = magnitude of free fall acceleration
Circular Motion
If an object moves in a circle with constant speed, then it is characterized as having
uniform circular motion. The object will be accelerating towards the center of the circle
(centripetal acceleration) with the following magnitude: acp = v2/r. The so-called
centripetal force is the net force on the object which will be macp. The centripetal force is
not a new force to add to the list of forces such as N, W, etc.; it is a force or combination
of forces whose vector sum points towards the center of a circle.
An object that moves in a circle or rotates also has a characteristic called angular
velocity, ω, which can be calculated with the following formula:
ω = angular distance/time
Note that in this formula, the angular distance must be measured in radians. The angular
velocity can be related to linear quantities as follows:
v = Rω (R = radius)
acp = Rω2
Rotation
Rotation is caused by torque, τ. Torque can be calculated with the following formula:
τ = l*F
l = lever arm = perpendicular distance from the force vector to the axis of rotation
Rotation is inversely proportional to the moment of inertia of an object which typically
has the following formula:
I = cmR2
c = constant which quantifies whether or not the matter is close to the axis
If there is no net torque on a system, then the law of conservation of angular momentum
applies. The angular momentum of an extended body is as follows:
L = Iω
For an object whose size is significantly less than its distance to the axis of rotation, the
angular momentum is the following:
L = mvR
The law of conservation of angular momentum states that the total angular momentum of
a system before and after an event is the same.
Work and Energy
Energy methods are useful when a time calculation is not needed and there are variable
forces such as springs or normal forces from a curved path.
The definition of work is the following:
W = F*d*cos(theta)
W = work
F = magnitude of force
d = distance the object moves
theta = angle between force and direction of motion
This formula can be used with an individual force or the net force to calculate the
individual work or total work. Calculating the total work can be useful because of the
work energy theorem.
Wtotal = ΔKE = KEfinal – KEinitial
The definition of kinetic energy is the following:
KE = ½mv2
The definition of kinetic energy can be combined with the work-energy theorem:
Wtotal = ½mvfinal2 – ½mvinitial2
An alternative to calculating work is to use the idea of potential energy. When there are
no significant forces such as friction or drag which degrade the energy, the principle of
conservation of energy can be used as follows:
PEinitial + KEinitial = PEfinal + KEfinal
PE = gravitational potential energy = mgh
Momentum
The definition of momentum is p = mv. This is a useful quantity to analyze when objects
interact with one another with forces that cannot easily be calculated, such as in a
collision.
Total momentum for an object or system of objects will be conserved (no change) if the
net external force is zero or insignificant compared to the internal forces. The relationship
can be written mathematically for two objects:
Σpi = Σpf
m1v1i + m2v2i = m1v1f + m2v2f
Since momentum and velocity are vectors, be sure to indicate direction by + or – signs.
There are certain types of collisions which are special. A collision (not an object) can be
characterized as completely inelastic, elastic, or neither. If the collision is completely
inelastic, then the velocities of all objects will be the same after the collision (the objects
stick together). If the collision is completely elastic, then the total kinetic energy of the
system does not change: ΣKEi = ΣKEf. Do not assume that a collision must be
completely inelastic or elastic.
Impulse has the following definition:
J = F*t
F = force
t = time
Newton’s second law combined with the definitions of impulse and momentum yields the
impulse-momentum theorem:
J = Δp
Fluids (Liquids and Gases)
See above for information on the buoyant force in a static fluid. The definition of density
is the following:
ρ = m/V (1.29 kg/m3 for typical air, 1000 kg/m3 for water)
m = mass
V = volume
The definition of pressure is the following:
P = F/A
F = force
A = area
The pressure in a static fluid can be calculated with the following formula:
P = Patm + ρgd
Patm = atmospheric pressure = 101,000 Pa
ρ = density of fluid
g = magnitude of free fall acceleration
d = depth (almost always positive)
For dynamic fluids in a streamline (such as in a pipe), don’t worry about quantification.
The following general principles hold:
Regions with smaller cross sectional area have greater speed.
Regions with greater speed have lower pressure.
For static gases that are not at low temperature or high pressure, the ideal gas law is a
useful approximation:
PV = nRT
V = volume
n = number of moles
R = ideal gas constant = 8.31 J/(mole*K)
T = temperature in kelvin
Temperature and Heat
Temperature (T) is a measure of the average translational kinetic energy of molecules in a
system. Heat (Q) is the spontaneous transfer of energy from a higher temperature system
to a lower temperature system. If only heat (no work) is transferred to a system and that
energy is not used to change phase, then the following formula holds:
Q = mcΔT
m = mass of object
c = specific heat capacity of object = 4190 J/(kg*K) for water
ΔT = change in temperature
If only heat (no work) is transferred to a system and that energy is used to change phase
(this occurs at specific temperatures and pressures for a given substance), then the
following formula holds:
Q = ±mL
L = heat of fusion or vaporization = 3.34E5 or 2.26E6 J/kg for water
The plus or minus in the above formula depends on the direction of the phase change. It
is positive for melting and vaporizing. It is negative for freezing and condensing.
Laws of Thermodynamics
The first law of thermodynamics is a statement of conservation of energy. If a system
gains or loses internal energy, then that gain or loss is from a process and the energy must
come from or go somewhere. There are two general ways to gain or lose energy. Work is
a macroscopic transfer of energy while heat is a microscopic transfer of energy. The
following is the first law of thermodynamics stated mathematically:
ΔE = Q + W
ΔE = change in energy of a system
Q = heat transferred to the system (negative if transferred out)
W = work done on the system (negative if work done by the system)
Historically, the field of thermodynamics has been used to study heat engines, which
have net work output and heat input. The ratio of these two quantities is the definition
efficiency:
e = Wtotal/Qinput
The second law of thermodynamics provides an upper limit to the efficiency of an engine
operating between two temperature extremes:
emax = 1 – Th/Tc
Th = highest temperature
Tc = lowest temperature
The second law has other equivalent statements, one of which is that the total entropy (a
rough measure of disorder) of an isolated system cannot decrease.
Misuses of the Laws of Thermodynamics
There are commonly held views among the general public that the laws of
thermodynamics falsify both cosmic and biological evolution. These views are not
supported by data.
One view unsupported by data is that the big bang couldn’t possibly be correct because it
violates the first law of thermodynamics. The claim is that there is obviously a lot of
energy now and there couldn’t be any energy before the big bang. The total energy
apparently increased thus violating the first law of thermodynamics.
For this alleged violation to be true, we must know both the total energy of the universe
both before and after the big bang and show that they are different (putting aside
objections that there is no such thing as “before the big bang”). No serious scientist
claims to know with certainty the total energy of the universe “before the big bang.” But
even if one assumes that the energy is zero “before the big bang”, a calculation of the
total energy of the universe based on classical physics also yields a total energy of zero!
This was shown in a paper written by E. Tryon in a 1973 article in the journal Nature
(and not refuted to date). How can that be so with all the stuff moving around (kinetic
energy), light energy, etc.? It turns out that the negative gravitational potential energy
balances out the positive energy and the net sum is zero. A relativistic calculation of the
current total energy of the universe is problematic, but it in no way yields a calculation
that refutes the big bang through the first law. Even if we assume we can know the total
energy of the universe both before the big bang and at the present time, there is no proven
violation of the first law of thermodynamics.
Another view unsupported by data is that the second law of thermodynamics prohibits the
evolution of chemicals to primitive organisms or of primitive organisms to more complex
organisms, an apparent decrease in entropy.
There are two important objections to the "life violates the second law" argument. First,
there is no calculation that shows that complex life forms are lower entropy than less
complex forms or non-living things. Hand waving metaphors are no substitute for a
proper calculation. Second, life forms are not closed systems. They continuously
exchange energy and particles with their environments, so the second law has nothing to
say about them unless you include their environments in the calculation. Evolution has
not been proven to violate the second law of thermodynamics.
List of things that are never true:
g = -9.8 m/s2
Formulas to use only if you are absolutely sure they are true and
useful:
vave = Δx/Δt (True, but not always useful)
N = mg (True only in a special case)
T = mg (True only in a special case)
Work = F*d (True only in a special case of force and motion parallel)
Q = mcΔT (True only if heat is the only energy transfer and no phase changes occur)