Problem Solving
Note: this is still an early-stage draft of this document.
©2006, Steve Carabello
Overview of a step-by-step procedure:
1. Get a clear picture (mental or sketch) of the problem. Usually, this is a physical process, so
imagine watching something happen over time, or relate it to things you observed in class or in
real life. In some cases (force problems, torque problems), it’s required to draw a new sketch of
your own, since it’s nearly impossible to solve correctly without such a sketch. Physics is not just
a special kind of algebra.
2. If possible, try to get a ballpark estimate of what your final answer should be. You may get good
at this after doing many physics problems, or by having a good feel for the way that sort of
physical system works.
3. Translate the information in the problem into variables, with the appropriate letters and units.
Make sure that you list prominently what you’re solving for. Note: some of the key information
might not be given as numbers, but instead as words in the text of the problem. Also: be careful
about minus signs, and vectors vs. scalars.
4. Try to decide which equations are true and useful for this problem. The best way to get good at
this is by doing a lot of problems, though a solid understanding of the meaning of each equation is
very useful no matter how many or how few problems you’ve done.
5. Choose an equation to try, and make sure you understand exactly what each of the variables in that
equation means. Without that sort of clear understanding, you’ll make mistakes more often than
not.
6. Look at what you know and what you’re trying to find, to make sure that the equation you chose is
indeed both true and useful for the problem you’re trying to solve. You may need to use multiple
equations, so it would be helpful to map out your strategy for getting your final answer.
7. Plug in your numbers, verifying again that the variable in the equation really does have the same
physical meaning as the number you’re using. This includes vector vs. scalar distinctions, being
careful with minus signs, being careful with units, etc.
8. Run through the algebra, using units in all steps.
9. Get an answer.
10. Look at your answer: are the units correct? Does the size of your answer make sense (especially
given your guess in step 2)? If not, check your work. It is actually more common to make an error
in the concept stage (steps 1-6) than in the algebra (8-9).
Note that only steps 7, 8, and 9 involve calculation. All of the other steps are important, and are usually
worth credit where partial credit is available. Those other steps are not a waste of time, they are necessary
steps toward understanding processes and getting correct answers. Physics is a way of understanding
physical systems, not just a process of plug-and-chug through equations.
The main equations:
Equation
The 4 kinematic equations.
e.g. Δx = vixΔt + ½ ax(Δt)2
Δx = ½(vix + vfx) Δt
vfx = vix + axΔt
vfx2 = vix2 + 2 axΔx
Is only true when...
1. you are dealing with a purely linear
problem, or you are dealing with
one component of 2- or 3dimensional motion, AND
2. the acceleration in the direction you
are considering is constant.
Δy = viyΔt + ½ ay(Δt)2
etc.
The 4 rotational kinematic
equations.
e.g. Δθ = ωiΔt + ½ α(Δt)2
the angular acceleration of your system
is constant.
Newton’s Laws
I. If ΣF = 0 then a = 0
II. ΣF = ma or Fnet = dp/dt
III. FAB = – FBA
Newton’s Laws for Torque
Στ = Iα or τnet = dL/dt
τAB = – τBA
are always true, as long as you are
completely clear about what object(s)
the forces are acting on.
Work and Energy:
Wnc + Ei = Ef
1. are always true, as long as you are
clear about what object(s) the
torques are acting on, AND
2. α must use radians as the angular
unit
is always true.
May be useful when it is true, and...
1. You have some “initial”
situation and some “final”
situation that you care about.
2. You have enough information to
get any 3 of the 5 variables (e.g.
Δx, vix, vfx, ax, Δt) for either
direction.
3. The net force is constant
(therefore acceleration is
constant too).
The net torque is constant (therefore
angular acceleration is constant too).
Other
You must split this up into
components.
1. You have some way of knowing
what Wnc is (it may be zero, but
you need to be sure).
2. You have some “initial”
situation and some “final”
situation that you care about.
3. You have changes in speed and
changes in height.
4. You have a mass and a spring.
5. You have something rolling
without slipping.
1. Wnc is zero if you have
an elastic collision (and
for no other
collisions).
2. Wnc is zero for cases of
rolling without slipping
(as long as there are no
applied forces)
You may use whatever
angular units you want, as
long as you stay consistent
with them.
You must draw free body
diagrams.
You must split this up into
components.
You must draw free body
diagrams.
Conservation of Momentum
Ptot_i = Ptot_f
Conservation of Angular
Momentum
Ltot_i = Ltot_f
Supporting equations
Equation
Centripetal Acceleration
ac = ar = v2/r = rω2
Tangential Acceleration
at = d|v|/dt
Rotation and Translation
Δs = rΔθ
v = rω
at = rα
Uniform Circular Motion
v = (2πr)/T
1. ...the net force on your system of
objects is zero OR
2. ...the net force times the time
between “initial” and “final” is
sufficiently small. This will be true
any time something is called a
“collision” and you have linear types
of motion (not spinning) OR
3. ...the net force on your system along
a certain direction is zero, and you
only apply this equation for the
component of momentum along that
direction.
1. ...the net torque on your system of
objects is zero OR
2. ...the net torque times the time
between “initial” and “final” is
sufficiently small. This will be true
any time something is called a
“collision” and you have spinning
happening.
1. You have different objects
pushing/pulling/hitting on a
frictionless surface.
2. You see the word “collision.”
Is only true when...
1. ... you have circular motion, so that
you can define a radius AND
2. ... ω is in radians
you have circular motion
Is often used when/with...
1. you have a circular orbit
2. Newton’s Laws
3. rotational kinematics
something going around in a circle is
changing speed
1. rotational kinematics
2. energy conservation
3. angular momentum
Other
“Centripetal acceleration”
and “radial acceleration”
mean the same thing.
If the speed |v| isn’t
changing, then at is zero.
The angular unit MUST be
radians.
1. you have a circular orbit
2. the words “uniform circular
motion” are used
In uniform circular motion,
α and at are zero, ac (or ar)
is constant but nonzero.
1. you have a thread unwinding
without slipping from a pulley OR
2. you have an object rolling without
slipping on a surface
you have uniform circular motion (that
is, motion around a circle at constant
speed)
You must split this up into
components.
1. You have different objects
pushing/pulling/hitting with a
frictionless axle.
2. You see the word “collision” and
you clearly have some rotation at
some time.
1. you have any 2 surfaces in contact
2. N is the normal force between the
two surfaces being considered.
3. fs for static friction, only when no
slipping going on.
4. fk for kinetic friction, only when
there is slipping.
1. you have a Hooke’s Law spring
2. |x| is the magnitude of the distance
the spring has been stretched from
its equilibrium position.
1. Newton’s Laws
2. Work and energy
Be careful about direction
of the force: kinetic friction
opposes the actual slipping,
static friction opposes the
tendency to slip.
1. Newton’s Laws
2. Spring potential energy
Newton’s Law of Gravitation:

Fgrav  Gmr 1m2 2
1. you have two point masses, or two
spherical masses.
2. r12 is the distance between the
centers of mass of the 2 objects (not
the radius of either object).
1. Newton’s Laws
2. Centripetal Acceleration
Torque:
a. τ = r×F
b. τ = rF sinθ = Fd = Fl
a) r is the vector starting from your
1. Newton’s Laws for Torque
chosen pivot point, to the location where 2. Rotational work and energy
the force F starts.
b) d or l mean the same thing: the
shortest distance between the pivot
point, and the line along which the force
acts.
Moment of Inertia:
a) only for one or more point masses
1. k is called the spring
constant. It has units
N/m.
2. You must draw a free
body diagram.
3. The direction of the
force from the spring
onto the object pulling
it is opposite to the
direction the spring was
stretched from its
equilibrium postion.
1. Gravity is always an
attractive force.
2. If you have more than 2
masses, you must
calculate the strength of
each gravitational force
separately, then do a
vector sum.
1. θ is the angle between
the vectors r and F.
Make a side sketch to
be sure you have the
correct angle.
2. Be sure you remember
how to apply the right
hand rule.
1. The moment of inertia
Friction:
fs ≤ μsN
fk = μkN
Spring Forces:
|Fspring| = k |x|
12
1. Torque
a) I = Σmiri2
b) ICM = KMR2
c) I = ICM + Md2
where ri is the radius of the circle that
2. Rotational Kinetic Energy
that mass makes. Any object with a size 3. Angular Momentum
much smaller than the radius of the
circle it sweeps out may be considered a
point mass.
b) only for a shape about the center of
mass. The equation at left is a
shorthand for the moments of inertia
given in the figure in the text. K is
some constant, M is the total mass of the
shape, and R is the radius of the shape.
e.g. for a solid cylinder, I = ½MR2.
c) only for an axis of rotation other than
the center of mass, where you know the
moment of inertia of that shape for an
axis through the center of mass and
parallel to your actual axis. (“Parallel
axis theorem”). d is the distance from
the actual axis of rotation to the center
of mass of the shape.
of multiple objects is
just the sum of the
moments of inertia of
each alone.
2. Be careful about the
difference between the
radius of an object, and
the radius of the circle
an object sweeps out.
May be useful when...
Other
Wave Speed:
v  f  k
Angular Frequency (SHM):
mass spring  k m
simple _ pendulum  g L
 physical _ pendulum  mgd I
Definitions equations
Equation
vavg = Δx/Δt v=dr/dt
aavg = Δv/Δt a=dv/dt
ωavg = Δθ/Δt ω = dθ/dt
Is often used with...
αavg = Δω/Δt α = dω/dt



vP A  vP B  vB A
mi xi
mi y i
X CM  iM , YCM  iM
p = mv
L = r×p L = Iω
Ktr = ½mv2
Krot = ½Iω2
Ug = mgh
Gm m
U grav   r112 2
Uspr = ½kx2
sf
 
W   F  ds  Fs cos
si
W = τ Δθ
 
P  Wt  F  v  Fv cos
P = τω
f  T1
ω = (2π rad)f = (2π rad)/T
k = (2π rad)/λ
Other notes:
Know the trig. functions, for splitting up vectors into their components. Draw triangles if necessary.
Know how to take dot products and cross products, when given magnitudes and directions, and when given vector components or unit vector
notation.
Know how to apply the right hand rule, to get the direction of the result of a cross product.