- Good morning.
While I am not a fan of memorization
and I do my best to avoid having my students
memorized just about anything,
there are a few equations which
are not on the equation sheet provided
for the AP Physics C: Mechanics Exam,
which I do suggest you memorize.
♫ Flipping Physics ♫
- This is a fourth uniformly accelerated motion equation
which can be helpful to know.
Call me crazy,
- [Bo] Crazy.
- but the equation for force of gravity
or weight is not on the equation sheet.
When an object is on an incline,
we will often, though not always,
break the force of gravity into its components
in the parallel and perpendicular directions.
The theta here is the angle of the incline
and it's easy to remember which of these two equations
is which because the force of gravity parallel
on a flat surface is equal to zero,
and the sign of zero degrees is zero.
The equation, the change in energy
of the system equals the total energy
transferred into or out of the system,
is where most of our equations,
which relate to work and energy, come from.
Net work equals change in kinetic energy.
Bobby, when is this equation valid?
Net work equals change in kinetic energy is always true.
Correct.
Bo, when can we use the work done
by friction equals change in mechanical energy equation?
That equation is true when there
is no work added to or removed from the system by a force.
Also correct.
Conservation of mechanical energy.
Billy, when is this equation valid?
Conservation of mechanical energy requires
that there be no work added to or removed
from the system by a force, and it also requires
no work be done by a non-conservative force,
which is basically friction.
Very nice.
When a force is conservative, that force
is equal to the negative of the derivative,
with respect to position, of the potential energy
associated with that force.
This equation is almost guaranteed
to come up on the exam.
Every derivative is an integral
and every integral is a derivative.
For example, using the equation
for conservative force, we can multiply both
sides by the dx and then take the definite integral
of the whole equation.
Which means on the left-hand side,
we have the definite integral of force
with respect to position, which class
is the equation for what?
Impulse.
No, that is the equation for work.
Impulse is the integral of force
with respect to time, not position.
Yeah, you are right and thank you.
You're welcome.
Which means the work done by a force
is equal to the negative of the change
in potential energy associated with that force.
For example, as I raise this book,
the work done by the force of gravity
is equal to the force of gravity times
the displacement of the book, times
the cosine of the angle between those two vectors.
The force of gravity equals mass times
the acceleration due to gravity.
The displacement is equal to the change
in height of the book.
And the angle between the force
of gravity and the displacement of the book
is 180 degrees, the cosine of which is negative one.
In other words, the work done by the force
of gravity as I raise this book equals
the negative of the change in gravitational
potential energy of the book.
Conservation of momentum.
Now, it might seem like a pretty obvious equation,
but you need to remember how to derive
it and therefore when you can use it,
which is something we've already done,
so click on the link if you want to review it.
The same is true for conservation of angular momentum.
Again, you need to remember how
to derive it so you can remember when you can use it.
And we've already done that, so click on this link
if you want to review.
The center of mass of a rigid object with shape
is equal to one over the total mass times
the integral of r with respect to mass,
where r is defined as the distance each
infinitesimally small piece, dm,
of the object is from the zero reference line.
To go along with that, we have two
equations for mass density.
Volumetric mass density equals
the total mass divided by the total volume.
And linear mass density equals the total mass
divided by the total length.
I did not include surface mass density here,
even though I did during the review,
because I have yet to see that
on the AP Physics C: Mechanics Exam.
The equations for arc length and tangential acceleration
are not on the equation sheet,
even though the equation for tangential velocity
is on the equation sheet.
Actually, the equation for arc length,
I found it on the equation sheet,
it's just not where, quite where you would expect it to be.
It's in the Geometry and Trigonometry section.
But that says S equals r theta.
Yeah, well where did the delta go?
Okay, S equals r theta assumes
the initial angular position is zero
instead of using the change in angular position.
Thank you Bobby for pointing that out,
I had actually never noticed that equation there before.
Although it is missing the delta,
which I do consider very important.
Remember, when you take the integral
of the tangential velocity equation,
you get the arc length equation.
And when you take the derivative
of the tangential velocity equation,
you get the tangential acceleration equation.
Once you have the previous three equations,
it is easy to remember the equations
for the velocity and acceleration of the center
of mass of an object, which is rolling without slipping.
And, remember the R in these equations
is capitalized because it is the radius of the object.
You probably remember that one revolution
equals 360 degrees, which equals two pi radians.
But it is worth noting that you do have to remember that.
There are two uniformly
angularly-accelerated motion equations,
- [Group] U pushing M.
- which are not on the equation sheet.
The parallel axis theorem is the way you determine
the moment of inertia of an object
with uniform density about a location
which is not its center of mass.
When you already know the moment
of inertia of that object about its center of mass.
The equation which describes the conditions
for an object to be in simple harmonic motion
is, the second derivative of the position
of that object with respect to time
or the acceleration of that object
is equal to the negative of the square
of the angular frequency of that object times
the position of that object
relative the equilibrium position.
These are two equations which I recommend
you memorize, which come directly
from the acceleration and velocity equations
for simple harmonic motion which are not
on the equation sheet.
They are the maximum velocity is equal
to the amplitude times the angular frequency,
and the maximum acceleration is equal
to the amplitude times the angular frequency squared.
It's just helpful to have those memorized.
I would also like to talk about some things
which you should not memorize.
These are the two equations for the resistive force,
or the force of drag acting
on an object as it moves through a fluid.
If the AP Exam writers want you to use
one of these equations, they will either give
it to you, or they'll tell you
the force is proportional to the velocity,
or the force is proportional to the square
of the velocity, or something like that.
But please, do not memorize these equations.
The relationship between watts and horsepower.
Again, if they want you to use it,
they will give it to you.
The value for the universal gravitational constant,
do not memorize it.
In fact, do not memorize any of the constants.
Instead, you should become familiar
with the table of information,
which you will get on the AP Exam,
which has all sorts of really useful information on it,
like unit symbols, prefixes, and conversion factors.
You should also familiarize yourself
with the general math formulas,
which will be on another page you will get
on the AP Exam.
There are common geometry, trig, and calculus equations.
Now, sure you may have many of these memorized,
however, if you do freak out on the exam,
find comfort in the knowledge that these equations
are there for you.
Now let's talk about equations which I suggest
you know how to derive.
Again, do not memorize them, instead,
know how to derive them.
And these are all derivations we did during
our AP Physics C: Mechanics review videos.
The velocity and acceleration of the center
of mass of a system of particles
to get these two equations, take
the derivative with respect to time,
once and then once again of the equation
for the position of the center of mass
of a system of particles.
The terminal velocity of an object
falling near the surface of a planet.
Notice how specific this is, this is the equation
for the terminal velocity of an object
falling near the surface of a planet.
It is not true in other situations,
it's only true in that specific situation.
But if you know how to derive this terminal velocity,
you will know how to derive the terminal velocity
in other situations, which is most likely
what they're going to do on the AP Exam.
- It's okay Mr. P, we believe you.
- Absolutely we believe you.
The binding energy of an object to a planet.
Actually, it's the binding energy
of any two objects to one another.
The escape velocity of an object from a planet.
The total mechanical energy of an object
in orbit around a planet.
Kepler's third law, or the relationship
between the orbital period of an object
and the radius of the circular orbit of the object.
The velocity and the acceleration as a function of time
of an object in simple harmonic motion.
The moment of inertia of a uniform thin hoop
or a cylinder around its long cylindrical axis.
The moment of inertia of a uniform thin rod
about its center of mass.
The moment of inertia of a uniform solid disc
or a cylinder about its long cylindrical axis?
We did not do that one in the review.
You're right, I'm sorry about that.
That's okay, we did it in class.
A quick note about moments of inertia
and the AP Exam, if you need a specific equation
for a moment of inertia of an object
to solve a problem, it will be provided for you.
Now you do need to be able to look
at objects of varying shapes and be able
to compare the relative moments of inertia,
but you should be able to do this based
on the basic equation for moment of inertia.
For example, if we have a hollow sphere
and a solid sphere of the same radius
and the same mass, about their centers of mass,
the hollow sphere will have a larger moment
of inertia than the solid sphere,
because the hollow sphere has more
of its mass concentrated farther
from the axis of rotation.
In other words, the average r value
for a hollow sphere is greater than
the average r value for a solid sphere
of the same mass and the same radius.
Therefore, the moment of inertia
about its center of mass for the same mass
and radius of a hollow sphere will be greater
than the moment of inertia of a solid sphere.
Which brings us to the end of my review
of the entire AP Physics C: Mechanics curriculum,
which means I need to ask for your help.
- You need our help?
- You need my help?
- [mr.p] No, I need your help.
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Thank you very much for learning with me today.
I enjoyed learning with you, and good luck on the AP Exam.