Download FREQUENTLY ASKED QUESTIONS Content Questions

FREQUENTLY ASKED QUESTIONS
March 31, 2015
Content Questions
What is the right answer to problem 2 on the midterm?
The best answer is at the end of the solutions posted on Sakai (which I
didn’t have in class today...) Assuming the spool is a disk, the moment of
inertia to use is I = M (3R)2 . In the formula given, the R value refers to a
generic radius, not the specific R given in the problem. However, as given,
the problem was ambiguous— the spool could in principle have some shape
giving it an actual I of M R2 . I’m told that answers using either assumption
were counted as correct.
In problem 2c on the midterm, I got points off for writing fs =
µs M g, although that was in terms of quantities given. Why?
The reason is that fs = µs M g is an expression for maximum static frictional
force. It doesn’t apply in general— in general, an inequality fs ≤ µs M g
holds. The equality can only be used in the last part of the problem where
maximum force is specified.
Can you explain the vector nature of gravitational force? When is
it negative and when is it positive?
Gravitational force is always attractive, so the force on a given mass by
another mass is always in the direction pointing directly towards the other
mass. Whether this direction is positive or negative depends on how you
have defined your coordinate axes (note that it might not even be along
an axis and the gravitational force might have both positive and negative
components).
The notation for this is: F~12 = Gmr21 m2 r̂,
12
where F~12 is the force on 1 due to 2, and r̂ is the unit vector in the
direction going from 1 to 2.
You can always keep track of the gravitational force direction by remembering that it’s always attractive.
What’s the difference between G and g?
The quantity “little-g” is the acceleration of gravity near the surface of the
Earth. It’s a constant, or very nearly a constant so long as you are near
the surface of the Earth. “Big-G” is a much more fundamental quantity.
It’s the constant that appears in Newton’s Universal Law of Gravitation,
|F~ | = Gmr21 m2 . This univeral law is completely general, applying to any two
12
massive bodies. You can derive the quantity g from the Universal Law by
considering m1 and m2 to be the masses of the Earth and an object near the
surface of the Earth, and plugging in the radius of the Earth (since r12 in the
Universal Law is the distance between the object and the center of the Earth).
Are there two definitions of potential energy? How do I know
whether to use mgh or −GM m/r?
Yes, we have two forms of gravitational potential energy which are used in
different contexts. The gravitational potential energy U = mgh can only be
used when the acceleration of gravity can be treated as a constant. This is
the case near the surface of the Earth, where the force of gravity changes
negligibly over typical distances. You use mgh for e.g., an object lifted a few
meters, or even kilometers, above the ground.
The gravitational potential energy U = −GM m/r is the more general
formula. This is the one to use when you are dealing with distances such
that the force of gravity varies appreciably over a typical change in separation. So you would use this for, e.g., a planet orbiting the Sun.
Why do we set U = 0 when r = ∞?
This might seem a little strange at first, but it turns out to be convenient.
From integrating the gravitational force to get the potential energy, we get
∆U = −Gm1 m2 ( x1f − x10 ). If you just take U = − Gmx1 m2 , then zero is at infinity and you can choose to keep it that way. You can always add a constant
to U , but why bother? If you keep this form of U , with U = 0 at infinity,
then we get some simplifications: for instance if U = 0 and K = 0 both at
infinity, then |U | = |K| at every point (if WN C = 0).
How does potential energy increase when you pull two objects
apart? It should lessen.
No– force lessens but potential energy should increase as the separation between two objects increases. If you move an object away from the surface of
the Earth, potential energy increases.
If you have two objects separated, and at rest, and you release them, they
will accelerate towards each other. The gain in kinetic energy must match
the loss in potential energy. So the objects must have more potential energy
when they are separated.
Another way of thinking: you have to do work to separate objects, and
that work becomes potential energy.
Can you explain escape velocity? Why is Kf = 0?
Escape velocity (really speed) is the speed you need to impart to an object
at the surface of the Earth in order for the object to “just escape”: in other
words, the object won’t come back down after being thrown up. What happens when you launch something away from the Earth surface with some
initial velocity? It slows down as kinetic energy gets converted to gravitational potential energy. The more initial velocity you give the object, the
higher it will get before coming to a stop and then falling back down. What
if you can give it enough initial velocity that it actually gets to infinity by
the time it comes to a stop (infinity meaning “really far away”)? Then it
never comes back down– it’s escaped! So if, at an infinite separation, the
final velocity is zero, then you’ve achieved escape velocity.
Joke of the Week
Q: What’s yellow and imaginary? A: The square root of a negative banana.