Study Tips for Exam 2
The purpose of this is to give you a little guidance on what I want and expect you
to know for the second exam. Again I want to emphasize that this course is not about just
putting numbers into equations. We only have a few equations at this point – what you
should be developing in this course is problem solving skills related to solving mechanics
problems using those few equations.
On the next exam, I will give you the following equations:
∆x = 1 a x t 2 + v ix t
v 2fx − v xi2 = 2a x ∆x
2
2
r
r
vt
F
=
m
a
=
p=mv
a
∑
centripetal
r
Knowing what to do with those equations (and the few that I’m not giving you,
such as the definitions of velocity and acceleration) is not a straightforward task. If you
have not understood everything on the activities, don’t assume that you can just cram the
night before the exam and do well – it won’t work. You need practice with solving
problems to gradually build your skill at doing that – it’s not something you can cram in
the night before an exam.
To be well prepared for the exam, here is what I expect you to know:
1. Everything from Exam 1. Go over the solutions to the first exam. Kinematics will not
be the primary focus of this exam, but it can show up.
2. Go through both practice exams (one under Files in Canvas, the other is a homework
assignment). When going over solutions - this doesn’t mean just look at the solutions.
Solve the problems on paper without looking at the solution if you can (if you get stuck,
check the solution for help getting started). Then check your answer. If you made a
mistake, go through the solution until you understand what you did wrong, and then solve
it again until you are comfortable that you could solve each problem without needing the
solution to help.
From my own experience as a student, I eventually (in grad school) realized that
just reading through a solution did not really prepare me well for exams, since I wasn’t
actually solving the problem myself. Instead, actually get out a pencil and paper and
solve the problems, using the solutions as a check until you are comfortable that you
could solve that problem without any help. This forces you to think through the entire
process yourself, and is therefore much more effective. You likely won’t see the same
exact problems on the exam, so instead make sure you fully understand the process and
why each step is valid.
3. Be aware that Newton’s 2nd Law may contradict what your intuition tells you. Review
activities 14, 15 and 16. These activities highlight the fact that our intuition is often
wrong when it comes to how forces affect motion. We tend to think force is proportional
to velocity, not acceleration. An important lesson from these activities is not just that
force is in fact proportional to acceleration, but also that you should not rely on your
intuition to try to analyze situations, since our intuitions often mislead us here.
Since our intuition can mislead you - ALWAYS draw free body diagrams (even if
a problem is not asking you to calculate anything, since it will help you think about what
is going on), and when identifying forces, keep in mind that (for now) every force other
than gravity is a contact force (i.e. I can’t exert a force on a ball after it has left my
hand). So look at where an object is being touched by other things, and that is where
there should be forces.
4. Understand Newton’s 3rd Law. There are no exceptions to Newton’s 3rd law – period.
If A pushes on B with force FAB, then B pushes back on A with an equal and opposite
force FBA. Regardless of what the accelerations, velocities, or masses of each object are,
FAB must equal FBA. Go through the solutions of activities 19 and 20, making sure you are
comfortable with them in their entirety.
6. (there is no number 5) Be able to analyze dynamic situations – i.e. draw FBDs of all
objects of interest, explicitly identifying (and giving unique names to) all of the forces,
keeping track of 3rd law pairs between objects of interest, and applying Newton’s 2nd Law
(in both directions). Don’t skip steps, it will result in mistakes. Explicitly draw your
coordinates (labeling what is your +x and +y direction) for each FBD, since you can
choose different ones for each object (this is primarily helpful when dealing with coupled
systems). For example, see Activities 17, 19, 22, 25, and both 26s.
7. Understand friction. Kinetic friction is always equal to fk=μkN, where N is the normal
force between the two surfaces that are sliding relative to each other. Kinetic friction
always acts to resist the motion. DO NOT assume the frictional force is fk=μkmg, where
m is the mass of the object on top. The normal force is very often not just equal to the
weight of the upper object. Review activity 16 until you are comfortable with this idea,
and with solving dynamic problems on inclined planes.
The force of static friction is only as big as it has to be in order to keep one object
from sliding relative to another. It can not get any bigger though than fs-max=μsN. The
force of static friction always points in the direction it must point to keep one object from
sliding relative to another.
8. Be comfortable with coupled systems. Your approach to analyzing these should be the
same with any other dynamic system – draw FBDs, define coordinate systems, write out
Newton’s 2nd Law, etc.. The thing to keep in mind here is that if you have two objects
coupled together in some way (i.e. two objects connected by a rope), what does that mean
about their accelerations? If their accelerations must be the same, you need to be careful
in how you define your coordinate systems in each FBD, so that your acceleration can be
the same for both object (i.e. you don’t want it to be positive for one object and negative
for another). When dealing with ropes, keep in mind that for now the tension is the same
on both sides of a pulley – a rope will pull on each object with the same tension force
(call it T). When you have multiple objects connected by a rope, the Newton’s 2nd Law
equations for those objects will be coupled mathematically because the same tension will
be in both equations, and they will have the same acceleration. Review Activity 22 and
similar problems on the homework and practice exams until you are completely
comfortable with this.
9. Be comfortable with analyzing objects moving along a circular path. If something is
moving along a circular path, it must be accelerating inward towards the center of the
circle with a particular acceleration to keep it on that circular path. The particular
acceleration that an object must have to move in a circle is called the “centripetal
acceleration” (centripetal means “center seeking”). If it does not have that inward
acceleration, then the object’s path will not follow the circle. Exactly how big the
centripetal acceleration must be (to keep it on a circular path) depends on how fast the
object is moving along the circular path (vt), and how tight the circle is (r). As an
2
vt
equation, the centripetal acceleration equals a c =
, directed towards the center of
r
the circle.
When dealing with objects moving along a circular path, draw a FBD and define
your axes such that one axis points radially inward towards the center of the circle, since
2
v
you know the acceleration in that direction is a c = t
. The object may or may not be
r
accelerating tangent to the path (the tangential speed may be constant or changing). What
is certain though is that for it to remain on that circular path, the radial acceleration must
equal that special value that we call the centripetal acceleration, ar=ac. So, when you
apply Newton’s 2nd Law in the radial direction, you know that the radial acceleration
2
v
. This means that the net force acting on the object must result in that
must be a c = t
r
acceleration. DO NOT add extra forces like “centrifugal force” or “centripetal force” to
your FBD – those are not additional forces acting on something. The NET force just must
be sufficient to give the centripetal acceleration. Go over the circular motion activity
(25), and both Activity 26s until you are comfortable with analyzing problems involving
centripetal acceleration.
So, this illustrates that there is a lot to know – a lot more than just the four
equations at the start of this. This is also why you can not expect to wait until the night
before the exam to cram all of this in, and do well. Start preparing well before the exam,
so you don’t have to cram the night before, and get a good night’s sleep so you are well
rested..
General Tips:
1. Work symbolically as much as possible (it’s a good habit to get into).
2. I prefer this nomenclature:
T = Tension
N (or n) = Normal force
f = force of friction, subscript s for static and k for kinetic
We often need subscripts for distinguishing forces (i.e. if you have multiple
objects, they may all have normal forces on them, so you can’t just call them all N.
Ideally, you should use notation like NAB to indicate the normal force that A exerts on B,
which should be in the FBD for B, so then NBA should show up in the FBD for A).
Because we need subscripts for those also, using things like FN for normal force,
FT for tension, and Ff for friction can start getting messy because of all the subscripts.
3. Remember that in the FBD for some object, only forces acting on THAT object should
show up in its FBD, and should be included in the net force acting on that object. If block
B is pushing A with a force NBA, then that force shows up in the FBD for A, and when
you apply Newton’s 2nd Law to A. The 3rd law pair force, NAB, should only show up in
the FBD for block B (since that force is acting on B), it should not show up in the FBD of
block A. But, Newton’s 3rd law says those two forces must be equal and opposite – so
they should point opposite directions, and in magnitude NAB=NBA
4. When you start putting numbers in, keep in mind that g=9.8 m/s2, NOT -9.8 m/s2. If
you put in g as negative, it will effectively reverse the direction of all of your forces that
depend on g (such as weight and friction).
5. With coupled systems, be very careful in how you define your coordinate system. You
can have different coordinate systems for each object – choose them to make things as
simple as possible.
What I find most convenient is this: define your +x direction for each object such
that it points in the direction of the acceleration (or the direction you expect the objects to
accelerate in). Review the Coupled Systems activity (22) until you are comfortable with
this.