Transcript Step 2. Draw a free-body diagram with all forces shown as vectors

Solving Problems
With Newton’s Laws
I want to remind you again of Newton’s third law…
4.5 Newton’s Third Law of Motion
If object A exerts a force on object B, then object B exerts and
equal and opposite force on object A.
Examples:
(http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html)
identify all the actionreaction pairs
Example:
(http://hyperphysics.phy-astr.gsu.edu/hbase/incpl.html#c1)
Let’s just look at the “rope” connecting the two masses. We
will learn how to solve this problem soon.
I’ve introduced the concept of “tension” in this example.
Read about on page 92 of your text.
Newton’s third law gives us one new OSE:
Fab = -Fba
How to remember the order of the subscripts: we will be
particularly interested on the forces ON an object. Forces on
come first. Therefore, you may write (if you wish)
Fon
a by b
= -Fon
b by a
I also want to remind you about…
The Normal Force
Normal force example: the table exerts an
upward force on the lamp. The force is
normal to the surface of the table.
Normal force example: the ramp exerts
a force on the block. The force is
normal to the surface of the ramp.
N
N
4.7 Solving Problems with Newton’s Laws:
Vector Forces and Free-Body Diagrams
Remember Newton’s second law:
I could also write this
F = m a .
Ftotal = Fnet = m a .
The acceleration of any object is proportional to the vector
sum of all the forces on the object.
In all of our problems, we will consider the different vector
components one at a time, rather than all at once.
As an example of a force problem, I’ll work the inclined plane
problem I showed you a few slides back. I’ll follow along with
the Litany for Force Problems.
You should work out other examples in section 4.7 (and your
homework, of course) using the Litany.
I’ll introduce several important concepts in this example, so
stop me if something doesn’t make sense.
Example: a wedge makes an angle θ with
the horizontal. The surface of the wedge is
frictionless. A frictionless pulley is attached
at the raised end. Two masses, M and m,
are connected by a massless cord, as
shown. Calculate the acceleration of the
system.
M
m
Step 1: draw a basic representative sketch.
We have a sketch, but we should
illustrate the forces on the sketch.
T
N
M
T
m
W
w
Forces on mass M are the tension in the cord, its weight, and
the normal force due to the wedge.
You could write W=Mg beside W if you wanted.
Forces on mass m are the tension in the cord, and its weight.
Note: same T as on M, different direction.
You could write w=mg beside W if you wanted.
In a minute we are going to define axes and show force
components. Before I do that…
Hint: exaggerate the shape of your inclined plane so you don’t
have to use trig to figure out which angle goes where.
θ
W
θ
?
I will do the rest of the work on the blackboard, listing the
steps here.
Step 2. Draw a free-body diagram with all forces
shown as vectors originating from the labeled point
mass you are considering.
Show an appropriate acceleration vector next to the point
mass (or write "a = 0" if the acceleration is zero).
This can be superimposed on the sketch if it can be done
clearly, with the vectors drawn darker than the lines of the
sketch. Each of these vectors must be distinctly and
legibly drawn.
Step 3. Label each vector with an appropriate symbol.
Use subscripts if necessary, e.g. T1 and T2, to
distinguish between two different vectors of one type
quantity.
Step 4. Lightly draw an appropriate coordinate axissystem near or on the free-body diagram, with an
arrow at one end of each line indicating the positive
direction of that axis. Choose the orientation of the
axes to fit the problem.
If the direction of acceleration is constant and known, it is
almost always best to choose one of the axes to be in the
direction of (or at least parallel to) the acceleration vector.
Otherwise, choose axis orientation to handle force
components conveniently.
Step 5. Lightly draw in vector projections of all force
vectors that are not parallel to a coordinate axis. Make
sure there are arrows at the end of these vectors to
indicate their direction.
Step 6. As an initial mathematical step, you MUST
begin with an appropriate Official Starting Equation.
All subsequent steps must mathematically follow from
this beginning point and reference to your diagram.
Step 7. Write out the sum of force components
explicitly, with the number of terms matching the
number of force vectors in your free body diagram.
Sometimes you may want to leave the component as a
symbol (maybe you don’t know its direction), but make sure
it has the proper axis-subscript, e.g., x in T1x.
Step 8. Solve for the desired quantity algebraically. If
you use a symbol in an equation, it must appear in your
diagram.
Hold off on the substitution of numerical values for the
symbols until the end of the solution.
Draw a box around your final answer.
If you don’t know the direction of a vector, e.g., acceleration in
this example, it is best to leave it in component form. Why?
If you choose to skip steps, you dramatically increase your
chances of making mistakes.