Physics I
Class 06
Newton’s Second Law
for More Complex Cases
06-1
Newton’s Second Law Yet Another Review!
Newton’s Second Law:
 
 F  Fnet

 Fnet

 m a or a 
m
In a complex situation, we will need to apply this to more than one
object and/or in more than one dimension.
When we do this, we will get a set of linear equations and we will solve
them for the unknown quantities.
06-2
Using Newton’s Second Law to
Solve Complex Problems
1.
2.
3.
4.
5.
6.
Identify all forces acting on the object.
Today: Gravity, normal, and ropes/strings.
Choose a coordinate system.
If you know the direction of acceleration, one
coordinate axis should be in that direction.
Draw a “Free-Body Diagram.”
We will use two dimensions today.
Express the force vectors in components.
We will use trigonometry today.
Use Newton’s Second Law to write one
equation for each direction considered.
Solve the equations.
06-3
A Common Example:
Atwood’s Machine
a
W = mg
X
W = Mg
T
T
m
w = mg
a
X
M
W = Mg
06-4
Solution to Atwood’s Machine
T  mg  ma
Mg  T  Ma
Add these to eliminate T .
Mg  mg  Ma  ma
( M  m) g  ( M  m)a
M  m

a
g
 M  m
06-5
Inclined Plane
06-6
Coordinate Systems and
Free-Body Diagrams
Y
X
a
X
Use trigonometry to determine X & Y components
of forces not aligned with coordinate system.
06-7
Solving for Acceleration
For mass 1:
T  m1g sin( )  m1 a
X:
Y:
N  m1g cos( )  0
For mass 2:
m 2g  T  m 2 a
X:
To solve for a, add the two X equations:
T  m1g sin( )  m 2 g  T  m1 a  m 2 a
m  m1 sin( )
a 2
g
m1  m 2
What would it mean if we found a < 0 after plugging in the values?
If  = 0, does the “inclined” plane resemble something in class?
06-8
Class #6
Take-Away Concepts
1.
2.
3.
4.
Keep using the six-step process for doing Newton’s Second Law
problems for complex cases, it will help you keep things straight.
For each dimension and each object, you will get one equation. You
may or may not need to know the forces in the “normal” direction.
You should have the same number of unknowns as equations.
The easiest way to solve is usually to add the two equations resulting
from opposite ends of a rope or string, or opposite sides of a contact
surface where two objects push on each other.
06-9
Activity #6 - Forces and Motion in
Coupled Systems
Objectives of the Activity:
1.
2.
Making detailed theoretical predictions and comparing
with measured data.
Understanding forces and motion in coupled systems.
06-10