Download equilibrium

Chapter 5
Two Dimensional Forces
Equilibrium


An object either at rest or moving
with a constant velocity is said to
be in equilibrium
The net force acting on the object
is zero (since the acceleration is
zero)
F  0
Equilibrium cont.

Easier to work with the equation in
terms of its components:
F
x
 0 and
F
y
0
Inclined Planes


Choose the
coordinate
system with x
along the incline
and y
perpendicular to
the incline
Replace the force
of gravity with its
components
Forces of Friction

When an object is in motion on a
surface or through a viscous
medium, there will be a resistance
to the motion


This is due to the interactions
between the object and its
environment
This is resistance is called friction
More About Friction





Friction is proportional to the normal
force
The force of static friction is generally
greater than the force of kinetic friction
The coefficient of friction (µ) depends
on the surfaces in contact
The direction of the frictional force is
opposite the direction of motion
The coefficients of friction are nearly
independent of the area of contact
Applications Involving Friction, Inclines
On a microscopic scale, most surfaces
are rough. The exact details are not
yet known, but the force can be
modeled in a simple way.
For kinetic – sliding – friction,
we write:
is the coefficient of
kinetic friction, and is
different for every pair of
surfaces.
Kinetic Friction, ƒk


The force of
kinetic friction
acts when the
object is in
motion
ƒk = µ n

Variations of the
coefficient with
speed will be
ignored
4-8 Applications Involving Friction, Inclines
Static Friction, ƒs




Static friction acts
to keep the object
from moving
If F increases, so
does ƒs
If F decreases, so
does ƒs
ƒs  µ n
Applications Involving Friction, Inclines
The static frictional force increases as the applied force
increases, until it reaches its maximum. Then the
object starts to move, and the kinetic frictional force
takes over.
Block on a Ramp, Example



Axes are rotated as
usual on an incline
The direction of
impending motion
would be down the
plane
Friction acts up the
plane


Opposes the motion
Apply Newton’s Laws
and solve equations
Atwood Machine
Atwood Machine

Let’s take a look
at the forces on
each mass…
Atwood Machine

Atwood Machine

Static Equilibrium


Condition of an object when
net forces equal zero
Object is motionless
Free Body Diagram
Hanging sign f.b.d.
T1
1
T2
2
mg
Since the sign is not
accelerating in any
direction, it’s in
equilibrium. Since
it’s not moving
either, we call it
Static Equilibrium.
Thus, red + green + black = 0.
Components & Scalar
T2
Equations
T1
(Y component)
T1 sin 1
(y component)
T2 sin 2
1
T1 cos 1
(x component)
2
T2 cos 2
(X component)
If in Equilibrium……..the
following would be true
Vertical:
T1 sin 1 + T2 sin 2 = mg
mg
Horizontal:
T1 cos 1 = T2 cos 2
Sample Problem
A mother and daughter are outside playing on
the swings. The mother pulls the daughter
and swing (total mass 55.0 kg) back so that
the swing makes an angle of 40.0° with the
vertical (50.0 ° from horizontal)

What is the tension in each
chain holding the swing seat
and the daughter?
A. 703N
Sample Problem…


Is this a case of
equilibrium?
Calculate the
magnitude of the
net force
160N
40 °
150N
45°
75N
Equilibrium Example –
Free Body Diagrams