Lesson 3
Forces in Two Dimensions
Focus Question
How can you analyze forces on a train as
it climbs a hill?
New Vocabulary
equilibrant
Review Vocabulary
equilibrium: the condition in which the net force
on an object is zero.
Equilibrium Revisited
• Recall that when the net force on an object is zero,
the object is in equilibrium.
• Three forces are acting on a ring and the individual
vectors are shown below.
• The sum of these forces is zero, and the object is in
equilibrium.
Equilibrium Revisited
• A force that puts an object in equilibrium is called an
equilibrant.
• To find the equilibrant, first find the resultant of the
forces being exerted on the object.
• The equilibrant has the same magnitude as the
resultant force, but it is in the opposite direction.
Inclined Planes
• To determine net force when motion is not horizontal
or vertical, choose your coordinate system carefully.
• In the example below, the girl’s acceleration is parallel
to the incline, and therefore one axis — the x-axis —
should be in that direction.
Inclined Planes
Use with Example Problem 5.
Problem
Jeff, who weighs 640.0 N, sits on the slope
of a hill that descends at an angle of 35.0°
from horizontal. What are the components
of his weight parallel and perpendicular to
the surface of the hill?
Response
KNOWN
Fg = 640.0 N
θ = 35.0°
UNKNOWN
+y
+x
35.0°
Fgy
Fgx
Fg
Fgx = ?
Fgy = ?
SOLVE FOR THE UNKNOWN
• Use trigonometry to find the components of
weight.
SKETCH AND ANALYZE THE PROBLEM
Fgx  640.0 Nsin 35.0  367 N
• Choose axes such that +y is downward
Fgy  640.0 Ncos 35.0  524 N
and normal to the slope, and +x is down
the slope.
• Draw a vector diagram and indicate the EVALUATE THE ANSWER
• Force is in newtons, so the units are correct.
components of Fg.
• The components of Fg are smaller than Fg itself, as
• List the knowns and unknowns.
expected.
Ff
KNOWN
Inclined Planes
μ = 0.18
θ = 30.0°
Use with Example Problem 6.
UNKNOWN
Problem
ax = ?
FN
Fgy
Fg
Fgx
30.0°
A child slides from rest at the top of a
playground slide. If the slide is inclined
SOLVE FOR THE UNKNOWN
at 30.0° and the coefficient of kinetic
• Use Newton’s second law to write equations for motion in
friction between the slide and the child’s
the x- and y-directions.
pants is 0.18, what is the child’s
max  Fgx  Ff
may  Fgy  FN  0
acceleration?
max  mg sin   FN FN  Fgy  mg cos
Response
• Combine the two equations and solve for the acceleration.
SKETCH AND ANALYZE THE PROBLEM
• Choose axes such that +y is upward
ax  g sin   μg cos 
and normal to the slope, and +x is
 9.8 N/kgsin 30.0  0.189.8 N/kgcos 30.0 
down the slope.
• Draw a vector diagram and indicate
 3.4 m/s2
the components of Fg.
EVALUATE THE ANSWER
• List the knowns and unknowns.
• Force is in newtons, so the units are correct.
•
The acceleration is less than g, which was expected.
Quiz
1. When an object is at equilibrium, which is always true?
A
The object is not moving.
B
The object is accelerating at a constant rate.
C
The net force on the object is zero.
D
The object is oscillating.
CORRECT
Quiz
2. Which is the term for a force that puts an
object in equilibrium?
A
normal force
C
equilibrant
CORRECT
B
static friction force
D
resultant force
Quiz
3. The equilibrant force has ______ magnitude compared
to the resultant force, and it is in the _______ direction.
A
a greater / same
B
the same / opposite
C
the same / same
D
a smaller / opposite
CORRECT