Motion and Forces in
Two Dimensions
1.
2.
3.
4.
A Review of Vector Addition
Forces on an Inclined Plane
How to find an Equilibrant Vector
Projectile Motion
Sec. 7.1
Forces in Two Dimensions
• Objectives
– Determine the force that produces equilibrium
when many forces act on an object
– Analyze the motion of an object on an inclined
plane with and without friction
Using Vector Resolution
to Add Vectors
A Review of Vector Addition
Weight Forces on an Inclined
Plane
Direction Matters!
Using Vector Resolution
to Add Vectors
Here’s how we can use
vector resolution to
add vectors A and B
to find the magnitude
and direction of the
resultant vector, C.
The process of
breaking a vector
into its x and y
components is called
VECTOR
RESOLUTION.
We can use it to add
vectors together.
Using Vector Resolution
to Add Vectors
We break A into its
components, Ax
and Ay.
Ax = A cos θA
Ay = A sin θA
We break B into its
components, Bx
and By.
Bx = B cos θB
By = B sin θB
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Using Vector Resolution
to Add Vectors
Using Vector Resolution
to Add Vectors
We then add the xcomponents:
We then can determine the
magnitude of vector C,
by adding together its
components (vector
resolution in reverse):
Ax + Bx = Cx (or “Rx”)
And we add the ycomponents:
C x + Cy = C
Ay + By = Cy (or “Ry”)
R x + Ry = R
or, aka
Of course, we are adding perpendicular vectors here, so
we must use the Pythagorean Theorem: R2 = Rx2 + Ry2
Using Vector Resolution
to Add Vectors
Using Vector Resolution
to Add Vectors
What about the direction of
vector C?
Traps and Pitfalls:
• Be aware of how you define θ; it may not
be counterclockwise from east!
Look at the purple triangle:
we now have a right angle,
formed from Cx and Cy.
• Be aware of signs (+ and -), especially
when adding vectors in different
quadrants!
We can determine the
direction with some right
angle trig: (tan-1).
θ = tan-1 (Cy/Cx)
Components of Vectors
So far, we have usually
dealt with vectors in the first
quadrant, and θ has been
measured from the x axis.
In that case,
• A = √ (Ax + Ay )
• Ax =A cos θ
• Ay = A sin θ
Why are we
reviewing this?
A
Ay
θ
Ax
But soon we will need to be
proficient with applying trig
functions to different situations…
A projectile’s
instantaneous
velocity can be
resolved into
horizontal and
vertical
components.
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Let’s Practice:
Let’s Practice:
Vector Addition Review:
Vector Addition Review:
Two forces act on an object: one pulls
eastward with 12.2 N, and a second
force pulls south at 14.0 N.
What is the net force on the object?
Two forces act on an object: one pulls
eastward with 24.0 N, and a second
force pulls 55 degrees north of east
with 18.0 N.
What is the net force on the object?
Let’s Practice:
Let’s Practice:
Newton’s Laws and Friction Review:
Vector Addition Review:
Two forces act on an object: one pulls
eastward with 24.0 N, and a second
force pulls due northwest with 18.0 N.
What is the net force on the object?
Forces and Motion on an
Inclined Plane
The coefficient of kinetic friction
between a 125 kg crate and a
concrete floor is 0.12. What force
must be applied to make the crate
move at a constant speed of 1.2 m/s?
Inclined Plane Problems: Strategies
• Draw a free body
diagram.
• Choose a
coordinate system:
Make plane’s
surface be x axis.
Make the y axis be
perpendicular to the
plane.
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Inclined Plane Problems:
Strategies
NOTE:
FN is not equal and opposite to Fg!
The inclined plane
exerts an upward force
perpendicular to its
surface, this is the
NORMAL FORCE.
Inclined Plane Problems:
Strategies
What is the magnitude of FN?
Since the box has no
acceleration in the ydirection all forces in
that direction must
balance. Therefore we
get the following
equations:
FN
FN + Fgy = 0
FN = - Fgy
It is 90o to the surface,
not 90o to the weight
force!
Fgy = Fg cos 
FN = - Fg cos 
Practice
• A 55-kg box is at rest on a 26o inclined
plane. What is the normal force on the
box?
Practice
• A 55-kg box is sliding at constant velocity
on a 26o inclined plane. What is the net
force on the box?
Practice
• A 55-kg box is at rest on a 26o inclined
plane. What is the net force on the box?
Practice
• A 55-kg box is at placed on a frictionless
surface which is a 26o inclined plane.
What is the net force on the box?
acceleration = ?
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Practice
• A 75-kg skier is sliding down a 16o inclined
plane at constant speed. What is the
friction force between the slope on the
skis?
Important Concepts and
Vocabulary
Resultant Force – vector sum of 2 or more vectors.
Equilibrium – condition in which net force on an object is zero. When
the net force is zero the object is in equilibrium.
Equilibrant Force – The force needed to bring an object into equilibrium.
Force that is applied to produce equilibrium. We will use this for the lab.
It is the single additional force that if applied to the same point as the
other forces, will produce equilibrium.
To find the equilibrant find the Resultant Force. The equilibrant force is
equal in magnitude to the resultant but opposite in direction. So
add 180°.
Equilibrant
• We know that for an object to be at rest, Fnet must = 0.
How to find an
Equilibrant Vector
• We often must ask, “What force additional force must
be applied to make Fnet must = 0 ?”
• This force , the force exerted on an object to produce
equilibrium, is called the equilibrant.
• It has the same magnitude as the resultant or net
force, but it is opposite in direction
Finding the Equilibrant Force
• Find the net force, or the resultant force.
(Use vector addition)
• Change the direction by 180o!
Finding the Equilibrant Force
Try it: A 12.0 N force is applied at 0.00o (east) to a
15.0 N object. What would the equilibrant be?
Note: this object is not on a surface!
FR = 19.2 N; 308.7
FE = 19.2 N; 128.7
Our strategy:
• Find the net force, or the resultant force.
• Change the direction 180o!
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