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Chapter 5
Forces in Two Dimensions
5.1 Vectors
 Vector problem from Chapter 4:
 If you pushed on a table with 40 N of
force and your friend pushed with 40 N
of force in the same direction, the
resultant force would be:
 80 N
 If you pushed on a table with 40 N of
force and your friend pushed with 60
N of force in the opposite direction,
the resultant force would be:
 20 N towards you
Vectors in Multiple Directions:
 To add vectors that are not at right
angles to each other:
 Create a vector diagram
 If they are at right angles:
 Create a vector diagram OR
 Resolve algebraically using Pythagorean’s
Theorem and SOH, CAH, TOA
 Pythagorean’s Theorem: R2 = A2 + B2
Practice Problem #1
 A person walks 50 km east and then
turns down a street that is 75o south
of east and travels another 50 km.
 What is the person’s total distance
walked?
 What is the person’s resulting
displacement from the starting point?
Practice Problem #2
 An airplane travels east at 200 m/s.
A wind blows towards the north at
50 m/s. What is the resulting velocity
of the plane?
Components of Vectors
 A single vector may be thought of as
a resultant of 2 vectors which are
called perpendicular components
 There is one horizontal and one
vertical component for every vector
 Vector resolution – breaking a vector
down into its components
Adding Vectors at Any Angle
 Resolve each vector into its horizontal and
vertical components
 Vx=V cos q
 Vy=V sin q
 Sum the results for each
 Find the magnitude using the Pythagorean
Theorem
 Find angle by the following formula:
tan q = Vy (sum)
Vx (sum)
Boat Problem!
 A boat heads east across a river that is 2.8
km wide with a velocity of 25 km/h. The
river flows south with a velocity of 7.2
km/h.
 What is the resultant velocity of the boat?
 How long does it take the boat to cross the
river?
 How far upstream is the boat when it
reaches the opposite side?
5.2 Friction
 Static – starting friction; works against the
start of motion
 Kinetic – sliding friction; works against
keeping an object in motion
 Ff = m F N
 If the object is moving with a constant
velocity, then the applied force (often called
horizontal force) is equal to the frictional
force so… FH = Ff
m – the coefficient of friction
 Greater for rougher surfaces
 Lesser for smoother surfaces
 Has no units!
Is a number between 0 and 1
Try This!
 A 64-N box is pulled across a rough
horizontal surface. What is the force
necessary to keep the box moving at
a constant speed if the coefficient of
friction between the box and the floor
is 0.81?
…and this…
 A 9.0 kg crate is resting on a floor. A
61-N force is required to just start
motion of the crate across the floor.
What is the coefficient of friction
between the floor and the crate?
5.3 Force in Two Dimensions
 Equilibrium
 When the sum of all forces acting on an object is
zero
 Equilibrant Force
 The force that will put all other forces in
equilibrium
 To calculate:
 Find the resultant
 The equilibrant is equal in magnitude and
opposite in direction
 Use the same force and add or subtract 180o to
the direction
Try This:
 Two forces act on an object. One is
125 N pulling toward 57o. The other
is 182 N pulling toward 124o. Find
the one force that would put the
other two in equilibrium. You may
use any method that you want.
Motion Along An Inclined Plane
 A skier has several
forces working on
him as he moves
down a hill:
 Gravitational force
toward center of
the earth
 Normal force
perpendicular to
the hill
 Frictional force
parallel to the hill
Calculating the Components of
Weight on an Inclined Plane:
 Fgx = Fg sin q (Parallel to the inclined
plane)
 Fgy = Fg cos q (Perpendicular to the
inclined plane)