FORCES AND
CIRCULAR MOTION
I. FORCE
A . Definition: a push or pull acting on a mass
1 . Force is a vector quantity with both magnitude (numeric
value) and direction
2. Force can be broken down into horizontal and vertical
components
3. Symbol: F
4. Units: Newtons ! (N)
B. Concurrent Forces: many forces acting on the same object at
the same time.
1 . Treat all forces individually to find a resultant force (break
into components)
2. This resultant of all concurrent forces is called the Net
Force
Symbol : Fnet
C. Free Body Diagram:
represents concurrent
forces acting on an
object
EXAMPLE 1: NET FORCE
 Two Physics students try pushing a car to see who is stronger.
One student pushes west with a force of 500 Newtons. The
other pushes East with a force of 700 Newtons.
 Draw a free body diagram of the situation.
 What is the Net Force ?
 What way does the car move?
EXAMPLE 2: NET FORCE
 Two Physics students are again arguing and this time are in a
tug of war. They are pulling on a box. One student pulls 30
degrees toward the northeast with a force of 400 Newtons
and the other pulls at 20 degrees toward the Northwest with
500 Newtons.
 Draw a free body diagram of the situation.
 What is the Net Force ?
JOURNAL #11
10/1
 What is net force on a box of mice being pulled with
a force of 20 Newtons due West toward a snake pit
and another force of 30 Newtons pulling due East
toward an alley filled with cats, a 50 Newton force
pulling due North toward a cliff, and a 50 Newton
force pulling due South toward a large pond?
 Draw a Free Body Diagram 1 st !!!
D. Static Equilibrium:
reached when the
resultant of all forces
acting on an object is
ZERO (balanced)
1 . At Equilibrium, objects
remain at rest or constant
velocity.
Fnet ≠= Zero
Zero
2. Net Force is equal to ZERO in static situations
Fnet =0
EXAMPLE: STATIC EQUILIBRIUM
 What forces MUST be added in order to produce static
equilibrium in the free body diagram below?
II. DYNAMICS
Ef fects of forces acting on objects
(Newtons Laws of Motion)
A . Newton’s First Law: An object maintains a state of
equilibrium unless acted on by an unbalanced force. (at rest
or constant velocity)
1 . Any unbalanced force ( F net ≠ 0) will produce a change in
an object’s velocity…either speed, direction, or both.
• the object will ACCELERATE
2. Newton’s First Law is also known as the Law of Inertia
• Inertia: the resistance of an object to a change in its
motion
 More Mass = More Inertia
• Masses resist changes in motion…
EXAMPLES: INERTIA
 What has more inertia? A 10 kg bag of feathers sitting still or
a 5 kg gold bar moving along at 10 m/s?
 What has more inertia? A 20 kg baseball sitting on a stand, or
a 5 kg bowling ball moving along at 30 m/s?
B. Newton’s Second Law: the acceleration of an object is
directly proportional to the net external force acting on an
object and inversely proportional to the object’s mass.
• force is related to mass and acceleration using the
famous expression:
Fnet  ma
• acceleration is produced by force(s )
• increasing force will increase the acceleration
1 . Units for Force…Yay!! Dimensional Analysis!
a. Newtons are the SI unit of force and are a derived unit
(combination of fundamental units)
Fnet  ma  kg 
m
s2
 1N
b. 1 Newton is equal to the force required to accelerate a 1
kilogram mass 1 meter per second squared
2. Increasing mass will increase the force needed to accelerate
that mass
Fnet  ma
larger
m  larger F net
*The equation must balance!
3. If the force is constant, then increasing the mass of an object
will decrease the resulting acceleration
Fnet
a
m
 F net   a
ma
F net = ma :
Direct
Relationship:
Increasing Force
produces more
acceleration
Acceleration (m/s 2 )
4. Graphing
Force (N)
EXAMPLE: NEWTON’S 2 ND LAW
A capybara with a mass of 100kg is tackled by a Jaguar with
a steady force of 100 N along the ground. Assuming no
friction, what is the acceleration of the rodent?
5. On HORIZONTAL surfaces… only the HORIZONTAL component
of a force will accelerate an object.
C. Newton’s Third Law: when one object exerts a force on a
second object, the second object exerts a force on the first
that is equal in magnitude, but opposite in direction .
 For every action there is an equal and opposite reaction!
• What happens to a
boat when you step
onto a dock?
Newton’s 3 rd Law!!!
 Newton’s 3rd Law also applies in space when
making objects move
III. NATURAL FORCES
A . Weight: gravitational force exerted on a small mass by a
planet/large body
1 . Weight CHANGES based on what planet/object you are
on… MASS does NOT CHANGE
2. Symbol:
Fg
3. Units:
Newtons !
4. Equation:
(N)
Fg  mg
How much do you weigh?
EXAMPLE: WEIGHT
 The fattest, ugliest Capybara has a mass of 66 kg. What is
the weight of the rodent on Earth?
 Convert the mass to pounds
if 1 kilogram = 2.2 pounds
B. Newton’s Universal Law of Gravitation: Describes the force of
attraction between dif ferent masses.
 Any two bodies attract each other with a force that is directly
proportional to the product of their masses, and inversely
proportional to the square of the distance between them
Gm1m2
Fg 
2
r
Gm1m2
Fg 
2
r
Fg = Gravitational Force
G = Universal Gravitational Constant = 6.67 x 10 -11 N•m 2/kg 2
m1 = mass of object 1
m2 = mass of object 2
r = distance between the two masses
On Your
Reference
Tables!!
(Front Cover)
• Graphical Representation:
EXAMPLE: NEWTON’S UNIVERSAL LAW
OF GRAVITATION
What is the force of gravitational attraction
between the Earth and the Moon?
Gm1m2
Fg 
2
r
m1 = Earth = 5.98 x 10 24 kg
m2 = Moon = 7.35 x 10 22 kg
r = 3.84 x 10 8 m
G = 6.77 x 10 -11 N•m 2 /kg 2
Gm1m2

Fg 
2
r
(6.67  10
11 N m 2
kg 2
)(5.98  1024 kg )(7.35  1022 kg )
(3.84  108 m)2
(2.93  1037 N  m2  kg 2 )

1.47  1017 m2  kg 2
Fg  1.99  10 N
20
JOURNAL #12
10/7
A 200 kg box of rodents is sitting on the road. A truck pulls the
crate with a force of 300 Newtons to the East while another
truck pulls the crate with a force of 150 Newtons to the West.
 Draw a free body diagram and label all forces…
 What is the net force on the crate?
 Disregarding friction, what will the acceleration of the crate
be?
 Calculate the weight of the crate of the rodents?
 What is the force of gravitational attraction between the crate
of rodents and the Moon?
2. Gravitational Fields: vectors are used to show
gravitational force
 A “unit test mass” will
accelerate along gravitational
field lines, toward the center
of the source of gravity
C. The Normal Force: force exerted on an object perpendicular
to the surface of contact
a. Prevents objects from accelerating due to
gravity
b. Usually denoted
FN Fn
or
N
c. For horizontal surfaces, the normal force perfectly
balances/cancels
Fg (weight)
90°
d. If the object is on an incline, the normal force is
angled from vertical the same amount as the incline
θ
θ
e. The components of the Normal force
Fgx = Fg sinq = mgsinq
Fg  Fg cos   mg cos 
y
Fg that is perpendicular to the
surface always balances FN
f. Component of
g. If the object “slides” down the surface , the force
responsible is the component of gravity acting along
the ramp (if it doesn’t slide, it must have another force
balancing gravity!)
D. Applied Force: the actual direction of the push or pull
• an object may accelerate in the direction of the applied
force if it is strong enough
Fapplied
E. Force of Friction: opposes applied force or relative motion
of two objects in contact with each other
Symbol:
Ff
a. If applied force exceeds frictional force, the object will
accelerate in the direction of the applied force
b. Factors Af fecting Force of Friction
1. Compression: Increasing the force pressing
objects together increases force of friction
 on Earth, weight and normal force
usually compress objects
friction is directly proportional
to the normal force and weight
 BEWARE: normal force and weight
are only equal on horizontal, flat surfaces!
2. Surface Area of contact between surfaces of
materials has NO ef fect on force of friction because an
object’s weight is distributed over the surface of
contact regardless of size
3. Type of Surface
Roughness = Friction
4. Speed vs. Stationary: Moving objects
experience less resistance while continuing
movement than to start moving
JOURNAL #14
10/9
A 20 kg crate is sitting on a 35 degree incline. There is friction
between the ramp and the crate.
 Draw a free body diagram and label all forces…
 Calculate the weight of the crate
 What is is the normal force of the incline on the crate?
(Break apart Gravity!)
c. Static Friction: opposes applied force on objects at
rest and resists motion
d. Kinetic Friction: opposes motion of objects in
contact that are already moving
e. Kinetic friction is LESS than Static friction due to
action of molecular forces
f. Coef ficient of Friction: ratio of frictional force to
normal force – denoted
μ

Ff
FN
 so the force of friction is equal to:
F f  FN
 essentially, the force of friction is a fraction
of the normal force
EXAMPLE: FRICTION!
 A large wooden crate of (you fill in the blank) is sitting on a
level wooden plank. The mass of the crate is 100 kg. A super
strong physics teacher pushes on the crate with a force of 500
Newtons East. FRICTION opposes that force.
 Sketch a free body diagram of the situation showing ALL
forces acting on the crate.
 What is the Weight of the crate?
 What is the Normal Force on the crate from the roadway?
 What is the force of Static Friction on the crate?
 Why does the crate move?
 What is the new Force of Friction on the moving crate?
 What is the Net Force on the moving crate?
 What is the acceleration of the crate?
1 . Coef ficients of friction are listed in the Reference
Tables!!
Reference Tables – Front Cover!!
2. Since the force of static friction is greater than the
force of kinetic friction, the coef ficients change as well
3. Equations become:
s 
Ff
s
FN
k 
Ff
k
FN
Ff s   s FN Ff  k FN
k
• treat static and kinetic situations separately
EXAMPLE: FRICTION!
A 25 kg crate of dead rodents some material, initially at rest
on a horizontal floor requires a 75 N horizontal force to the
West to set it in motion
FN
 Draw a free body diagram
 Find the coef ficient of static friction
between the crate and the floor.
F
app
Ffs = 75N
Fg
m = 25.0kg
g = 9.81 sm2
FN = ?
ms = ?
FN = Fg = (25.0kg ´ 9.81 sm2 ) = 245N
Ff
75 N

s 
 0.30
FN 245 N
s
Ff
s
g. Friction and Motion
1 . Any force problems require us to
think of Net Force!
2. Force of friction only opposes motion along a
surface!!!
3. Break down gravity and simplify diagrams
Ff
Re-orient 
θ
4. Then “tally” forces in x and y directions
Fnetx = Fg sinq + Ff
Fnety = Fg cosq + FN = 0
5. Remember: if there is no net force in a direction,
then acceleration in that direction is zero
EXAMPLE: FRICTION ON AN INCLINE
 A 150 kg GOLD plated guinea pig sits on a ramp at 40
degrees to the horizontal.
 Draw a free body diagram and label all forces…
 What is the weight of the trophy?
 What is the normal force?
 What is magnitude of the force of friction preventing the
trophy from moving? (It is equal to the part of gravity pulling
it down the ramp!)
 What is the coef ficient of Friction between the trophy and the
ramp?
F. Tension: The force in ropes, strings, or wires
1 . Symbol:
FT
or
T
2. Tension is equal to weight F g if the object hangs
vertically
3. Objects moving the opposite direction as tension can
reduce the tension in a rope (elevators moving down)
4. Objects being forced UPWARD increase the tension in
a rope based on the force applied
5. Tension can be split by angled rope
F. UNIFORM CIRCULAR MOTION
1 . For an object moving at a constant velocity, a force that
acts perpendicular to the direction of the velocity will change
the direction of the velocity, causing uniform circular motion
2. Remember: velocity is a vector…and can be changed in
both magnitude and direction
a. A change in velocity is an acceleration!
vi is always
b. Therefore, an objectv finUCM
v circular

a
toward the center of its
path
accelerating
t
***Note that speed can
remain the same but object
is still accelerating***
c. The constant acceleration of an object toward the
center of its circular path is called centripetal
acceleration.
d. Magnitude of centripetal acceleration can
found using the following equation:
2
v
ac 
r
ac
= centripetal acceleration
v
= magnitude of velocity
r
= radius of circular path
be
3. The force directed toward the center of a circular path
that causes a centripetal acceleration is called the
centripetal force.
a. Centripetal force is denoted
Fc
b. Newton’s 2nd Law can be used to determine
centripetal force:
becomes
F  ma
Fc  mac
c. Using the formula for centripetal acceleration:
2
v
ac 
r
Fc  mac
can be re-written
mv
Fc 
r
2
4. What about “centrifugal force”?
a. This is a FICTICIOUS FORCE
(does not exist)
b. Misconception comes from the apparent
motion of an object tangent to the circular
path…
c. Tangential motion only occurs if the
force goes to zero!
 then the object will obey
Newton’s Law of Inertia
With Centripetal Force
object continues in circular path
centripetal
Centripetal Force Removed
object continues in a path tangent
to its initial velocity
5. DIAGRAM!
6. Example: A rope is tied around a 0.5 kg rat which is then
twirled around overhead. The rope is 0.75 m and it makes a
full circle in 0.55 seconds. Calculate the speed, centripetal
acceleration, and centripetal force.
JOURNAL #X
A fuzzy bunny rabbit hanging from a tree with a rope 2.1 m in
length. Assuming the mass is 10 kg and swinging with a
tangential speed of 2.5 m/s…
 What is the magnitude of the centripetal acceleration?
 What is the magnitude of the centripetal force?
 What supplies the centripetal force and what is it’s direction?