Concurrent and Parallel Forces

1
Concurrent and Parallel Forces

• Describing Forces
• Force A physical quantity that can affect the
  motion of an object.
• Examples Force of hand on door
• Pushing a door shut
• Force of door on hand
• Force of you on chair
• Sitting in a chair
• Force of chair on you

2

• Forces can be applied through both physical
  touching and without any physical contact.
• With Contact see above examples
• Without Contact Earth/Moon system, a freely
  falling body and earth.

3
Characteristics of Forces

• 1. A net force will change the state of motion of
  an object.
• - Apply a net force to an object at rest, the
  object will start to accelerate.
• - Apply a net force to an object in motion,
  Fma the object will either accelerate or
  decelerate. (Depending on the direction of the
  force)

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Characteristics of Forces

• 2. Forces can be exerted through long distances.
• Gravitational forces
• Magnetic Forces
• 3. Forces always occur in pairs
• Newtons 3rd Law
• Action/Reaction Forces

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Characteristics of Forces

• 4. In each pair of forces, the two forces act in
  exactly opposite directions.

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Combining Force Vectors

• Forces have both magnitude AND direction, making
  them vector quantities.
• The magnitude of the force being applied is
  represented by the length of the arrow.
• When drawing vectors on a free body force
  diagram, you MUST find the direction of the force.

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EXAMPLE

• A tugboat applies a force of 10,000 N to a barge
  through a towline.
• The force is to the left.

towline
Tugboat
Barge
F 10,000 N
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Vector Addition and Subtraction

• Two tugboats applying a force tot he barge

10,000 N
Resultant force of 19,000 N
9,000 N
Resultant force of 1,000 N to the left
9,000 N
10,000 N
9

• You pick positive and negative directions!
• You label the diagram properly so I know what
  youre doing!
• In both examples, the forces are considered to be
  concurrent.
• Concurrent forces- Two or more forces acting on
  the same point at the same time. (The Barge as a
  whole is considered to be a singular point.)

10
Net Forces

• If forces are NOT equal and opposite, there is a
  net force which causes an acceleration.
• Example
• A rock climber pulls his equipment (mass5.4 kg)
  up the side of a cliff using a rope. What total
  force must be applied to give the equipment an
  acceleration of 2.3 m/s2?

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Free Body Force Diagram
The equipment weighs 52.92 N. FW mg
(5.4)(9.8) To cause an acceleration, there must
be a net force in excess of the weight of the
equipment. FTOTAL FW FNET FNET ma
(5.4)(2.3) So the total force is 12.42N 52.92 N
65.34 N
FNET
FTOTAL
FW
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Resultant Force

• Resultant Force - A single force that has the
  same effect as two or more concurrent forces.
• Just as in the vector unit (because force is a
  vector)

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Law of Sines and Law of Cosines

• Two or more forces at angles other than 90º
• To find the resultant graphically, use the
  parallelogram method.
• To find the resultant mathematically, use the law
  of sines and/or law of cosines

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Law of Sines
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Law of Cosines

• a ? b2c2 - 2bc cos A
• b ? c2a2 - 2ca cos B
• c ? a2b2 2ab cos C
• Example In ?ABC, a 10, b 13, and ?C 70?.
  Find c.
• c2 a2 b2 2ab cos C
• c ? a2 b2 2ab cos C
• c ? 102 132 2(10)(13) cos 70
• c 100 169 188.92
• c 180.08
• c 13.42

A
13.42
13
B
C
10
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Example

• A force of 3.50 N acts North on Point O. A
  second force of 8.75 N acts concurrently on Point
  O, but at an angle of 30.0? West of North.
  Calculate the magnitude and direction of the
  resultant force.
• Solution
• FA 8.75 N
• FE 3.50 N
• ? 30?
• ? 150?

o
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• FR FA2 FE2 -2FEFA cos ?
• FR (8.75)2 (3.50)2 2(3.5)(8.75)(cos 150)
• FR 88.81 (-53.04)
• FR 141.85 11.91 N
• The Resultant Force is 11.91 Newtons.

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• FA FR
• sin ? sin ?
• FA sin ? FR sin ?
• sin ? FA sin ?
• FR
• sin ? (8.75) sin 150
• 11.91
• ? sin 1 (.03673)
• ? 21.55?
• The angle is 21.55? or 21.6? West of North.

11.91 N
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The Equilibrant Force

• Equilibrium is the state of a body in which there
  is no change in its motion
• - needs to have both translational and
  rotational equilibrium
• Translational - linear motion
• Rotational - circular/rotary motion
• In equilibrium, a body is either at rest or
  moving with a constant speed in a straight line.
  (constant velocity)

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First Condition of Equilibrium

• No unbalanced (net) forces acting on a body
• translational equilibrium
• Vector sum of forces on body equals zero
• ?F 0 OR Vector sum of forces 0

Pull a box with 15 N
Push a box with 15 N
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• To have the vector sum of the forces equal zero,
  there has to be two forces (action and reaction)
• Action Force push the box
• Reaction Force box pushing back on you
• The reaction force is known as the equilibrant.

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Equilibrant Force

• When two or more forces act concurrently at a
  point, the equilibrant force is that single force
  that if applied on the same point produces
  equilibrium.

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To find the Equilibrant Force

• 1. Find the magnitude of the resultant vector
• 2. Find the angle of the resultant vector
• 3. Draw in the equilibrant 180 from the
  resultant vector with the same magnitude

Equilibrant Force
Resultant Vector
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RESOLVING GRAVITATIONAL FORCES
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RESOLVING GRAVITATIONAL FORCES

• When an object is level on the ground and in
  translational equilibrium, all of its weight is
  being supported by the earth there are no other
  forces.
• FN is called the normal force.
• It is equal in magnitude and
• opposite in direction of FW
• when an object is level.

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Inclined Planes

• On an inclined plane, it is different. Examples
  of inclined planes include hills, ski slopes,
  sliding boards, etc. In this case, part of the
  force acts down the plane while the other part
  acts towards the center of the earth (due to
  gravity).
•  

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Inclined Planes

• The plane prevents the motion in the direction of
  FW because it pushes back up on the block. This
  force is not as much as the weight of the object
  since part of the force is down the plane. As
  the plane increases in steepness, this component
  decreases. We will call this the normal force
  and label it FN on a free body force diagram.
•  

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Inclined Planes

• Remember, FN is not only the force the box
  pushes down with, but also the force that the
  plane pushes up with
•   FN (BOX) FN (PLANE)
•  So, putting everything together,

FP
FN
FW
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Inclined Planes

• FN Normal force (or the force that the incline
  is
• holding up the mass/object). It is equal in
• magnitude and opposite in direction to the
• normal force.
•  FP Force of the object moving down the incline
• (ignoring friction)
•  FW Force of gravity on the object (weight)
•  The angle between the weight (FW) and normal
  force (FN) is the same as the angle of the
  incline.

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Inclined Planes

• From SOHCAHTOA
•  
• Sin ? FP / FW or FP FWsin?
•  
• Cos ? FN / FW or FN FWcos?
•  
• The steeper the incline, the less FN and greater
  FP.

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EXAMPLE OF INCLINED PLANES

• 1. A box is sitting motionless on a plane
  inclined 25? to the horizontal. If its mass is 75
  kg, find the following
•         a. its weight
•       b. the normal force
•        c. the force down the plane
•        d. the magnitude of the force holding
  the box at rest
•  

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FREE BODY FORCE DIAGRAM

• a.      FW mg (75 kg)(9.8 m/s2) 735 N
• b.      FN FWcos? 735cos25? 666.14 N
• c.      FP FWsin? 735 sin 25? 310.62 N
• d.      FHOLDING FORCE 310.62 N up the plane

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• NOTE This force has to be equal in magnitude
  and opposite in direction of FP to stay in
  translational equilibrium.
•  
• Objects at rest have only the forces mentioned
  before. However, there can be more forces
  present. Pushing or pulling the object up or
  down the plane will result in more forces
  present, so you NEED A FREE BODY FORCE DIAGRAM!
•  

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EXAMPLE

• A box is pulled with constant velocity up an
  inclined plane of 12? to the horizontal.
• _at_ constant velocity, the box is in
  equilibrium ??FY 0
• No movement up or down
• ?FX FH FP Ff 0   

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• The variables are as follows
• FH force of the push/pull up the plane
• When this force is present, we will label up the
  plane as positive.
• FP force down the plane
• It opposes the force up the plane, so it is
  negative
• Ff the force due to friction
• Friction always opposes motion, so it too is
  negative

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THE NATURE OF FRICTION

• When an object is in motion, we will now be
  concerned with friction. There is no more ideal
  case.
•  Friction a force that resists motion
• -        involves objects that are in contact
  with each other
•  Irregularities in surfaces cause friction. No
  surface is perfectly smooth.
•  When talking about friction, we are no longer
  dealing with ideal situations.

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THE NATURE OF FRICTION

• REAL LIFE CASES OF FRICTION
• -        tires (rubber) on road (asphalt)
• -        shoes on ground
• -        going down a sliding board in the
  summertime
•  
• Measuring Friction
•      -we will only be dealing with solid objects
  in this course.

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5 Rules of Friction

• 1. Friction acts parallel to the surfaces that
  are in contact and in a direction opposite to the
  direction of motion of the object or to the net
  force tending to produce such motion.

FN
FFRICTION
FMOTION
FW
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5 Rules of Friction

• 2. Friction depends on the nature of the
  materials in contact and the smoothness of their
  surfaces.
•  
• TYPES OF MATERIAL
• Lots of friction dry rubber on dry road Steel on
  steel
• Not much friction wet rubber on wet
  road Teflon on Teflon
• SMOOTHNESS OF SURFACES
• Lots of friction finger on a metal
  file Sandpaper on wood
• Not much friction ice skates on ice You on a
  waterslide
•  

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5 Rules of Friction

•  3. Sliding friction is less than or equal to
  starting friction
• -   sliding friction is the frictional force
  between objects that are sliding with respect to
  one another
•    Starting friction is the maximum frictional
  force needed to get an object moving. Both
  objects are originally stationary.
•   It is always easier to keep things moving than
  to first get it started. Greater starting
  friction hard to get rolling. Less sliding
  friction easy to keep moving.
•          Example trying to move a stalled car.
  
•  

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5 Rules of Friction

•  4. Friction is practically independent of the
  area of contact.
•  

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5 Rules of Friction

• 5. Starting or sliding friction is directly
  proportional to the force pressing the two
  surfaces together.

2m
m
Ff
2Ff
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Coefficient of Friction

• Tests have been performed to find both starting
  and sliding friction for different substances by
  using the ration of the force of friction to the
  weight of the object.
• We use the Greek letter mu, ?, to stand for the
  coefficient of friction.
•  
• ? Ff / FN

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CHANGING FRICTION

•  
• The amount of friction between two objects can be
  changed (either increase or decrease friction).
•  
• 1.    Add salt or sand to an icy sidewalk to
  increase friction.
• 2.    Add Teflon to a metal surface to decrease
  friction

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Center of Gravity

• Gravity acts on ALL points of an object, not just
  one ideal point.
• Taking the vector sum of all of the gravitational
  forces on every point, we can find one resultant
  vector.
• This one resultant vector is called the objects
  center of gravity (COG).

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Center of Gravity

• Gravity pulls down at We say that all of its
  weight all points of the rock is concentrated
  at the center

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Center of Gravity

• Finding the center of gravity of a uniform object
  is easyit is in the middle.

FN
FW
Center of Gravity (FW)
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Torque

• When an equal and opposite force is applied to
  the bar, it is considered to be in equilibrium.
• If an excess force is applied to one side of the
  pivot point (center of gravity), then a torque
  will be produced.
• Torque - a force that will take an object out of
  rotational equilibrium. It is measured in
  Newton-meters (Nm) or meter-Newton (mN).

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Torque

• Rotational Equilibrium - the sum of all clockwise
  torques equals the sum of the counterclockwise
  torques about any pivot point.
• - also known as the second condition of
  equilibrium (can be stationary OR moving with
  constant angular velocity

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Applying a Torque
A force on the left of the pivot point going up
or to the right going down causes the bar to
spin clockwise
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Applying a Torque
A force on the right of the pivot point going up
or to the left going down causes the bar to
spin counterclockwise
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Applying a Torque

• In both cases, a torque has been applied. The
  magnitude of the torque depends on two things
• 1. The amount of force applied (F)
• 2. The torque arm
• - the distance (in meters) from
• the pivot point to the force
• or torque (force)(distance) ? Fd
• The torque arm MUST be measured perpendicular to
  the direction of the applied force!

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Coupled Forces

• Coupled forces are a pair of forces of equal
  magnitude that act in opposite directions in the
  same plane, but not at the same point.

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EXAMPLES (on board)

• 1 from notes
• 2 from notes
• Forces at angles to bar