Chapter 2: Motion

1
Chapter 2 Motion

• Alyssa Jean-Mary

2
Speed

• The speed of something is the rate at which it
  covers distance
• If the speed is high, it travels faster and it
  covers more distance in a given period of time
• Speed (v) distance (d) / time (t)
• Common units to express speed are meters/second
  (m/s)
• When we dont know how something moved, its speed
  is referred to as the average speed i.e. it
  might have moved faster at some times and slower
  at other times
• For example, if a car has an average speed of 50
  mi/h, it doesnt mean that it was moving at a
  constant speed of 50 mi/h during the entire time
  when it slowed down for traffic or stopped at a
  stop sign, it was obviously moving slower than 50
  mi/h
• The instantaneous speed is how fast something is
  going at any given moment
• In a car, what the speedometer reads is the
  instantaneous speed of the car

3
Steps to Solving a Problem

• Step 1 Identify the information that is given in
  the problem.
• Step 2 Identify what information the problem is
  looking for.
• Step 3 Identify the equation that is needed to
  obtain the information you are looking for, while
  using the information you were given.
• Step 4 Put the given information into the
  equation found in Step 3 and solve for the
  information you are looking for.

4
Example Calculation of Speed

• Example What is the speed of an object that
  traveled 34 meters in 65 seconds?
• Answer
• 1. Given 34 meters, 65 seconds
• 2. Looking for speed
• 3. Equation v d/t
• 4. Solution v d/t 34 meters/65 seconds
  0.52 m/s

5
Distance and Time

• The equation for speed can be rewritten in order
  to calculate
• the distance (d) something travels at a given
  speed (v) in a given period of time (t)
• distance (d) speed (v) x time (t)
• the time (t) something takes to travel a given
  distance (d) going a given speed (v)
• time (t) distance (d) / speed (v)
• Distance is commonly expressed in meters (m) and
  time is commonly expressed in seconds (s)

6
Distance Solving the Equation and Example
Calculation

• Solving the Equation
• 1. Start with the equation for speed v d/t
• 2. Get d on a side by itself by multiplying both
  sides by t, thus obtaining the equation for
  distance d vt

• Example How far did an object move in 76 seconds
  if it was traveling at a speed of 2.5 m/s?
• Answer
• 1. Given 76 seconds, 2.5m/s
• 2. Looking for how far distance
• 3. Equation d vt
• 4. Solution d vt 2.5 m/s x 76 s 190 m

7
Time Solving the Equation and Example Calculation

• Solving the Equation
• 1. Start with the equation for speed v d/t
• 2. Get t on top (it is now on the bottom) by
  multiplying both sides by t, thus the equation
  is vt d
• 3. Get t on a side by itself by dividing both
  sides by v, thus obtaining the equation for time
  t d/v

• Example How long did it take for an object move
  230 meters if it was traveling at a speed of 14.7
  m/s?
• Answer
• 1. Given 230 meters, 14.7m/s
• 2. Looking for how long time
• 3. Equation t d/v
• 4. Solution t d/v 230 m / 14.7 m/s 15.6 s

8
Scalar Quantities and Vector Quantities

• Scalar quantities are quantities that contain a
  number and a unit. Scalar quantities give only
  the magnitude (how large it is), not the
  direction
• Examples of scalar quantities
• Speed
• Vector quantities, on the other hand, are
  quantities that contain both a magnitude and a
  direction
• Examples of vector quantities
• Displacement, which is change in position
• Force
• Velocity, which is the speed of something and the
  direction of this speed

9
Representing Vector Quantities

• Vector quantities are represented by a vector, a
  straight line that has an arrowhead at one end to
  show the direction of the quantity
• The length of the line is scaled for the
  magnitude of the quantity
• Vector quantities are usually printed in bold
  (i.e. Force F), while scalar quantities are
  usually printed in italics (i.e. Speed v). To
  represent the magnitude of a vector quantity,
  italics is also used (i.e. the magnitude of a
  Force, F, is F). When a vector quantity is
  handwritten, an arrow over the symbol is used to
  indicate that it is a vector quantity.

10
Adding Scalar Quantities and Vector Quantities

• To add scalar quantities, ordinary arithmetic is
  used
• To add vector quantities,
• if they are going in the same direction, ordinary
  arithmetic is used
• if they are not going in the same direction, the
  following steps need to be followed
• 1. Draw the vectors that are to be added together
  tail to head.
• In this example, to add vector B to vector A,
  draw vector B with its tail at the head of vector
  A
• 2. Draw a vector from the tail of the first
  vector to the head of the last vector to be added
  this vector is the addition of all the vectors
• 2. Draw a vector C from the tail of vector A to
  the head of vector B vector C is the addition
  of vector B to vector A

11
Pythagorean Theorem

• A right triangle is a triangle in which two of
  its sides are perpendicular to each other in
  other words, its when the two sides meet at a
  90 angle
• The Pythagorean Theorem states that, in a right
  triangle, the square of each of the short sides
  added together is equal to the square of the
  longest side (i.e. the hypotenuse)
• If the short sides are A and B and the long side
  is C, the Pythagorean Theorem can be shown as the
  following equation
• A2 B2 C2
• When the Pythagorean Theorem is applied to adding
  vectors, A, B, and C are the magnitudes of the
  vectors A, B, and C
• The following equations can be used to find one
  of the sides if the length of the other two sides
  are known
• A v(C2 B2)
• B v(C2 A2)
• C v(A2 B2)

12
Example Calculations Using the Pythagorean Theorem

• Example 1 What is the length of side A if side B
  is 2.3 m and side C is 7.6 m?
• Answer
• 1. Given side B 2.3m, side C 7.6m
• 2. Looking for length of side A
• 3. Equation A v(C2 B2)
• 4. Solution A v(C2 B2) v(7.6m2 2.3m2)
• Example 2 What is the length of side B if side A
  is 4.3 m and side C is 10.5 m?
• Answer
• 1. Given side A 4.3m, side C 10.5m
• 2. Looking for length of side B
• 3. Equation B v(C2 A2)
• 4. Solution B v(C2 A2) v(10.5m2 4.3m2)
• Example 3 What is the length of side C if side A
  is 8.7 m and side B is 3.1 m?
• Answer
• 1. Given side A 8.7m, side B 3.1m
• 2. Looking for length of side C
• 3. Equation C v(A2 B2)
• 4. Solution C v(A2 B2) v(8.7m2 3.1m2)

13
Acceleration

• An object that has an acceleration is one whose
  velocity is changing
• The change in velocity can be one of three
  things
• An increase in speed (i.e. going faster)
• A decrease in speed (i.e. going slower)
• A change in direction of the speed (i.e. turning)
• Acceleration is a vector quantity
• When acceleration is in a straight line
• Acceleration change in speed / time interval
• OR
• a (v2 - v1) / t, where a is the acceleration,
  v2 is the final speed, v1 is the initial speed,
  and t is the time interval
• Common units to express acceleration are m/s2
• When acceleration is from a decrease in speed, it
  is called negative acceleration or deceleration
• We will assume that acceleration is constant,
  although that is not always the case

14
Example Calculations With Acceleration

• Example If an objects initial speed is 30m/s
  and its final speed is 55m/s, what is its
  acceleration over 571 seconds?
• Answer
• 1. Given initial speed 30m/s, final speed
  55m/s, 571 seconds
• 2. Looking for acceleration
• 3. Equation a (v2 - v1) / t
• 4. Solution a (v2 - v1) / t (55m/s 30m/s)
  / 571 s 0.0438 m/s2

15
Using Acceleration to Calculate Final Speed

• To calculate the final speed of something, the
  equation for acceleration (a (v2 - v1) / t )
  needs to be rewritten
• 1. First bring t over to the other side of the
  equation by multiplying it on both sides
• 2. Then, separate v2 from v1 by adding v1 to both
  sides
• The equation to calculate final speed is v2 v1
  at

• Example If an object has an initial speed of 60
  m/s with an acceleration of 3.2m/s2, what is its
  final speed after 43 seconds?
• Answer
• 1. Given initial speed 60m/s, acceleration
  3.2m/s2, 43 seconds
• 2. Looking for final speed
• 3. Equation v2 v1 at
• 4. Solution v2 v1 at 60m/s
  (3.2m/s2)(43s) 197.6m/s

16
How Far?

• To calculate the average speed of an object
  during uniform acceleration
• average speed (v1 v2) / 2
• The equation to calculate the distance the object
  moves is d vt
• If the speed, v, is an average speed, then the
  equation for distance is d (average speed)t
• If we insert the equation for the average speed
  into this equation, then the equation for
  distance is d (v1 v2)/2t (v1t/2)
  (v2t/2)
• If we insert the equation for the final speed
  into this equation, then the equation for
  distance is d (v1t/2) ((v1 at)t/2)
  (v1t/2) (v1t/2) (at2/2) v1t (at2/2)
• So, d v1t (at2/2)
• If the object starts at rest, then v1 0, so d
  (at2/2)

17
Example Calculations for Average Speed and
Distance

• Average Speed Example What is the average speed
  of an object that was moving at some times 45m/s
  and at other times 35m/s?
• Answer
• 1. Given speed 1 45m/s, speed 2 35m/s
• 2. Looking for average speed
• 3. Equation average speed (v1 v2) / 2
• 4. Solution average speed (v1 v2) / 2
  (45m/s 35m/s)/2 40m/s

• Distance Example 1 If an object has an initial
  speed of 10 m/s with an acceleration of 1.5m/s2,
  how far did it travel in 76 seconds?
• Answer
• 1. Given initial speed 10m/s, acceleration
  1.5m/s2, 76 seconds
• 2. Looking for how far distance
• 3. Equation d v1t (at2/2)
• 4. Solution d v1t (at2/2) (10m/s)(76s)
  (1.5m/s2)(76s)2/2 5092m
• Distance Example 2 If an object starts from rest
  with an acceleration of 0.45m/s2, how far did it
  travel in 5.6 seconds?
• Answer
• 1. Given acceleration 0.45m/s2, 5.6 seconds,
  initial speed 0 m/s (at rest)
• 2. Looking for how far distance
• 3. Equation d (at2/2)
• 4. Solution d (at2/2) (0.45m/s2)(5.6s)2/2
  7.06m

18
Acceleration of Gravity

• Before Galileo, philosophers tried to answer
  questions about the movement of objects due to
  gravity by creating concepts that were so obvious
  that there was no need to test them
• Aristotle, an ancient Greek thinker, was one of
  these famous philosophers
• He thought that all falling bodies followed this
  basic concept every material has a natural
  place where it belonged and toward where it tried
  to move
• For example, stones are from the earth, so they
  naturally fell downward, toward the earth

19
Free Fall

• Galileo, an Italian physicist, experimented with
  falling bodies 2000 years after Aristotle
• He performed his experiments using objects
  rolling down an inclined plane instead of falling
  in free fall, but his results apply to objects in
  free fall as well
• He found that the higher an object is dropped,
  the greater its speed when it reaches the ground,
  meaning that the object has an acceleration
• He also found that no matter the size of the
  object, whether it is big or small, the
  acceleration is the same
• Galileo found that if an object that is falling
  near the earths surface doesnt have to push its
  way through any air, it has an acceleration of
  9.8 m/s2 this acceleration is known as the
  acceleration of gravity, and is represented by g
• When an object is falling from rest, its downward
  speed can be calculated by the following
  equation vdownward gt, where g is 9.8 m/s2 and
  t is the time

20
How Far Does a Falling Object Fall?

• To determine how far (h) a falling object falls
  from rest in a given time
• 1. Start with d at2/2
• 2. If you replace d, distance, with h, height,
  the equation is h at2/2
• 3. If you replace a, acceleration, with g, the
  acceleration of gravity, the equation is h
  gt2/2

21
A Falling Object Speed vs. Height

• The downward speed is calculated by the following
  equation vdownward gt
• This equation shows that the speed is directly
  proportional to the time i.e., the speed at 5
  seconds is 5 times the speed at 1 second
• The height is calculated by the following
  equation h gt2/2
• This equation shows that the height is directly
  proportional to the time squared (t2) - i.e., the
  speed at 5 seconds is 25 times the speed at 1
  second
• The height thus increases faster with increasing
  time than the downward speed

22
Example Calculations of Downward Speed and Height

• Downward Speed Example What is the speed of an
  object that is falling for 34 seconds?
• Answer
• 1. Given 34 seconds, a g
• 2. Looking for speed
• 3. Equation vdownward gt
• 4. Solution vdownward gt (9.8m/s2)(34s)
  333.2m/s

• Height Example How far does an object fall after
  4.9 seconds?
• Answer
• 1. Given 4.9 seconds, a g
• 2. Looking for how far height
• 3. Equation h gt2/2
• 4. Solution h gt2/2 (9.8m/s2)(4.9s)2/2
  117.6m

23
Thrown Objects

• The acceleration of gravity, g, is the same
  whether an object is
• just dropped
• thrown downward
• thrown upward
• thrown sideways
• If a ball is just dropped, it goes faster and
  faster as it drops until it hits the ground
• If a ball is thrown sideways, it has a curved
  path that becomes steeper and steeper as it drops
• If a ball is thrown upward, gravity first reduces
  the upward speed of the ball until its upward
  speed is zero. When its upward speed is zero, the
  ball is at the top of its path and is momentarily
  at rest. Then the ball starts falling toward the
  ground, faster and faster, until it hits the
  ground.
• If a ball is thrown downward, the original speed
  of the ball is steadily increased by the downward
  acceleration (g)

24
Parabolas

• A parabola is a curved path
• When a ball is thrown upward at an angle to the
  ground, a parabola occurs
• If there is no air resistance, the maximum
  distance a ball can travel horizontally occurs
  when the ball is thrown upward at an angle of 45
  to the ground
• If the angle is greater than 45 or less than
  45, then the ball travels less than it would
  have if it had be thrown at an angle of 45
• There is one angle above 45 and one angle below
  45 that if the ball is thrown, it will land in
  the same place

25
Air Resistance

• Air resistance keeps things from developing the
  full acceleration of gravity without air
  resistance, anything, including a light shower,
  would be dangerous
• The faster something moves, the more air
  resistance it encounters - i.e. the more the air
  slows down its motion
• For a falling object, the air resistance
  increases with speed until the air resistance
  equals the force of gravity on the object. Once
  this occurs, the object falls at a constant
  terminal speed. The terminal speed of an object
  depends on its size, its shape, and how heavy it
  is

26
Air Resistance for Objects Thrown Upward at an
Angle

• If there is no air resistance (i.e., in a
  vacuum), the maximum distance a ball can travel
  horizontally occurs when the ball is thrown
  upward at an angle of 45 to the ground
• However, if there is air resistance, the maximum
  distance a ball can travel horizontally doesnt
  occur when the ball is thrown upward at an angle
  of 45 to the ground it occurs when the ball is
  thrown upward at an angle less than 45 to the
  ground

27
Falling Air vs. Vacuum

• In air, a stone falls faster than a feather
  because the air resistance affects the feather
  more than the stone
• In a vacuum, since there is no air resistance,
  both a stone and a feather fall with the same
  acceleration, g (9.8 m/s2)

28
The First Law of Motion

• If a ball is left alone on the floor, it will not
  move unless someone pushes it
• If there is a perfectly round ball and a
  perfectly smooth floor, a ball that is pushed
  will continue to roll forever in the same
  direction with the same speed until someone
  pushes to make it roll slower, to make it roll
  faster, or to change its direction of motion
• The First Law of Motion If no net force acts on
  it, an object at rest remains at rest and an
  object in motion remains in motion at constant
  velocity (that is, at constant speed in a
  straight line).

29
Force

• A force is any influence that can change the
  speed or direction of motion of an object.
• Examples of forces the force of gravity (pulling
  us downward), a magnet pulling a piece of iron, a
  person lifting a box, a car pulling a trailer,
  etc.
• Just because a force is applied to an object,
  doesnt mean that the object will move - a force
  has to be of a certain amount to be able to move
  an object
• For example, if you push on a building, you will
  not be able to move it because you dont have
  enough force
• Thus, every force does not result in accelerating
  an object but every acceleration does result from
  a force
• An object continues to accelerate from the
  application of a force only as long as the force
  is not balanced out by any other forces i.e. it
  continues to accelerate as long as there is a net
  force

30
Inertia and Mass

• Inertia is the reluctance of an object to change
  its state of rest or of uniform motion in a
  straight line i.e. inertia is what keeps an
  object at rest or an object at constant speed in
  a straight line
• For example, when a car starts, you feel like you
  are being pushed back in your seat because your
  body wants to remain at rest, and, when a car
  stops, you feel like you are being pushed forward
  because your body wants to remain in motion
• The mass of an object is the amount of matter it
  contains. Mass is the property of matter that
  shows itself as inertia if something has more
  mass, it has more inertia (i.e., a bowling ball
  has more inertia than a baseball because it has
  more mass than a baseball). In the SI System, the
  unit for mass is the kilogram (kg).

31
The Second Law of Motion

• The greater the force applied to an object, the
  greater the acceleration of that object from that
  force AND the smaller the force applied to an
  object, the smaller the acceleration of that
  object from that force
• Thus, the acceleration is directly proportional
  to the force that is, if the force is doubled,
  the acceleration is also doubled
• If the same force is applied to two objects with
  different masses, the object with the smaller
  mass will have a greater acceleration than the
  object with the larger mass
• Thus, the acceleration is inversely proportional
  to the mass that is, if the mass is doubled,
  the acceleration is halved
• These two statements give the following equation
• acceleration (a) Force (F) / mass (m) --- a
  F/m
• This equation can also be expressed in terms of
  the force
• Force (F) mass (m) x acceleration (a) --- F
  ma
• The direction of the acceleration is always in
  the direction of the force applied to it i.e.,
  so if the force is applied to the east, the
  object the force is applied to will accelerate to
  the east
• The Second Law of Motion The net force on an
  object equals the product of the mass and the
  acceleration of the object. The direction of the
  force is the same as that of the acceleration.

32
The Newton

• The Force is equal to the mass times the
  acceleration F ma
• The units for force are therefore the units of
  mass times the units of acceleration (kg) x
  (m/s2)
• The Newton (N) is the unit for force and it is
  therefore equal to (kg) x (m/s2)
• In the British System, the unit of force is the
  pound
• 1 N 0.225 lb OR 1 lb 4.45 N

33
Example Calculations using the Second Law of
Motion

• Acceleration Example If a 430N force is applied
  to an object, what is its acceleration if it has
  a mass of 71kg?
• Answer
• 1. Given 340N, 71kg
• 2. Looking for acceleration
• 3. Equation a F/m
• 4. Solution a F/m 340N/71kg 4.8m/s2

• Force Example What is the force on an object
  that has an acceleration of 1.4m/s2 and a mass of
  29kg?
• Answer
• 1. Given 1.4 m/s2, 29kg
• 2. Looking for force
• 3. Equation F ma
• 4. Solution F ma 29kg x 1.4 m/s2 41N

34
Weight and Mass

• The weight of an object is the force with which
  it is attracted by the earths gravitational pull
• For example, if you weigh 140 pounds, the earth
  (i.e. gravity) is pulling you down with a force
  of 140 pounds
• Thus, weight is different than mass, which is the
  amount of matter it contains i.e., the weight
  of an object depends on the gravity, while the
  mass doesnt thus, your mass is the same on
  every planet, but your weight is different on
  every planet because the gravity on every planet
  is different
• When an object is at the earths surface, an
  objects weight (w) is the force that is exerted
  on it. This force of gravity causes an object to
  fall with constant acceleration due to gravity or
  g 9.8 m/s2, as long as no other force acts on
  the object. Thus, if we start with F ma and
  substitute w for F and g for a, we obtain the
  equation for weight
• weight (w) mass (m) x acceleration of gravity
  (g)
• This equation shows weight is directly
  proportional to mass i.e., if the mass is
  greater, the weight is greater AND if the mass is
  less, the weight is less
• The weight of an object depends on where the
  object is located i.e. the pull of gravity is
  not the same everywhere on the earth
• The pull of gravity is greater at sea level than
  on a mountaintop AND the pull of gravity is
  greater near the north and south pole than at the
  equator

35
Example Calculations using Weight

• Example What is the weight of an object that has
  a mass of 54kg?
• Answer
• 1. Given 54kg
• 2. Looking for weight
• 3. Equation w mg
• 4. Solution w mg 54kg x 9.8 m/s2 529N

36
The Third Law of Motion

• If you push on a heavy object and it doesnt
  move, it is because it is pushing back on you
  with an equal force. The object stays in place
  because there is an opposing force of friction
  between the object and the floor it is on. You
  stay in place because, just like on the object,
  there is an opposing force of friction between
  you and the floor you are on.
• If you push on a heavy object that is on a
  surface without friction, the object moves in the
  direction that you push it. Since the object also
  pushes back on you, you will move in the opposite
  direction that you pushed the object in i.e. in
  the direction that the object pushed on you.
• The Third Law of Motion When one object exerts
  a force on a second object, the second object
  exerts an equal force in the opposite direction
  on the second object.
• In other words, the Third Law of Motion states
  that NO force occurs singly i.e. all forces
  occur in pairs
• For example, you push downward on the earth with
  the same force that the earth is pushing up on
  you with. Also, a fruit that falls from a tree
  due to the pull that the earth has on it has an
  equal pull on the earth, even though it doesnt
  appear so since the mass of the fruit is so small
  when compared to the mass of the earth.
• The Third Law of Motion applies to TWO forces
  that are on TWO different objects
• On one of the objects, there is the action force.
  This is the force that the first object exerts on
  the second object.
• The force that the second object exerts on the
  first object is called the reaction force, which
  is opposite in direction to the action force.
• This law allows us to walk walking is not due
  to you pushing on the earth, but instead on the
  earth pushing back on you i.e. as you move
  forward, the earth is actually moving backward,
  even though the amount is so small when you
  compare it to its mass

37
Circular Motion

• The moon circles around the earth and the planets
  circle round the sun because of a force being
  applied to them
• An inward force must be applied to keep an object
  moving in a curved path this force is called
  centripetal force
• The centripetal force points towards the center
  of the objects curved path thus, the force is
  perpendicular (at a right angle to) the direction
  in which the object is moving

38
Centripetal Force

• Centripetal force is calculated from the
  following equation
• Fc mv2/r
• where Fc is the centripetal force, m is the mass,
  v is the speed, and r is the radius of the circle
• The equation for centripetal force shows that it
  is directly proportional to mass and speed and
  inversely proportional to the radius of the
  circle
• The greater the mass of an object, the greater
  the force needed to keep it in circular motion
• The greater the speed of an object, the greater
  the force needed to keep it in circular motion
• The larger the radius of the circle (i.e., the
  larger the circle), less force is needed to keep
  it in circular motion

39
Example Calculations using Centripetal Force

• Example What is the centripetal force on an
  object that is moving in a circle with a radius
  of 4.7m if its mass is 80kg and its velocity is
  42m/s?
• Answer
• Given 4.7m, 80kg, 42m/s
• Looking for centripetal force
• Equation Fc mv2/r
• Solution Fc mv2/r ((80kg)(42m/s)2)/4.7m
  30026N

40
Newtons Path to the Law of Gravity

• If the planets orbit is a circle, with the sun
  at the center of the circle, the centripetal
  force on the planet is from the sun. Thus, the
  force of gravity between the planet and the sun
  acts along a line between them. Keplers Second
  Law, which concludes that a planet travels faster
  when it is near the sun and slower when it is
  away from the sun, was used by Newton to verify
  that this conclusion works even though the
  planets are in elliptical orbits.
• To find the force of gravity between the planet
  and the sun, Newton combined Keplers Third Law,
  which states the following (period of a
  planet)2/ (average orbit radius)3 is the same
  value for every planet, with the equation for
  centripetal force. He found that the force of
  gravity (F) varies inversely with the square of
  the distance (R) between the planet and the sun
  (i.e. F varies with 1/R2). Thus, if a planet is
  twice the distance from the sun, the force of
  gravity or the gravitational attraction of the
  planet is 1/22 or 1/4, AND if a planet is half
  the distance from the sun, the force of gravity
  or the gravitational attraction of the planet is
  1/(1/2)2 or 1/(1/4) or 4. These conclusions are
  supported by Keplers First Law, which states
  that the planets move in ellipses around the sun,
  with the sun at the center of the ellipse,
  because only a force like the one described here
  can keep the planets moving in a circular orbit
• The final step was from Galileos work on falling
  bodies, which states that an object in free fall
  at the earths surface has the acceleration g.
  Because of this, the objects weight, which is
  the force of gravity on it, is proportional to
  its mass since w mg. In his Third Law of
  Motion, he states that every action force has a
  reaction force i.e. if the earth attracts an
  object, that means that the object is attracting
  the earth. Since the earths attraction for the
  object i.e. the objects weight depends on
  the objects mass (w mg), the objects
  attraction for the earth also depends on the
  earths mass. Thus, the gravitational force
  between two bodies is proportional to the masses
  of both of the bodies.
• To summarize Every object in the universe
  attracts every other object with a force
  proportional to the square of the distance
  between them

41
The Equation of Newtons Law of Gravity

• The Law of Gravity can be shown in the following
  equation
• F (Gm1m2)/R2
• where F is the gravitational force, G is a
  constant of nature (i.e. the same value
  everywhere), which equals 6.670 x 10-11 Nm2/kg2,
  m1 is the mass of one of the objects, m2 is the
  mass of the other object, and R is the distance
  between the two objects
• R is measured from the objects center of mass.
  The location of the center of mass of an object
  depends on its shape and how its mass is
  distributed.
• The equation from the Law of Gravity shows that
  the force is directly proportional to the masses
  of the objects and inversely proportional to the
  distance between the two objects (R)
• Thus, if the mass of the one of the objects
  increases, then the force of gravity is larger
• Also, if the distance between the two objects
  increases, the force of gravity is smaller
• On the earth, an astronaut weighs 600N. When the
  astronaut is 100 times farther from the earth
  (i.e. 640,000km), the astronaut weights 1/10O2 or
  1/10,000 times as much as she did on earth, i.e.
  0.06N, which is the weight of a cigar on the
  earths surface

42
Example Calculations using Newtons Law of Gravity

• Example What is the gravitational force between
  an object that weighs 45kg and an object that
  weighs 329kg if the distance between the two
  objects is 43m?
• Answer
• Given 45kg, 329kg, 43m
• Looking for gravitational force
• Equation F (Gm1m2)/R2
• Solution F (Gm1m2)/R2 ((6.670x10-11
  Nm2/kg2)(45kg)(329kg))/(43m)2 5.3 x 10-10N

43
Artificial Satellites

• The first artificial satellite, Sputnik I, was
  launched in 1957 by the Soviet Union since
  then, many other artificial satellites were
  launched since then, mostly by the United States
  and the former Soviet Union
• In 1961, the first person, a Soviet cosmonaut,
  circled the earth at a height of 240 km since
  then, many other people have been in orbit
• The satellites that are circling the earth range
  from 130 km above the earth to 36,000 km above
  the earth
• The satellites that are closer to the earth are
  called eyes in the sky. These satellites
  monitor the surface of the earth for military
  purposes and to provide information on weather
  and earth resources (i.e. mineral deposits,
  crops, water)
• There are 24 satellites at 17,600 km above the
  earth that are used for the Global Positioning
  System (GPS) that was developed by the United
  States.
• GPS allows a person to find their position,
  including their altitude, to within a few meters,
  anywhere in the world at any time
• The satellites that are farthest from the earth
  circle the earth only once a day they remain in
  place indefinitely over a particular location on
  the earth. They are in what are called
  geostationary orbits. These satellites can see
  large areas of the earths surface. About 200 of
  these satellites are used to relay telephone,
  data, and television communications from one
  place to another

44
Why Satellites Dont Fall Down

• Satellites are actually falling down, but they
  are falling down at such a rate that allows them
  to maintain a stable orbit around the earth, just
  like the moon, which is a natural satellite
• The word stable for the orbit of a satellite is
  a relative word because the satellite will
  eventually fall down because there is friction
  due to the extremely thin atmosphere present
  where the satellite is located. The length of
  time a satellite spends circling the earth ranges
  from a matter of days to hundreds of years.
• The gravitational force on a satellite is the
  same as the gravitational force on us i.e. its
  gravitational force is its weight, mg, where g is
  the acceleration of gravity at the satellites
  location above the earth. The value of g
  decreases with increasing distance from the earth
  i.e. a satellite that has an orbit father from
  the earth has a lower g than a satellite that has
  an orbit closer to the earth.

45
The Speed Needed to Orbit the Earth

• For a satellite to circle the earth, it needs a
  centripetal force (Fc mv2/r) and since the
  force of gravity of the earth (w mg) provides
  the centripetal force, the following can be
  written
• Fc w, so (mv2)/r mg
• If we solve this equation for v2, by multiplying
  by r and dividing by m, we obtain the equation
  v2 rg, so v v(rg), where, in this case, v is
  the speed of the satellite
• The second equation shows that the mass of the
  satellite does not matter for the speed of the
  satellite
• For a satellite to circle the earth only a few
  kilometers above the earth, the satellite needs a
  speed of about 28,400 km/hour. If the speed of
  the satellite is less than this, the satellite
  would just fall back to earth. If the speed of
  the satellite is more than this, the satellite
  would orbit the earth in an elliptical orbit
  instead of a circular one. For a satellite that
  initially has an elliptical orbit to have a
  circular orbit, a small rocket motor gives it a
  push at the required distance from the earth.
• If the speed of an object is greater than 40,000
  km/hour, it can escape entirely from the earth.
  This speed is the escape speed that an object
  needs to leave the earth. The escape speed is the
  speed that is required for something to leave the
  gravitational influence of an astronomical body
  permanently. The ratio between the escape speed
  and the minimum orbit speed is v2 or 1.41.