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Mechanics

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• 1. Topic 2 Mechanics
• 2. Contents
• Kinematic Concepts: Displacement Speed vs. Velocity Acceleration Changing Units Instantaneous vs. Average
• Frames of Reference Graphical Representation of Position
• Graphical Representation of Velocity
• Graphical Representation of Acceleration
• Equations for Constant Acceleration
• Vertical Motion of Objects
• Air Resistance
• Force and Mass: Springs
• 3. Mechanics
• Mechanics is the study of :
• motion,
• force and ,
• energy.
• 4. Kinematic Concepts
• Kinematics is the part of mechanics that describe s how objects move.
• The why objects move as they do is called dynamics .
• 5. Kinematic Concepts
• Displacement
• When a person moves over a time interval,
• they change their position in space.
• This change in position is called the :
• displacement .
• It is a vector quantity .
• 6. Kinematic Concepts
• The further it moves from its starting position,
• the greater it’s displacement.
• 7. Kinematic Concepts
• If the object moves in the opposite direction ;
• as defined,
• its displacement will be negative.
• 8. Kinematic Concepts
• Displacement is given the symbol s ;
• or sometimes  x ,
• it’s S.I. unit is the (m)etre.
• 9. Kinematic Concepts
• The term distance is a scalar :
• the symbol is d .
• Distance is more useful when purchasing a car.
• The distance travelled is important ;
• while the direction it travelled in is not.
• 10. Kinematic Concepts
• A person who walks 100 m east and then ;
• 100 m west has travelled a distance of ,
• 200 m.
• Their displacement however is 0 m.
• They have ended up at their starting point.
• Distance vs Displacement
• 11. Kinematic Concepts
• Speed vs Velocity
• Although speed and velocity are used interchangeably in everyday life,
• both terms have specific meanings.
• 12. Kinematic Concepts
• Speed is defined as ;
• the distance travelled by an object in a given time interval.
• This will give us the average speed.
• Mathematically, it can be represented by:
• 13. Kinematic Concepts
• As distance is one of the variables,
• speed must be a scalar.
• Velocity is used to signify both magnitude and direction hence ;
• it is a vector.
• 14. Kinematic Concepts
• The average velocity is defined as :
• the change in position of the object ;
• in a given time interval.
• Mathematically, it can be described as:
• 15. Kinematic Concepts
• The S.I. unit for both speed and velocity is m s -1 .
• Always include a direction when giving the value of the velocity.
• 16. Kinematic Concepts
• Acceleration.
• An object whose velocity is changing is :
• accelerating.
• Acceleration is defined as :
• the rate of change in velocity.
• 17. Kinematic Concepts
• Which cars below are acceleration and which cars are traveling at constant velocity?
• Which car accelerates fastest?
• 18. Kinematic Concepts
• The red car is traveling at constant velocity.
• The blue car is accelerating fastest.
• Its rate of change of velocity is greatest.
• 19. Kinematic Concepts
• Since an object always accelerates in a given direction,
• acceleration is a vector quantity.
• The S.I. units are m s -2
• 20. Kinematic Concepts
• Changing Units
• When using kinematic equations,
• the data given is not always given in S.I. units.
• The data needs to be converted to S.I. Units;
• before they can be substituted into an equation.
• 21. Kinematic Concepts
• An example is speed.
• Very often speed is given in km h -1 .
• The kilometres need to be converted to metres and the hours into seconds.
• There are 1000 m in 1 km
• 3600 s in 1 hr (60 x 60).
• 22. Kinematic Concepts
• There is doubt over whether the correct answer should be :
• 30 m s -1 ,
• 28 m s -1 or ,
• 27.7 m s -1
• due to the number of significant figures given.
• 23. Kinematic Concepts
• To avoid this problem, always give your data using scientific notation.
• In the above example,
• 1.00 x 10 2 m s -1 would eliminate the problem
• hence the answer would be 27.7 m s -1 .
• 24. Kinematic Concepts
• Instantaneous vs Average
• If you drive a car for 240 km in 3 hrs,
• your average speed is 80 km h -1 .
• It is unlikely that for every part of the journey,
• you would be travelling at 80 km h -1 .
• At each instant, your speed would change.
• 25. Kinematic Concepts
• The speedometer in the car gives ;
• the instantaneous speed.
• The instantaneous speed is defined as :
• the average speed over an indefinitely short time interval .
• 26. Kinematic Concepts
• The same definition can be used for velocity.
• The formula for velocity needs to be changed to :
• accommodate the difference between average ,
• and instantaneous velocity.
• 27. Kinematic Concepts
• Average velocity:
• Instantaneous velocity:
• 28. Kinematic Concepts
• As  t becomes very small,
• approaching zero,
•  s approaches zero as well.
• However, the ratio approaches a definite value.
• 29. Kinematic Concepts
• This definite value is known as the instantaneous velocity.
• The symbol for instantaneous velocity is :
• v no av or bar above the v .
• 30. Kinematic Concepts
• The same procedure can be used for acceleration and so the equations become:
• Average acceleration:
• Instantaneous acceleration:
• 31. Kinematic Concepts
• Relationship between Velocity & Acceleration
• 32. Kinematic Concepts
• Frames of Reference
• The simple question ‘how fast are you moving’ is quite complex.
• In your seat now, you are moving at :
• 0 m s -1 relative to the room.
• If the room were on the equator,
• you would be moving at 1700 km h -1 .
• 33. Kinematic Concepts
• At the pole you would be spinning on the spot.
• This however is not true relative to the Sun.
• Two planes moving at 1000 km h -1 relative to the ground :
• may be stationary relative to each other.
• 34. Kinematic Concepts
• All motion is relative .
• We must choose our frame of reference in each description of motion we give.
• Relative Motion (frame of reference)
• 35. Kinematic Concepts
• Usually the motion of objects is considered relative to the earth .
• This is called:
• the inertial frame of reference .
• Unless otherwise stated,
• this will be assumed.
• Inertial Frame
• 36. Graphical Representation of Motion
• Graphical Representation of Position
• Multiflash photographs of an object in motion ;
• can be taken and ,
• data collected from it.
• An example of a multiflash photo is the toy car on the next slide.
• 37. Graphical Representation of Motion
• To take this photo ;
• the film needs to be exposed for a period of time in darkness ,
• with a strobe flashing at a known rate.
• 38. Graphical Representation of Motion
• From the data collected,
• graphs can be drawn.
• By plotting position on the vertical axis and ;
• time on the horizontal,
• the graph will look like:
• 39. Graphical Representation of Motion
• 40. Graphical Representation of Motion
• A number of facts can be ascertained from this graph.
• As the line is straight,
• the change in position per unit time, is constant.
• This means the velocity is constant.
• 41. Graphical Representation of Motion
• 2. The magnitude of the velocity can be obtained by the slope;
• 42. Graphical Representation of Motion
• If the graph is steep, it means :
• there is a greater change in position ,
• per unit time and the object ,
• is moving relatively fast.
• 43. Graphical Representation of Motion
• 44. Graphical Representation of Motion
• What happens when two cars traveling at different speeds but with constant velocity?
• 45. Graphical Representation of Motion
• If the graph is horizontal :
• the object is stationary.
• If the slope is negative :
• the object is moving back towards its starting position with ,
• constant velocity .
• 46. Graphical Representation of Motion
• 47. Graphical Representation of Motion
• In graph a);
• The person has moved at constant velocity over the interval t 1 to point p
• They then remained stationary for the period t 2
• The person then returned to the original starting position at a constant, but slower velocity over the interval t 3 .
• 48. Graphical Representation of Motion
• In graph b);
• The person has moved at constant velocity over the interval t 1 to point p
• They then remained stationary for the period t 2
• The person then continued in the original direction for the same distance p as in interval t 1 at a constant, but slower velocity over the interval t 3 .
• 49. Graphical Representation of Motion
• Graphical Representation of Velocity
• Consider graph a) from above.
• From the information from the graph, a graph of the person’s velocity can be drawn.
• 50. Graphical Representation of Motion
• 51. Graphical Representation of Motion
• The graph is slightly idealised as :
• the person could not travel at the constant velocity ,
• at every instant of the journey.
• 52. Graphical Representation of Motion
• In a real situation ;
• velocity does not change in zero time and a more likely description is ,
• the one shown above.
• 53. Graphical Representation of Motion
• What about v/t graphs for the two cars previously?
• 54. Graphical Representation of Motion
• V vs t graph animation
• 55. Graphical Representation of Motion
• Multiflash photography can be used to obtain :
• direct values of average velocity.
• This is because time intervals are :
• very short and ,
• constant.
• 56. Graphical Representation of Motion
• The distance from one image to the next is :
• the change in position of the object in ,
• a specific interval of time.
• Changes in position in equal intervals of time are :
• direct measures of the average velocity ,
• over those intervals.
• 57. Graphical Representation of Motion
• The area under the Velocity vs. Time graph can :
• be used to obtain further information.
• This information relates to :
• the change in position of the object.
• 58. Graphical Representation of Motion
• Rearranging this equation :
• change in position is given by the product;
• average velocity x time taken
• The graph on the next slide deals with constant velocity .
• 59. Graphical Representation of Motion
• 60. Graphical Representation of Motion
• As the velocity is constant ;
• the average velocity must be v .
• Hence, the change in position :
• will be vt .
• This is also equal to the :
• area under graph in time interval t .
• This relationship holds for all v/t graphs ;
• as in the example on the next slide :
• 61. Graphical Representation of Motion
• 62. Graphical Representation of Motion
• The motion in the graph above describes :
• motion under non-zero constant acceleration
• eg motion under the force of gravity.
• 63. Graphical Representation of Motion
• Since velocity changes in a regular way ;
• the average velocity is ,
• the average of the initial velocity and ,
• the final velocity.
• The horizontal line through P represents this.
• 64. Graphical Representation of Motion
• The areas a and b are equal :
• one being under the graph and ,
• the other, above the line.
• 65. Graphical Representation of Motion
• The product of v av t is :
• the area under the graph and ,
• gives the change in position.
• 66. Graphical Representation of Motion
• Another way to calculate the change in position is :
• to divide the area under the graph into ,
• a rectangle and triangle and ,
• 67. Graphical Representation of Motion
• Any area below the axis corresponds to :
• motion in the opposite direction where ,
• the change in position is opposite in sign.
• 68. Graphical Representation of Motion
• Graphical Representation of Acceleration
• 69. Graphical Representation of Motion
• In the example, the slope,
• and therefore the rate at which velocity is changing,
• is constant.
• Constant Acceleration Graph
• 70. Graphical Representation of Motion
• Acceleration is defined as :
• the rate of change of velocity.
• Mathematically, this can be written as:
• 71. Graphical Representation of Motion
• The slope of the graph is given by: rise/run .
• In the example , the rise =  v and the run as  t.
• 72. Graphical Representation of Motion
• This gives the formula:
• This corresponds to our formula for acceleration.
• 73. Graphical Representation of Motion
• This corresponds to our formula for acceleration.
• The slope of a v/t graph gives :
• acceleration.
• Acceleration is measured in m s -2 and ;
• is a vector quantity so ,
• direction must always be included.
• 74. Graphical Representation of Motion
• A car is stationary at the lights when the lights change to green.
• Another car is moving when the lights turn green.
• What is the displacement of each car after 3 seconds?
• 75. Graphical Representation of Motion
• 76. Graphical Representation of Motion
• Red car
• Area of triangle = ½ b x h
• Displacement = ½ x 3 x 12
• Displacement = 18 m
• 77. Graphical Representation of Motion
• Blue car
• Area of rectangle = b x h
• Displacement = 3 x 10
• Displacement = 30 m
• 78. Graphical Representation of Motion
• Watch the animation again.
• What is the acceleration of the red car after 3 seconds?
• 79. Graphical Representation of Motion
• Slope = ryse/run
• Acceleration = 12/3
• Acceleration = 4 m s -1
• Watch the animation again and determine the displacement of both cars after 9 seconds.
• 80. Graphical Representation of Motion
• 81. Graphical Representation of Motion
• Red Car
• Displacement = area of triangle + area of rectangle.
• Displacement = (½ x 3 x 12) + (9 x 12)
• Displacement =18 + 72
• Displacement = 90 m
• 82. Graphical Representation of Motion
• Blue Car
• Displacement = area of rectangle
• Displacement = 9 x 10
• Displacement = 90 m
• Watch the animation again.
• When do the two cars pass each other?
• Does it agree with your calculations?
• 83. Graphical Representation of Motion
• 84. Graphical Representation of Motion
• Graphs of position, velocity and acceleration can be drawn for the same object.
• 85. Graphical Representation of Motion
• What happens if it is traveling backwards?
• 86. Graphical Representation of Motion
• What happens when it is being pushed forward?
• 87. Graphical Representation of Motion
• Traveling in the opposite direction?
• 88. Graphical Representation of Motion
• What happens when it is pulled backward?
• 89. Graphical Representation of Motion
• Traveling in the opposite direction?
• 90. Graphical Representation of Motion
• Very complicated motion can be studied using graphs.
• Watch the two stage rocket as it is launched, run out of fuel and returns to Earth.
• 91. Graphical Representation of Motion
• 92. Graphical Representation of Motion
• The Moving Man
• Motion Graphs
• If you would like more practice at drawing motion graphs diagrams, try this web site:
• http://www.glenbrook.k12.il.us/gbssci/phys/shwave/graph.html
• 93. Equations for Uniformly Accelerated Motion
• The previous equations used can :
• be applied to all types of motion.
• However, when acceleration is constant :
• our mathematical description can be taken further.
• 94. Equations for Uniformly Accelerated Motion
• 95. Equations for Uniformly Accelerated Motion
• In the graph on the previous slide , as the slope is constant ;
• the acceleration is also constant.
• If we let u and v be the velocity at the start and end of the time interval t ;
• average velocity can be described as;
• 96. Equations for Uniformly Accelerated Motion
• and from previously,
• Comparing these equations, we can say:
• 97. Equations for Uniformly Accelerated Motion
• The average acceleration can also be given as:
• Rearranging the equation gives:
• 98. Equations for Uniformly Accelerated Motion
• These two equations describe motion :
• at constant acceleration ,
• in terms of five variables;
• u , v , t , s , a .
• 99. Equations for Uniformly Accelerated Motion
• Equation  does not use a and ;
•  does not use s .
• Using algebra, we can derive 3 more equations,
• each one not using one variable listed above.
• 100. Equations for Uniformly Accelerated Motion
• rearranging gives;
• 101. Equations for Uniformly Accelerated Motion
• Equation 2: v = u + at rearranged becomes;
• substituting for t we have:
• 102. Equations for Uniformly Accelerated Motion
• v 2 - u 2 = 2as 
• 103. Equations for Uniformly Accelerated Motion
• Equation  v = u + at
• Substituting for v from  we get;
• 104. Equations for Uniformly Accelerated Motion s=ut + ½at 2 
• 105. Equations for Uniformly Accelerated Motion
• Exercise:
• Try and derive  , which is independent of u .
• 106. Equations for Uniformly Accelerated Motion VARIABLES EQUATION u v t s a u v t s a v = u + at u v t s a v 2 - u 2 = 2as u v t s a u v t s a
• 107. Vertical Motion of Objects
• The Earth, near its surface, has a uniform gravitational field.
• Objects that move perpendicular to the Earth’s surface,
• move parallel to the gravitational field.
• 108. Vertical Motion of Objects
• The acceleration experienced by the object will be constant.
• a = 9.81 m s -2 down.
• If the object is moving towards the Earth,
• the object’s speed will increase at the rate of,
• 9.81 m s -1 every second.
• This is assuming that air resistance is negligible.
• 109. Vertical Motion of Objects
• 110. Vertical Motion of Objects
• True or False?
• 1. The elephant encounters a smaller force of air resistance than the feather and therefore falls faster.
• 2. The elephant has a greater acceleration of gravity than the feather and therefore falls faster.
• 111. Vertical Motion of Objects
• 3. Both elephant and feather have the same force of gravity, yet the acceleration of gravity is greatest for the elephant.
• 4. Both elephant and feather have the same force of gravity, yet the feather experiences a greater air resistance.
• 112. Vertical Motion of Objects
• 5. Each object experiences the same amount of air resistance, yet the elephant experiences the greatest force of gravity.
• 6. Each object experiences the same amount of air resistance, yet the feather experiences the greatest force of gravity.
• 113. Vertical Motion of Objects
• 7. The feather weighs more than the elephant, and therefore will not accelerate as rapidly as the elephant.
• 8. Both elephant and feather weigh the same amount, yet the greater mass of the feather leads to a smaller acceleration.
• 114. Vertical Motion of Objects
• 9. The elephant experiences less air resistance and than the feather and thus reaches a larger terminal velocity.
• 10. The feather experiences more air resistance than the elephant and thus reaches a smaller terminal velocity.
• 115. Vertical Motion of Objects
• 11. The elephant and the feather encounter the same amount of air resistance, yet the elephant has a greater terminal velocity.
• If you answered True to any of these questions, you need to review your understanding.
• 116. Vertical Motion of Objects
• 117. Vertical Motion of Objects
• Objects which move vertically upwards:
• will slow down at the rate of 9.81 m s -1 ,
• every second until it is stationary.
• It will then start to accelerate towards the earth at:
• the rate of 9.81 m s -1 every second.
• Vertical Motion
• 118. Air Resistance
• Objects falling in the Earth’s uniform gravitational field have two opposing forces acting on it.
• Gravity acts towards the Earth, pulling the object downward.
• Any resistance force opposes motion.
• 119. Air Resistance
• This means, in this case ;
• air resistance acts upwards.
• The faster the object falls ;
• the greater the air resistance.
• As the object accelerates under the force of gravity ;
• the greater the air resistance.
• 120. Air Resistance
• This slows the rate at which the object ;
• accelerates towards the earth.
• Eventually, the two forces cancel each other out.
• This means there is no acceleration and ;
• the velocity becomes constant.
• This is known as terminal velocity.
• 121. Air Resistance
• The terminal velocity is different :
• for different objects.
• Sky diver’s have a terminal velocity of :
• about 150 to 200 km h -1 depending on ;
• the mass of the sky diver and ,
• their orientation.
• 122. Air Resistance
• Sky divers tend to try to:
• increase the air resistance thereby ,
• reducing the terminal velocity.
• This gives them a longer free fall.
• The parachute :
• greatly increases air resistance and ,
• cuts the terminal velocity to ,
• between 15 and 20 km h -1 .
• 123. Air Resistance
• 124. Forces and Dynamics
• To maintain the velocity of an object such as a wheelbarrow :
• a push or pull is required.
• To maintain acceleration :
• either by changing the speed of an object or changing its direction,
• a push or pull is required.
• 125. Forces and Dynamics
• The name we use for this push or pull is :
• force .
• A force is required to :
• keep things moving or ,
• change their motion.
• A force can also be used to :
• deform an object.
• This happens when a tennis racquet hits a tennis ball.
• 126. Forces and Dynamics
• Using an air track, we can eliminate (or very nearly) ;
• the force of friction on a glider.
• This leaves the force of gravity and :
• the force of the hand that pushes a glider.
• 127. Forces and Dynamics
• How can we determine the effect of a single force when ;
• there are two forces acting?
• We must make an assumption.
• Gravitational forces being vertical :
• have no effect on motion ,
• in the horizontal plane.
• 128. Forces and Dynamics
• Recall the air track practical :
• once the hand released the glider,
• it moved with a constant velocity.
• This indicates that the assumption is true.
• 129. Forces and Dynamics
• There are cases when more than one force acts on a object.
• These forces may oppose each other:
• Pushing an object forward and,
• Friction backward
• Or a tug of war
• 130. Forces and Dynamics
• Or they may act at some other angle:
• Running for the football and,
• being hip and shouldered.
• A diagram can be drawn to show these forces.
• It is called a Free-body Diagram.
• 131. Forces and Dynamics
• Consider a statue resting on a table.
• What forces are acting on it?
• As it is at rest:
• there are no unbalanced forces.
• Gravity ( F g ) is acting downwards.
• What force balances gravity?
• 132. Forces and Dynamics
• The table exerts an upward force.
• The table is compressed by the statue
• Due to its elasticity it pushes upwards.
• This is called a contact force.
• As it occurs when two objects are in contact.
• 133. Forces and Dynamics
• When the contact force acts:
• perpendicular to the common surface of contact, it is called,
• the Normal Force ( F N ).
• A diagram of the objects and their forces can be drawn.
• 134. Forces and Dynamics
• 135. Forces and Dynamics
• We are interested in the forces and so:
• we will only include vectors for the forces.
F g F N
• 136. Forces and Dynamics
• Notice:
• The vectors are of the same length:
• Indicating same magnitude of force.
• The vectors are in opposite directions:
• Indicating the forces oppose each other.
• The vectors have been labeled:
• with appropriate symbols.
• 137. Forces and Dynamics
• What happens when you add a third force?
• Consider a box at rest on a table.
• What forces are acting on it?
• F g and:
• F N
• 138. Forces and Dynamics F g F N
• 139. Forces and Dynamics
• What if you were to push down on the box with a force of 40 N?
F N F g 40 N
• 140. Forces and Dynamics
• Notice that the combined length of:
• the F g and 40 N vectors,
• equal the F N vector.
• This indicates the box is stationary.
• What happens if you pull up with a force of 40 N?
• 141. Forces and Dynamics F g F N 40 N
• 142. Forces and Dynamics
• Notice that the combined length of:
• the F N and 40 N vectors,
• equal the F g vector.
• This indicates the box is stationary.
• 143. Forces and Dynamics
• What forces act on a shopping trolley?
• 144. Forces and Dynamics
• 145. Forces and Dynamics
• As F N = F g
• The force making the trolley move is equal to:
• F p = Force supplied by the person.
• This is an unbalanced force and:
• it will cause an acceleration.
• 146. Forces and Dynamics
• What happens when the ( 10 kg) box in the earlier example is:
• connected to another box (12 kg),
• by a string and both,
• are pulled along a table?
• What do the Free Body Diagrams look like?
• 147. Forces and Dynamics
• 148. Forces and Dynamics Box 1 F N F g T F P
• 149. Forces and Dynamics
• Notice:
• The vectors F g and F N are of the same length:
• Indicating same magnitude of force.
• No vertical motion
• The vector F p is longer than T indicating:
• acceleration to the right
• 150. Forces and Dynamics Box 2 F N F g T
• 151. Forces and Dynamics
• Free Body Diagram
• Another one
• If you would like more practice at drawing free body diagrams, try this web site
• http://www.glenbrook.k12.il.us/gbssci/phys/shwave/fbd.html
• 152. Forces and Dynamics
• Suppose you wanted to see what was in the box in the earlier example.
• You pull it towards you:
• using a string attached with,
• a force of 40 N at,
• an angle of 30 o above the horizontal..
• 153. Forces and Dynamics
• 154. Forces and Dynamics
• This becomes more difficult to analyse .
• Notice F N  F g
• This is because some of the force supplied by the person is:
• acting in an upward direction.
• Forces can be resolved into components.
• 155. Forces and Dynamics
• Forces can be resolved into two components :
• horizontal and vertical.
• 156. Forces and Dynamics
• This is useful when forces are applied at an angle but :
• the effective force is in a particular direction ,
• such as pushing a roller
• 157. Forces and Dynamics
• Forces can combine to give very different results.
• A 10 N block hanging vertically from one horizontal string :
• when measured by a spring balance would read ,
• 10N.
• 158. Forces and Dynamics
• 159. Forces and Dynamics
• As the block is stationary :
• the force of the scale pulling up balances ,
• the force downwards supplied by ,
• the weight.
• If 2 spring balances support the block :
• the total weight will be the same.
• 160. Forces and Dynamics
• However each spring balance will support equal amounts :
• ie 5N.
• 161. Forces and Dynamics
• Again the weight down is balanced by :
• the two springs supplying a force up.
• In all situations the two forces vectors :
• one vertically downward and ,
• the other vertically upwards ,
• must balance.
• 162. Forces and Dynamics
• This looks logical for a vertical orientation but how do they work for non-vertical situations?
• 163. Forces and Dynamics
• The vertically up vector must still equal 10 N and so by vector addition :
• the two scales will read more than 5 N each ,
• perhaps 10 N.
• If the angle is increased to 60 o from the vertical :
• 120 o between the scales ,
• The reading is much higher.
• 164. Forces and Dynamics
• 165. Forces and Dynamics
• If we continue to do this until the angle is 90 o ,
• the horizontal rope must support a force that is ,
• much greater than the original weight.
• 166. Forces and Dynamics
• Gymnasts who hold their arms out horizontally from the rings :
• must supply a force that is considerably greater than their own body weight.
• This is an extreme test of strength.
• 167. Forces and Dynamics
• To determine exact values for the components:
• Trigonometry must be used.
• 168. Forces and Dynamics
• F vert = opposite side
• F horiz = adjacent side
• F = applied force
• sin  = opposite/hypotenuse
• Opposite = F vert = F sin 
• Adjacent = F horiz = F cos 
• 169. Forces and Dynamics
• Springs
• If you hang a weight from a spring:
• apply a force,
• it stretches.
• It stretches more
• Remove the weights:
• It returns to its original length.
• 170. Forces and Dynamics
• The spring is said to be elastic.
• By stretching a spring:
• or compressing it,
• The amount it is stretched is:
• Directly proportional to the force applied.
• This was first noticed by the British Physicist:
• Robert Hooke
• In the 17 th century
• 171. Forces and Dynamics
• It is now remembered as Hooke’s Law.
• The applied force F and:
• The extension (or compression) x
• Can be represented mathematically.
• F  x
• A graph of this can be shown.
• 172. Forces and Dynamics F x
• 173. Forces and Dynamics
• Compare the line to the equation of a straight line:
• y = mx + c
• y = F
• x = x
• c = 0 (as through the origin)
• To turn the proportionality into an equation:
• 174. Forces and Dynamics
• F = mx
• The slope is constant and so is given a special symbol:
• k
• All constants in Physics have special symbols.
• The equation now becomes:
• F = kx
• 175. Forces and Dynamics
• The full form of the equation is:
• F = -kx
• This is because the force is a restoring force.
• It always acts to try an return the spring to the original length.
• Hookes Law
• 176. Forces and Dynamics
• You will need to be able to:
• draw a graph
• Given data
• Determine the spring constant k :
• For a particular spring.
• 177. Forces and Dynamics
• Determine the force required to:
• extend a spring by a certain amount
• Determine the amount a spring is stretched by:
• for a given force.
• 178. Forces and Dynamics
• An object that moves with constant velocity :
• requires no force.
• Although this is one experiment, it can be shown to be true :
• in all situations
• whether it is a person ice-skating or ,
• a person slipping on a banana peel.
• 179. Forces and Dynamics
• Newton investigated this phenomenon,
• which he embodied in his first law:
• Provided no external force acts, the velocity of any object will remain constant unless an unbalanced force acts upon the object.
• 180. Forces and Dynamics
• A body is resistant to change.
• This resistance of a body to change is called inertia .
• Does this hold for a stationary object?
• A stationary object has zero velocity :
• and so it will remain at zero.
• 181. Forces and Dynamics
• Plates on a tablecloth are at rest.
• If you pull the tablecloth quickly enough from under the plates :
• t he small and brief force of friction ,
• between the plates and the tablecloth is ,
• not significant enough to ,
• appreciably move the dishes.
• 182. Forces and Dynamics
• Does this hold for tug of war contest?
• When the two sides are even :
• the forces are equal in magnitude and ,
• opposite direction.
• The resultant force will be zero.
• This implies that the two sides will not move.
• 183. Forces and Dynamics
• Does this apply to a car moving on a straight road ;
• at a constant velocity?
• The car will slow to a stop unless :
• the force applied by the engine continues.
• 184. Forces and Dynamics
• This appears to disobey the law :
• until friction is taken into account.
• We can ignore gravity because :
• it is acting in a vertical direction.
• Friction however is a retarding force acting :
• in the opposite direction to the motion.
• 185. Forces and Dynamics
• This is an example of translational equilibrium.
• Two forces are being applied in opposite directions.
• As the forces cancel each other out,
• no acceleration occurs.
• The object continues to move at:
• a constant velocity.