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Mechanics L2 NCEA Achievement Standard 2.4 Text Book reference: Chapters 7-13 Scalars and Vectors A scalar quantity is one that has a size (or magnitude) only Eg. Mass, energy, time A vector quantity is one that has a size and a direction Eg. Force, velocity, momentum Motion Distance (scalar) Symbol: d Displacement (vector) Symbol: d or s How far in total the object has moved Unit: m How far the object ends up from it’s starting position Unit: m Motion Speed (Scalar) How fast an object is travelling Symbol: s Unit: ms-1 Velocity (Vector) The speed and direction that an object is travelling Unit: ms-1 Symbol: v Motion Velocity is calculated by Dd velocity(v) Dt Where d is displacement, t is time and D means “the change in”. Velocity may refer to either average velocity or instantaneous velocity. Constant velocity means that neither the speed nor the direction of the objects motion is changing. Motion Acceleration (Vector) Symbol: a The rate at which the velocity of an object is changing Unit: ms-2 Acceleration can be calculated by Dv a Dt Acceleration is always in the direction of …………… The concorde flies at an average velocity of 1440 km hr-1. How long in seconds does it take to fly 100km? 1440km hr-1 is 400ms-1 and 100km is 100,000m Using v=d/t t=d/v 100,000/400 and t=250s How far will the concorde fly in a minute? 24km Anyone not sure how to get that? A car travels 500m to the right turns around and travels another 1000m to the left.The car travelled with a uniform speed and the time taken was 150s. Find: total distance travelled 1500m total displacement -500m average speed of the car 10ms-1 average velocity of the car -3.3ms-1 Motion Acceleration is used to describe motion where the object slows down as well as when it speeds up. Sometimes the word deceleration is used. Acceleration is given a negative value when the object is slowing down. Objects are accelerating when their direction changes, even though their speed may remain constant. A car accelerates from 10ms-1 to 20ms-1 in 4.0s Calculate its acceleration A=2.5ms-2 Does everyone know how that was solved? The same car can brake from 20ms-1 to rest in 5.0s Find the acceleration. a= 4.0ms-2 What is wrong with that answer? a=-4.0ms-2 Task • Measure the speed and velocity of your birds. • I need to see times – distances and calculations • The birds seem to move between 3-12 cms-1 Eric the snail moves at 3.0mms-1 N sees a bird and takes off West at 4mms-1. What is his change in velocity? Vectors A vector is drawn as a straight, arrowed line. The arrow points in the direction of the vector The length of the line represents the size of the quantity Vectors Vectors can be multiplied or divided by a scalar This will change the length of the vector A negative scalar will reverse the direction Eg Force F= So 2F= 1/2F= -3F= Vectors Vectors can be added together. This is done by drawing them “head to tail”. The result is a vector called a resultant. The resultant has the same effect as the 2 vectors combined. The order in which they are added does not matter. d1 d2 Eg d1+d2 d1+ d2 Sp p137 Vectors Vectors can be subtracted. This is done by adding a negative vector Order does matter. Eg. v1- v2 v1 v1- v2 v2 -v2 DV DV=Vf-Vi try the ball thing If the velocity changes this means the object is……… If the object is accelerating there must be a ………….. applied in the direction of the acceleration. Working out DV when Non Linear 1. Draw a vector representing each motion. 2. Draw the –Vi vector. 3. Draw a vector diagram of DV=Vf-Vi or DV=Vf +-Vi 4. Using trig and/or pythagaros find the magnitude and DIRECTION ofDV Example on the board Graphs of Motion Distance / diplacement versus time… Displacement(m) B A C D Time (s) E A=Constant velocity (slow) B=Constant velocity (faster) C=Stopped D=Constant velocity (backward) E=Constant velocity (backward past starting point) Graphs of Motion Speed / velocity versus time Velocity (ms-1) B A C D Time (s) E …area under the graph? A=Constant acceleration (low) B=Constant acceleration (high) C=Constant velocity D=Constant deceleration to stop E=Constant acceleration in opposite direction Kinematic Equations To solve problems involving objects moving in straight lines with constant acceleration. Terms used: d=distance/displacement (m) vi=initial velocity (ms-1) vf=final velocity (ms-1) a=acceleration (ms-2) t=time (s) Kinematic Equations vi v f d 2 v f vi at t If you know 3 variables you can work out the other 2 2 1 d vi t at 2 v v 2ad 2 f 2 i Tricks of the Trade It is assumed you know gravity in any problem which involves rising or falling. Look out for Vi=0 or Vf=0 in other words from rest or stops. Make sure you get the signs correct. A rising object will have –acceleration due to gravity acting in the opposite direction to motion. • A grasshopper’s legs extend by 2.0cm in 0.020s when jumping from rest. Assuming the jump is vertical: • What is the average acceleration of the grasshopper while extending it’s legs? • With what velocity does the grasshopper leave the ground? • What is the maximum height the grasshopper can jump? • A flea takes 1.0 millisecond to reach take off speed of 1.2 ms-1 in a jump. • What is it’s average acceleration? • Assuming vertical take off how high does the flea reach? • A jet plane lands on one end of a runway 1.0km long. It’s maximum stopping acceleration is -4.0ms-2 and it takes 20s to come to rest. Does the plane stop in time? Vectors Vectors can be resolved into components. This is done using SOHCAHTOA and/or a2+b2=c2 Vertical F Component 40° Horizontal component A ship sails from Lyttelton and sets a straight course of 130km in a direction N230E from the New Brighton pier. How far North of the pier is the ship? 130 cos 230 = 120km How far east of the pier is the ship? 130 sin230 = 51km A supermarket trolley is pushed with a force of 200N acting at an angle of 400 to the ground. Find the effective horizontal force pushing the trolley along. 200N 400 FH=Fcos =200 cos 400 = 153N How fast is the ball rising after being hit? How fast is the ball moving horizontally? Vv = V sin 47.10 = 52.0 sin 47.10 = 38.1ms-1 52.0ms-1 47.10 VH=Vcos 47.10 =52.0 cos 47.10 = 35.4ms-1 Kiwi Bobsled When a green light shows the team accelerates at 2.0ms-2 for 5.0s and then they all jump in. Acceleration still the same. How fast is it going after 5.0s? What is the distance after 5.0s? What is the average speed @ 5.0s? What distance is covered when v=40ms-1 R.McLeod the Cyclist • If he rides at 6.0ms-1 for 6.0s then 12ms-1 for 12s what is his average speed. •Clue: the answer is not 9.0ms-1 10ms-1 • Mr KK runs athletically up the stairs at 5.5ms-1. A bunch of chemists are lazily traveling on the esculator at 2.3ms-1. What is the relative speed of Mr KK w.r.t. : The chemists? The ground? A group of shoppers going down a similar esculator? • A train goes by at 95ms-1 • A Man is walking forward at 1.2ms-1 • How fast will the man be moving to an observer on the ground? In the train? • A train goes by at 95ms-1 • The Man is walking towards the back now at 1.2ms-1 • How fast will the man be moving to an observer on the ground? In the train? • A train goes by at 95ms-1 • A Man is walking forward at 1.2ms-1 • How fast will a bird flying 10ms-1 in the same direction see the man moving? How fast will the man see the bird flying? • A train goes by at 95ms-1 A bird is flying 10ms-1 in the same direction as the train. How fast will these people see the bird flying? • A train goes by at 95ms-1 A bird is flying 10ms-1 in the opposite direction as the train. How fast will these people see the bird flying? Relative Velocity The velocity of one object in relation to another object. The velocity an object appears to move at may change if the object measuring is also moving. The velocity of B relative to A can be calculated by doing this vector subtraction…. vBrelA vB v A (Do Page 49 Questions 3B) Projectile Motion Projectile motion is a parabolic shaped motion experienced by moving objects that have only the force due to gravity acting on them. Eg. Bullets,shotputs,netballs, water jets, rugby balls Projectile Motion When dealing with projectiles, the horizontal and vertical components are treated separately. The horizontal motion is constant velocity (as there are no forces acting in this direction). The vertical motion is constant acceleration of 10ms-2 due to the force of gravity. Kinematic Equations for vertical motion Do Page 89 Questions 6B The canon ball travels 25ms-1 when fired horizontally from the top of a 45m cliff. t=0s t=1.0s t=2.0s For each position find the horizontal, vertical and resultant velocity. t=3.0s t=1.0s t=0s t=2.0s t=3.0s Type 1 Questions What is a projectile path? A projectile path is the movement of an object under the action of gravity only. Explain the motion of the golf ball in the vertical direction. Give a reason for your explanation Type 1 Questions The ball is moving with a constant acceleration acting vertically downwards (constantly decelerating upwards). The golf ball’s acceleration is due to gravity. The golf ball’s weight is the unbalanced force acting on it. Back to Golf-Still Type 1 20 m s-1 32 m s-1 Draw clearly on the diagram a velocity vector to represent the size and direction of the initial velocity (U) of the golf ball. Golf Analysis -1 20 m s U 32 m s-1 What is the size and direction of the balls speed ? Quote the answer to the correct number of significant figures. Golf Analysis u2 = 202 + 322 u = 37.735925 = 38 m s1 tan = 20/32 = 320. What is the time taken for the ball to reach the top of its flight? vf = vi + at 0 = 20 – 10t t = 2.0 s Golf Analysis Calculate the maximum vertical height (H) reached by the golf ball at the top of its flight. Acceleration due to gravity is 10 m s-2. vf 2 = vi 2 + 2ad 02 = 202 + 2 x(- 10)H H = 20 m Golf Analysis Explain the motion of the golf ball in the horizontal direction, Give a reason for your explanation. Motion is constant velocity (speed and direction) as there is no unbalanced force acting on the golf ball in the horizontal direction. What is the velocity of the ball at the top of its flight? 32ms-1 horizontal Golf Analysis Calculate the horizontal distance (R) travelled by the golf ball. v = Dd / Dt 32 = R/2t where Dt = 2t R = 2 x 32 x 2 = 128m What is the velocity of the ball when it lands? 38ms-1 @320 to the ground What is a force? (type 1) Forces A force causes the motion or shape of an object to change. Force is a vector quantity so must have both a size and a direction Force is measured in Newtons N. A resultant (or net) force is produced when 2 or more forces act on an object. These forces can be added to find the resultant. Forces Newtons First Law Of Motion: An object will remain in it’s current state of motion until a force acts to change it. Newton’s Second Law Of Motion: The acceleration of an object is proportional to the net force applied. Law 2 can be written like this for short: Fnet ma Forces Newton’s Third Law Of Motion: For every action there is an equal and opposite reaction. Not What it seems!!! A 500kg hot air balloon rises at a rate of 0.75ms-2 in a cool Christchurch morning air. What is the total force lifting the balloon? 5375N 5000N to overcome gravity + 375N to cause the acceleration Type 1 Draw a force diagram of an aeroplane accelerating in level flight. Draw a force diagram of Jim standing with both feet on a skate board traveling between B and D block. What will be happening to a cyclist experiencing no net force? Forces Friction: Friction occurs when two surfaces move past each other. One of these surfaces could be air – eg air resistance is a frictional force. Friction is a force that always opposes the direction of the motion. Friction is sometimes called: drag, water resistance, air resistance or the retarding force. Forces Tension: This is the force that occurs in connecting strings and ropes Tension pulls in both directions along the string or rope. Weight: This is the force of gravity pulling downwards on an object. Weight can be calculated by: Fw mg g is acceleration due to gravity and has a value of 10ms-2 on Earth (Do Page 55 Questions 4A) Torque Torque causes things to spin. Symbol: t (Greek letter Tau) Units: Nm The size of a torque depends on the size of the force and the perpendicular distance from the pivot to where the force is applied. t Fd Equilibrium An object is at equilibrium if it is at rest or moving uniformly (First Law) Two conditions apply: (Do page 63 Questions 4B) Equilibrium All the forces acting on the object must add to zero SF=0 (Do page 63 Questions 4B) Equilibrium All the torques acting on the object must add to zero. St=0 (Do page 63 Questions 4B) Momentum The amount of “oooomph” an object has. Momentum depends on the mass of an object and it’s velocity. Symbol: p Unit: kgms-1 Momentum is a vector. Momentum can be calculated using: p mv Momentum If a force acts on an object, it’s momentum will change. The change in momentum can be calculated by subtracting vectors. Change in momentum =final momentum – initial momentum. Dp p f pi Impulse When a resultant force acts on an object, the amount it changes the object’s momentum by depends on how long the force acts for. The force multiplied by the time it acts for is called impulse. Units: Ns Impulse equals the change in momentum. FDt Dp (Do Page 73 Questions 5A) Conservation of Momentum The conservation of momentum principle states: Momentum is conserved in collisions and explosions as long as there is no net external force acting. This means the momentum before equals the momentum after. m1vi1 m2vi 2 m1v f 1 m2v f 2 Conservation of Momentum The same principle applies in 2 dimensions. The vector representing the sum of the momentums before must be the same vector as the one representing the sums after. Do Page 79 Questions 5B Circular Motion Period of Rotation T - time it takes to make one rotation (revolution, cycle) Measured in seconds s. Frequency f – number of rotations completed per second. Measured in Hertz Hz or s-1 T and f are inverses of each other. 1 T f Circular Motion Circumference – distance travelled in one rotation (m) Circumference 2r The speed of an object moving in a circle can be calculated by: d 2r Speed (v) t T Circular Motion An object moving in a circle may be travelling at constant speed, but because its direction is always changing, its velocity is changing…. If velocity is changing, the object is accelerating…. If an object is accelerating, there must be a net force acting on it…. Circular Motion The force acting on an object in circular motion is in towards the centre of the circle, changing the objects direction but not its speed. This is called centripetal force. This force causes a centripetal acceleration towards the centre of the circle. Circular Motion Centripetal force and acceleration can be calculated using the following formulae: v=speed(ms-1) r=radius of motion Do Page 98 Questions 7A 2 v ac r 2 v Fc m r Solve these The minute hand of the clock is 10cm, second hand 9.0cm, hour hand 7.0cm. How fast is the tip of each hand going? A Swift, the world’s fastest bird, of mass 0.10kg is seen to complete a circular turn of radius 10m without changing speed of 72kmhr-1. i) Find the centripetal acceleration of the bird and compare it to gravity. ii) Find the centripetal force on the bird. Solve These A record player has speeds of 33rpm, 45rpm and 78rpm. Find the period of revolution for a record played at each speed. If a fly lands 30cm from the centre of rotation. What is the tangential velocity if the record is going 33rpm? Type 1 Questions What are the units for momentum, torque, tangential speed, work, acceleration? A cricket ball is thrown from the boundary. Describe the path it takes. Draw a force diagram for the ball travelling through the air. What is the direction of the net force on the ball? Timy and Cameroon sit on a bench outside A2 at playtime. Draw a diagram showing their weight the weight of the bench and the support forces Through the bench legs Solve This A stone of mass 750g is tied to the end of a string and spun. The string has a breaking strain of 35N and is 1.0m long. It is spun in a plane horizontal to the earth at a rate of 60 times a minute. i) What is the tangential velocity of the stone? ii) What is the centripetal acceleration of the stone? iii)Show whether the string will break. iv)If the stone is now spun in a vertical plane at the same speed show whether the string will break now. Motion due to Gravity All objects accelerate towards the ground at (-) 10ms-2 because of gravity when dropped. This acceleration is fairly constant at the Earth’s surface, but varies at great altitudes or on other planets. Gravity is always an attractive force unlike magnetism or electric forces. Do Page 86 Questions 6A Energy W Fd The three kinds of mechanical energy are: kinetic, gravitational and elastic. Work W is the process of transforming energy from one kind to another. Energy E is measured in …………….. d is the …………. moved in the direction of the force. 1 of 6 on energy Energy If an object is lifted against gravity, work is done transforming chemical energy (muscles) into gravitational potential energy. The force needed is the weight force of the object, the distance moved is the change in height: W Fd so DE p (mg )Dh 2 of 6 on energy Energy Any moving object has kinetic energy. Doubling the speed increases the energy by four. (Squared relationship) When moving objects stop, this energy is transformed into other forms, eg sound, heat Ek mv 1 2 power 2 How many more man-3 Type 1 Questions When Timy and Cameroon sat on a bench outside A2 at playtime you could work out the forces acting. Discuss the Physics principles you would have to assume to work out the forces? Have to mention equilibrium That means the forces add up to 0, nothing, zip, nada The torques clockwise equal the anticlockwise torques Type 1 Questions Brody running with the union ball at 2.5ms-1 North changes direction and goes 3.2ms-1West to avoid a tackler. Draw labeled vector diagrams showing his initial and final velocity. Draw a vector diagram showing his change in velocity. He then crashes into another player and rebounds. Discuss the Physics principle that will allow you to calculate his rebound speed. Power Power P is the rate at which work is done. Measured in Watts W (or Js-1) W DE P t t conservation 4 of 6 get over it Conservation of Energy The conservation of energy principle states: Energy cannot be created or destroyed, only transformed from one kind to another. Efficiency/lost One more after this Energy Efficiency Often some of the forms it is transformed into are not useful. The energy is “lost” to us The efficiency of an object is a measure of the ratio of input energy to useful output energy Elastic collisions-stop talking and listen then make a note useful output energy Efficiency% 100 total input energy End of the energy story for today Solve This 80% of the electricity going into a light bulb gets turned into heat. How much energy does a 100W bulb use in 10 minutes and how much of this is turned into light? DE DE P 100 t 10 60 DE 60,000 60kJ 12kJ light Solve This A 60kg woman runs up a set of stairs in 15s. She rises 10m in her climb. Calculate her power. The important message with this problem is to realise that the majority of energy is used against gravity as opposed to the horizontal motion DE mgh P P t t 60 x10 x10 P 15 P 400W Solve This At what speed must a 50 gram squash ball travel if it is to have energy of 0.50J? 4.5ms-1 If the energy is doubled to 1.0J what must the speed be? 6.3ms-1 Solve This A sports car with mass ¾ of a tonne accelerates to 108 kmhr-1 in 8.0s. Ignoring friction what power is exerted by the motor? 1 2 W DE P t t 2 mv P t 1 2 x 750 x 30 2 8 42187 42kW Solve This A 200W motor is used to lift a 15kg bucket of cement 40m. How long will it take? W DE P t t mgh P t 15 x10 x 40 200 t t 30 s Springs Energy can be stored in a spring as elastic potential energy. Hookes Law: F=kx F=force k=spring constant (Nm-1) – a measure of how stiff or soft a spring is. x=extension (m) – the amount a spring is stretched or compressed when the force is applied. Springs Hooke’s Law as a graph: Force (N) Gradient = k Area under graph = energy stored in spring Extension (m) Springs Elastic potential energy can be found by calculating the area under a Hooke’s Law graph. Area b h 1 2 E Fx 1 2 ( F kx) and E kx so Do Page 107 Questions 8A 1 2 2 Solve This A person sits in a car with a suspension of spring constant 104 Nm-1. If the suspension is compressed 1.0 cm how much energy is stored in the springs? Ep=1/2kx2 =0.5 x 104 x (0.01)2 =0.5 J You know everything required to get an excellence in Thursdays test now But will you……………………. Revise the problems in your text and The 2.4 exam on the NCEA website for 2004.