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The Trebuchet Engineering 111 Bring your laptops to class on Wednesday. Complete Homework 8 before Wednesday. Class Question How do units differ from variables? List 10 clear examples of units and 10 clear examples of variables. Class Objectives Learn the relationship between position, velocity, and acceleration Learn and apply Newton’s First, Second, and Third Laws Learn some secrets of Calculus Chapter 10 Newton’s Laws 5 What are the Forces of Nature? Gravitational Force Electromagnetic Force – Electrostatic and Magnetic Nuclear Force – Strong and Weak Friction Spring Tension Push Newton’s First Law Law of Inertia Newton’s 1st Law Law of Inertia “Bodies remain at rest or in uniform motion in a straight line unless a net force acts on it.” Thought Questions Why don’t planets move in straight lines? Would they move it straight lines if there were no gravitational force? Newton’s Cannon Newton’s Second Law SF = m a “The amount of acceleration that a force produces depends on the mass of the object being accelerated.” Newton’s Second Law Example Questions: – What happens to acceleration if the force is doubled? Answer: – ….if the mass is double? Answer: Newton’s Third Law Action-Reaction “Whenever one body exerts a force on a second body, the second body exerts an equal and opposite force on the first body.” Newton’s Third Law - Action/Reaction Which exerts a stronger force on the other? Earth or Moon Mac Truck or VW Bug during Collision Car and a Mosquito during Collision Newton’s Laws 1st – Law of Inertia 2nd – SF=ma 3rd – Action Reaction Gravity Universal Gravitational Force m1m 2 FG 2 r Inverse Square Law G = 6.67 10-11 N·m2/kg2 Gravity Questions Did the Moon exert a gravitational force on the Apollo astronauts? What kind of objects can exert a gravitational force on other objects? The constant G is a rather small number. What kind of objects can exert strong gravitational forces? Gravity Questions If the distance between two objects in space is doubled, then what happens to the gravitational force between them? Find the gravitational force between a 90-kg person and the Earth using the inverse square law (Equation 10-37) and Table 10.6. Show your work and include units for your answer. Chapter 10 Newton’s Laws 18 Newton’s Laws 1st – Law of Inertia 2nd – SF=ma 3rd – Action Reaction – Demo Metal ball launcher and dropper Maximum Range and the 90-degree-rule Airplane flare: http://observe.phy.sfasu.edu/courses/phy101/lectures101/ Movies/Plane%20Drops%20Flare.wmv Definitions Vector Quantity – a quantity that has both magnitude and direction Vector – an arrow drawn to scale used to represent a vector quantity Scalar Quantity – a quantity that has magnitude but not direction Vector or Scalar? Speed……….. scalar Velocity……... vector Acceleration.. vector Time…………. scalar Force………… scalar Distance…….. it depends... Some Definitions Position - a location usually described by a graphic on a map or by a coordinate system 4 3 (4, 3) 2 1 0 0 -3 -2 -1 -1 1 2 3 4 5 (-2, -3) -2 -3 -4 Some Definitions Displacement -- change in position, where Δ r r2 r1 4 3 2 (4, 3) r2 1 0 Δr 0 -3 -2 -1 -1 1 2 3 4 5 r1 (-2, -3) -2 -3 -4 Example Write an expression for each vector below and find their magnitudes. 4 3 (4cm, 3cm) 2 r2 1 0 Δr 0 -3 -2 -1 -1 1 2 3 4 5 r1 -2 (-2cm, -3cm) -3 -4 Vector Notation Position Vector ˆ ˆ ˆ r x i yj zk Unit Vectors i jˆ ˆ, ˆ,k Magnitude r r x y z 2 2 2 Team Exercise 4.4 Some Definitions Average velocity rate of position change with time r Displaceme nt vector v ave t Elapsed Time scalar Some Definitions Instantaneous Velocity r dr v lim t 0 t dt Some Definitions Speed the magnitude of instantaneous velocity speed v Example Suppose that your drive around in a circle in a parking lot at 30mph. Consider the instant that your car is facing north. What is your speed? What is your instantaneous velocity ? What is average velocity? Some Definitions Average Acceleration - rate of velocity change with time v 2 v1 v aave t 2 t1 t Instantaneous Acceleration v dv a lim t 0 t dt Kinematic Equations Where did they come from? v x v x0 a xo t v y v yo a yo t 1 x x o v xo t a xo t 2 2 1 y y o v yo t a yo t 2 2 Chapter 10 Newton’s Laws 32 Derivatives The derivative is associated with the slope of a function. dr dv v a dt dt rise slope run Some Derivatives Powers d n x ? dx Trig Functions d sin( x ) ? dx Exponentials d x e ? dx Example Assume that r(t)=kt where k is a constant. Plot r versus t. Plot v versus t. Plot a versus t. Write an equation for v(t) and a(t). Example Assume that r(t)=Ct3 where C is a constant. Plot r versus t. Plot v versus t. Plot a versus t. Write an equation for v(t) and a(t). Example Assume that r(t)=rosin(wt) where ro and w are constants. Plot r versus t. Plot v versus t. Plot a versus t. Write an equation for v(t) and a(t). Project Two - Week Two Tuesday Wednesday – 11:00AM – 11:00AM – 11:30AM – 11:30AM – 12:00PM – 12:00PM – 12:30PM – 12:30PM – 1:00PM – 1:00PM N/A – 1:30PM – 1:30PM N/A – 2:00PM – 2:00PM N/A – 2:30PM – 2:30PM – 3:00PM – 3:00PM – 3:30PM – 3:30PM – 4:00PM – 4:00PM – 4:30PM – 4:30PM Example 10.1 Assume that you... – walk at 3mph for 15 minutes, – then drive at 40mph for 2 hours, and – then ride a bike at 10pm for 45 minutes. Plot v versus t. Plot r versus t. Do you differentiate or integrate to get r(t)? Integrals The integral is associated with the area under the curve. dr v dt dv a dt area length width Some Integrals Powers x dx ? n Trig Functions sin(x)dx ? e dx ? Exponentials x Example Assume that a(t)=k where k is a constant. Plot a versus t. Plot v versus t. Plot r versus t. Write an equation for v(t) and r(t). Team Exercise 43 Problem 10.2 A motorcycle moves with an initial velocity of 30m/s. When its brakes are applied, it decelerates at 5.0m/s2 until it stops. Plot the position, velocity and acceleration as a function of time. What is the position, velocity and acceleration 2 seconds after the brakes are applied? 1 v vo a t x x o v o t ao t 2 2 Multiple Directions The equations of motion can be written for each direction independently. Velocity v x v x a x t 0 o v y v yo a yo t Position 1 x x o v xo t a xo t 2 2 1 y y o v yo t a yo t 2 2 Problem 10.3 A girl shoots an arrow upward. It strikes the ground 10.0 seconds later. What was its initial velocity and what was the maximum height? 1 v y v yo a y t y y o v yo t a yo t 2 2 Problem 10.4 A bullet is fired vertically into the air and reached a maximum height of 15,000ft. What was the initial velocity? What assumptions must be made? 1 v y v yo a y t y y o v yo t a yo t 2 2 Problem 10.5 A man standing on a 200-ft tower throws a ball upward at 40 ft/s. How long does it take to hit the ground? Announcements Homework 12 due on Wednesday. Remember to get advising. Project Two is due on December 6th. – That’s next week today. Project Two Scoring: – 1 Point for each inch that the projectile travels. Instructions: – See course home page Homework 12 http://www.physics.sfasu.edu/astro/OnlineExa ms/FCI/fci_main.html Kinematic Equations Where did they come from? v x v x0 a xo t v y v yo a yo t 1 x x o v xo t a xo t 2 2 1 y y o v yo t a yo t 2 2 Calculus Basics Derivative d n x nx n 1 dx Indefinite Integral x n 1 x n 1 n Definite Integral n 1 b b n 1 a n 1 b x a x n dx n 1 a n 1 n 1 Constant Acceleration Equation of motion is dv a dt where acceleration is constant. a dt dv Integrate both sides dv a dt v vo a t The o in v subscript refers to the original or initial value at the beginning of the time interval of interest. dx v dt Arranging this equation v dt dx Substituting the velocity equation from the previous page dx v o ao t dt Integrating both sides dx v o ao t dt 1 x x o v o t ao t 2 2 Coil Gun Final Exam Question – How fast must the projectile be traveling when it leaves the straw if it is to travel 100 inches horizontally? Newton’s Laws -- Review First Law Law of Gravity Interia m1m 2 Second Law FG 2 F=ma r Third Law – Action/Reaction Team Exercise, 3 min. 1. The derivative of velocity with respect to time is: – position or acceleration 2. By integrating velocity with respect to time we get: – distance traveled or acceleration 3. The derivative of position with respect to time is: – acceleration or velocity 4. Integrating acceleration twice with respect to time is : – velocity squared or distance 5. The derivative is associated with the _________ while the integral is associated with _________ – area under the curve, slope Homework 12 5. According to Newton’s Second Law, what is the net force on a 2000-lb car if it travels at a constant 60mph for 2 hours? Homework 12 6. A 1500-kg automobile has a projected frontal area of 1.9 m2 and a drag coefficient of 0.35. It is traveling at 100km/h on a flat road when suddenly both the engine and brakes fail. What is the drag force on the automobile at the moment the brakes fail? The density of air is 1.3 g/liter. 1 F C d Av 2 2 Homework 12 Will the drag force increase, decrease or stay the same as time goes by? What would the drag force be if the medium was water rather than air? Problem 10.7 (modified) Assume that a projectile is fired upward at an angle and that air resistance is negligible. Find y as function of x instead of time using the equations below. What is the shape of the projectiles path? 1 1 x x o v xo t a xo t 2 y y o v yo t a yo t 2 2 2 Team Exercise (3 minutes) One dimensional motion – What is the distance traveled in 3 seconds? – What is the acceleration at 1.25 hours? 20 0 1 2 3 Time, hours Exercise - Newton’s Laws A pickup truck is moving with a constant speed of 30 mph along a city street. You are sitting in the back of the truck and you throw a ball straight upwards at a speed of 20 mph. Neglecting air resistance: – What will be the path of the ball with respect to the pickup truck and where will the ball land with respect to the truck? (i.e. what are the x and y values of the peak and final positions) Solution Step 1: Draw a Picture 20 mph 30 mph X=0 Y=0 Solution (cont’d) The ball follows a parabolic path and remains directly above the truck at all times. Is a horizontal force acting upon the ball? – Votes for yes? – Votes for no? Solution Answer: NO. There is no horizontal force acting on the ball. – The horizontal motion of the ball is the result of its own inertia. When thrown from the truck, the ball already possessed a horizontal motion, and thus will maintain this state of horizontal motion unless acted upon by a force with a horizontal component (Newton's first law). Summary: forces do not cause motion (velocity); rather, forces cause accelerations (change in velocity). Today’s Demos Inertia Center of Mass – Lead Brick and Hammer – State of Texas – Inertia Bars – Curious George – Disk and Ring Race Next Week – Rotating Platform – Bed of Nails – Metronome – Fluid Flasks – Balancing A Meter Stick – And more.... Rotation Adds Stability – Bicycle Tires – Moving Water – Spin Guns Trebuchet Problem (Modified 10.6) The projectile is fired at an angle of 35° relative to the ground. It is fired with a velocity of 100ft/s. v x v x ax t 0 o Find the following: vy v y ay t – vxo and vyo o o – axo and ayo How long was the projectile in the air? How high did it go? 1 How far did the projectile go? x x o v xo t a xo t 2 2 1 y y o v yo t a yo t 2 2 Section 10.3 - Forces Forces are vector quantities. ˆ ˆF ˆF k F Fx i y j z What is the magnitude of this force? F 10 N ˆ 10 N ˆj 5N k i ˆ Problem 10.1 A car starts from rest and travels northward. It accelerates at a constant rate for 30 seconds until it reaches a velocity of 55mph. Plot the acceleration, velocity and position as a function of time. Next Chapter… Locate Homework 12 Take 3 minutes to answer questions 1, 2 and 3. Final Trebuchet Tweaks Consider a trough Weight and test your trebuchet Allow for hole and pin adjustments Prepare for rain The Ideal Trebuchet Problem In the absence of air resistance, a projectile fired at an angle of 45° will have a maximum range. What initial speed must a projectile have in order to have a range of 20 yards? ...of 40 yards? 1 v x v x0 a xo t x x o v xo t a xo t 2 2 v y v yo a yo t 1 y y o v yo t a yo t 2 2 Question 1 – Drop Time A – ½ time for heavy ball B – ½ for lighter ball C – same for both balls D – considerably less time for heavy ball E – considerably less time for lighter ball Question 2 – Roll off landing position A – same horizontal distance for both B – heavier ball hits half horizontal distance C – lighter ball hits half horizontal distance D - heavier ball hits considerably closer to base E - lighter ball hits considerably closer to base Question 3 – Stone Drop Speed A – reaches maximum speed soon B – speeds up because gravity gets considerably stronger closer to the Earth C – speeds up because of an almost constant force of gravity D – falls because of the natural tendency of all objects to rest on the surface of the earth E - falls because of the combined effects of the force of gravity and the force of the air pushing it downward. Question 4 – Truck and Car Collide A – truck exerts more force on car B – car exerts more force on truck C – neither exerts a force on the other D - the truck exerts a force on the car but the car does not exert a force on the truck. E - the truck exerts the same amount of force on the car as the car exerts on the truck