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Physics B Classwork/1-D Motion Name:_______________________________________ Physics Period:_______ Practice Problem: If x is the displacement of a particle, and d is the distance the particle traveled during that displacement, which of the following is always a true statement? a) d = |x| b) d < |x| c) d > |x| d) d > |x| e) d < |x| Kinematics: the branch of mechanics that describes the motion of objects without necessarily discussing what causes the motion. Distance Definition: SI Unit: Practice Problem: A particle moves from x = 1.0 meter to x = -1.0 meter. a) What is the distance d traveled by the particle? Displacement (x) Definition: Equation: b) What is the displacement of the particle? SI Unit: Question: Does the odometer in your car measure distance or displacement? Practice Problem: You are driving a car on a circular track of diameter 40 meters. After you have driven around 2 ½ times, how far have you driven, and what is your displacement? Can you think of a circumstance when it would measure both distance and displacement? Practice Problem: Two tennis players approach the net to congratulate one another after a game. A 5m B 2 m a) Find the distance and displacement of player A. Average Speed Definition: Equation: b) Repeat for player B. SI unit: 6/18/2017 1 Bertrand/Perkins Average Velocity Definition: Graphical Problem What physical feature of the graph gives the constant velocity from A to B? x Equation: B A t SI unit: Practice Problem: How long will it take the sound of the starting gun to reach the ears of the sprinters if the starter is stationed at the finish line for a 100 m race? Assume that sound has a speed of about 340 m/s. Graphical Problem: Determine the average velocity from the graph. x (m) Practice Problem: You drive in a straight line at 10 m/s for 1.0 km, and then you drive in a straight line at 20 m/s for another 1.0 km. What is your average velocity? Graphical Problem: Determine the average velocity between 1 and 4 seconds. 6/18/2017 2 Bertrand/Perkins Instantaneous Velocity Definition: Acceleration in 1-D motion has a sign! If the sign of the velocity and the sign of the acceleration is the same, what happens? If the sign of the velocity and the sign of the acceleration are different, what happens? Practice Problem: Determine the instantaneous velocity at 1.0 second. Practice Problem: A 747 airliner reaches its takeoff speed of 180 mph in 30 seconds. What is its average acceleration? Practice Problem: A horse is running with an initial velocity of 11 m/s, and begins to accelerate at –1.81 m/s2. How long does it take the horse to stop? Acceleration Definition: What does the sign of the acceleration signify? What types of acceleration are there? Graphical Problem What physical feature of the graph gives the acceleration? Questions If acceleration is zero, what does this mean about the motion of an object? v Is it possible for a racecar circling a track to have zero acceleration? B A t Uniform (Constant) Acceleration Equation: SI unit: 6/18/2017 3 Bertrand/Perkins Practice Problem: Determine the acceleration from the graph. Draw Graphs for Stationary Particles x v t a t t Draw Graphs for Constant Non-zero Velocity x v t a t t Determine the displacement of the object from 0 to 4.0 seconds (using the graph above) Draw Graphs for Constant Non-zero Acceleration x How would you describe the motion of this particle? v t a t t Kinematic Equations Describe the motion Equation 1: Equation 2: Equation 3: Practice Problem: What must a particular Olympic sprinter’s acceleration be if he is able to attain his maximum speed in ½ of a second? 6/18/2017 4 Bertrand/Perkins Practice Problem: A plane is flying in a northwest direction when it lands, touching the end of the runway with a speed of 130 m/s. If the runway is 1.0 km long, what must the acceleration of the plane be if it is to stop while leaving ¼ of the runway remaining as a safety margin? Practice Problem: You are driving through town at 12.0 m/s when suddenly a ball rolls out in front of you. You apply the brakes and decelerate at 3.5 m/s2. a) How far do you travel before stopping? b) When you have traveled only half the stopping distance, what is your speed? Practice Problem: On a ride called the Detonator at Worlds of Fun in Kansas City, passengers accelerate straight downward from 0 to 20 m/s in 1.0 second. a) What is the average acceleration of the passengers on this ride? c) How long does it take you to stop? b) How fast would they be going if they accelerated for an additional second at this rate? d) Draw x vs t, v vs t, and a vs t graphs for this. c) Sketch approximate x-vs-t, v-vs-t and a-vs-t graphs for this ride. Free Fall Definition: Practice Problem: Air bags are designed to deploy in 10 ms. Estimate the acceleration of the front surface of the bag as it expands. Express your answer in terms of the acceleration of gravity g. Acceleration due to Gravity: 6/18/2017 5 Bertrand/Perkins c) What is the ball’s velocity when you catch it? Practice Problem: You drop a ball from rest off a 120 m high cliff. Assuming air resistance is negligible, a) how long is the ball in the air? d) Sketch approximate x-vs-t, v-vs-t, a-vs-t graphs for this situation. b) what is the ball’s speed and velocity when it strikes the ground at the base of the cliff? Symmetry in Free Fall When something is thrown straight upward under the influence of gravity, and then returns to the thrower, this is very symmetric. The object spends half its time traveling up; half traveling down. Velocity when it returns to the ground is the opposite of the velocity it was thrown upward with. 2 Acceleration is 9.8 m/s and directed DOWN the entire time the object is in the air! c) sketch approximate x-vs-t, v-vs-t, a-vs-t graphs for this situation. Practice Problem: You throw a ball straight upward into the air with a velocity of 20.0 m/s, and you catch the ball some time later. a) How long is the ball in the air? Homework Problem: Below is some data for a car in the Pinewood Derby. Using these data, work the following problem: Pinewood Derby x(m) t(s) b) How high does the ball go? 6/18/2017 0 0 2.3 1.0 9.2 2.0 20.7 3.0 36.8 4.0 57.5 5.0 On your graph paper, do the following. a) Draw a position vs time graph for the car. b) Draw tangent lines at three different points on the curve to determine the instantaneous velocity at all three points. c) On a separate graph, draw a velocity vs time graph using the instantaneous velocities you obtained in the step above. d)From your velocity vs time graph, determine the acceleration of the car. 6 Bertrand/Perkins Physics B Classwork: 2 and 3D Motion Homework problem Name:_____________________ Case 3: Ball A is dropped from rest at the top of the cliff at exactly the same time Ball B is thrown vertically upward with speed vo from the foot of the cliff such that Ball B will collide with Ball A. Derive an expression for the amount of time that will elapse before they collide. Case 1: Ball A is dropped from rest at the top of a cliff of height h as shown. Using g as the acceleration due to gravity, derive an expression for the time it will take for the ball to hit the ground. A h Case 4: Ball A is dropped from rest at the top of the cliff at exactly the same time Ball B is projected vertically upward with speed vo from the foot of the cliff directly beneath ball A. Derive an expression for how high above the ground they will collide. Case 2: Ball B is projected vertically upward from the foot of the cliff with an initial speed of vo. Derive an expression for the maximum height ymax reached by the ball. h vo B 6/18/2017 7 Bertrand 2-Dimensional Motion Definition: Sample Problem A roller coaster rolls down a 20o incline with an acceleration of 5.0 m/s2. a) How far horizontally has the coaster traveled in 10 seconds? Examples: Solving 2-D Problems Resolve all vectors into components. Work the problem as two one-dimensional problems. Re-combine the results for the two components at the end of the problem. b) How far vertically has the coaster traveled in 10 seconds? Sample Problem You run in a straight line at a speed of 5.0 m/s in a direction that is 40o south of west. a) How far west have you traveled in 2.5 minutes? Sample Problem A particle passes through the origin with a speed of 6.2 m/s in the positive y direction. If the particle accelerates in the negative x direction at 4.4 m/s2 a) What are the x and y positions at 5.0 seconds? b) How far south have you traveled in 2.5 minutes? b) What are the x and y components of velocity at this time? 6/18/2017 8 Bertrand Projectile Motion Something is fired, thrown, shot, or hurled near the earth’s surface. Vertical Component of Velocity Characteristics: Horizontal velocity is __________ Equations: Vertical velocity is __________ Air resistance is __________ 1-Dimensional Projectile Definition: Launch angle Definition: Examples: Why is the launch angle important? You calculate vertical motion only. The motion has no horizontal component. Zero Launch angle What does a zero launch angle imply? 2-Dimensional Projectile Definition: Sample Problem The Zambezi River flows over Victoria Falls in Africa. The falls are approximately 108 m high. If the river is flowing horizontally at 3.6 m/s just before going over the falls, what is the speed of the water when it hits the bottom? Assume the water is in freefall as it drops. Examples: You calculate vertical and horizontal motion. Horizontal Component of Velocity Characteristics: Equation: 6/18/2017 9 Bertrand Sample Problem An astronaut on the planet Zircon tosses a rock horizontally with a speed of 6.75 m/s. The rock falls a distance of 1.20 m and lands a horizontal distance of 8.95 m from the astronaut. What is the acceleration due to gravity on Zircon? Resolving the velocity Use speed and the launch angle to find horizontal and vertical velocity components. Then proceed to work problems just like you did with the zero launch angle problems. Sample problem A soccer ball is kicked with a speed of 9.50 m/s at an angle of 25o above the horizontal. If the ball lands at the same level from which is was kicked, how long was it in the air? Sample Problem Playing shortstop, you throw a ball horizontally to the second baseman with a speed of 22 m/s. The ball is caught by the second baseman 0.45 s later. a) How far were you from the second baseman? Sample problem Snowballs are thrown with a speed of 13 m/s from a roof 7.0 m above the ground. Snowball A is thrown straight downward; snowball B is thrown in a direction 25o above the horizontal. When the snowballs land, is the speed of A greater than, less than, or the same speed of B? Verify your answer by calculation of the landing speed of both snowballs. b) What is the distance of the vertical drop? 6/18/2017 10 Bertrand Projectiles launched over level ground These projectiles have highly symmetric characteristics of motion. Position graphs for 2-D projectiles Trajectory of a 2-D Projectile launched over level ground Sketch: Velocity graphs for 2-D projectiles Characteristics: Acceleration graphs for 2-D projectiles Range of a 2-D Projectile launched over level ground Definition: Maximum height of a projectile launched over level ground Notes: Sample problem A golfer tees off on level ground, giving the ball an initial speed of 42.0 m/s and an initial direction of 35o above the horizontal. a) How far from the golfer does the ball land? Acceleration of a projectile launched over level ground Notes: Velocity of a projectile launched over level ground Notes: b)The next golfer hits a ball with the same initial speed, but at a greater angle than 45o. The ball travels the same horizontal distance. What was the initial direction of motion? Time of flight for a projectile launched over level ground Notes: 6/18/2017 11 Bertrand Free Response Preparation #1 A cannonball is fired at an angle of 45o above the horizontal at an initial velocity of 77 m/s. The cannon is located at the top of a 120 m high cliff, and the cannonball is fired over the level plain below. a) Draw a representation of the trajectory of the cannonball from launch until it strikes the plain below the cliff. Label the following: A: The point where the projectile is traveling the slowest; B: The point where the projectile has the same speed as it does at launch; C: The point where the projectile is traveling the fastest. b) Calculate the total time from launch until the cannonball hits the plain below the cliff. c) Calculate the horizontal distance that the cannonball travels before it hits the plain below the cliff. d) Calculate the maximum height attained by the cannonball. 6/18/2017 12 Bertrand Free Response Preparation #2 A soccer player on Krypton kicks a ball directly toward a fence from a point 35 meters away. The initial velocity of the ball is 25 m/s at an angle of 40o above the horizontal. The top of the fence is 3.0 meters above the ground. The ball hits nothing while in flight, and, since Krypton has no atmosphere, air resistance is nonexistent. The acceleration due to gravity on Krypton is 12 m/s 2. a. Sketch the problem b. Determine the time it takes for the ball to reach the plane of the fence. c. Will the ball hit the fence? If so, how far below the top of the fence will it hit? If not, how far above the fence will it pass? d. Sketch the horizontal and vertical components of the ball’s velocity as functions of time until the ball reaches the plane of the fence. Draw and label your axes! e. What is the minimum speed the soccer player must give to the ball if it is to just hit the bottom of the fence at the same time it hits the ground? 6/18/2017 13 Bertrand Physics B Classwork and Notes: Newtons’ Laws I Name:_________________ Force Sample Problem Definition: a) A monkey hangs by its tail from a tree branch. Draw a force diagram representing all forces on the monkey. What does a force do? Newton’s First Law b) Now the monkey hangs by both hands from two vines. Each of the monkey’s arms are at a 45o from the vertical. Draw a force diagram representing all forces on the monkey. What happens if there is zero net force on a body? Mass and Inertia Definition of mass (physics definition): Does zero net force mean there is no force at all on a body? Definition of inertia: Sample Problem A heavy block hangs from a string attached to a rod. An identical string hangs down from the bottom of the block. Which string breaks a) when the lower string is pulled with a slowly increasing force? Draw a force diagram and a free-body diagram for a book resting on a table. b) when the lower string is pulled with a quick jerk? 6/18/2017 14 Bertrand Newton’s Second Law A catcher stops a 92 mph pitch in his glove, bringing it to rest in 0.15 m. If the force exerted by the catcher is 803 N, what is the mass of the ball? Definition: Equation: SI Units: A 747 jetliner lands and begins to slow to a stop as it moves along the runway. If its mass is 3.50 x 105 kg, its speed is 27.0 m/s, and the net braking force is 4.30 x 105 N a) what is its speed 7.50 s later? Working 2nd Law Problems 1. Draw a force or free body diagram. 2. Set up 2nd Law equations in each dimension. Fx = max and/or Fy = may 3. Identify numerical data. x-problem and/or y-problem 4. Substitute numbers into equations. “plug-n-chug” 5. Solve the equations. b) how far has it traveled in this time? Sample Problems Newton’s Third Law In a grocery store, you push a 14.5-kg cart with a force of 12.0 N. If the cart starts at rest, how far does it move in 3.00 seconds? 6/18/2017 Definition: 15 Bertrand Sample Problem You rest an empty glass on a table. a) How many forces act upon the glass? Newton’s 2nd Law in 2-D b) Identify these forces with a free body diagram. Identify all forces and draw a force diagram. Problem must be resolved into x- and yproblems. Follow the procedure for 1-D problems! Sample Problems A surfer “hangs ten”, and accelerates down the sloping face of a wave. If the surfer’s acceleration is 3.50 m/s2 and friction can be ignored, what is the angle at which the face of the wave is inclined above the horizontal? c) Are these forces equal and opposite? d) Are these forces an action-reaction pair? Sample Problem (similar to #17) A force of magnitude 7.50 N pushes three boxes with masses m1 = 1.30 kg, m2 = 3.20 kg, and m3 = 4.90 kg as shown. Find the contact force (a) between boxes 1 and 2. How long will it take a 1.0 kg block initially at rest to slide down a frictionless 20.0 m long ramp that is at a 15o angle with the horizontal? (b) between boxes 2 and 3. 6/18/2017 16 Bertrand Apparent weight An object acted on by three forces moves with constant velocity. One force acting on the object is in the positive x direction and has a magnitude of 6.5 N; a second force has a magnitude of 4.4 N and points in the negative y direction. Find the direction and magnitude of the third force acting on the object. Definition: Elevator rides Why do you feel lighter or heavier during parts of an elevator ride? Draw force diagrams for an elevator ride when you are ascending. Mass and Weight Definition of Weight: Equation for Weight: Stationary at Beginning the ground the ascent floor Constant velocity between floors Slowing at top floor Draw force diagrams for an elevator ride when you are descending. Sample Problem A man weighs 150 pounds on earth at sea level. Calculate his a) mass in kg. b) weight in Newtons. Stationary at Beginning the top floor the descent 6/18/2017 17 Constant velocity between floors Slowing at the ground floor Bertrand Sample Problem b) moving upward and slowing at 3.2 m/s2? An 85-kg person is standing on a bathroom scale in an elevator. What is the person’s apparent weight a) when the elevator accelerates upward at 2.0 m/s2? c) moving downward and speeding up at 3.2 m/s2? b) when the elevator is moving at constant velocity between floors? d) moving upward and speeding up at 3.2 m/s2? c) when the elevator begins to slow at the top floor at 2.0 m/s2? Normal force Definition: Sample Problem A 5-kg salmon is hanging from a fish scale in an elevator. What is the salmon’s apparent weight when the elevator is a) at rest? 6/18/2017 In what direction is the normal force relative to a surface? 18 Bertrand Normal force on flat surface Sample problem A 5.0-kg bag of potatoes sits on the bottom of a stationary shopping cart. Sketch a free-body diagram for the bag of potatoes. Now suppose the cart moves with a constant velocity. How does this affect the free-body diagram? Normal force on ramp Sample problem Find the normal force exerted on a 2.5-kg book resting on a surface inclined at 28o above the horizontal. Normal force not associated with weight. If the angle of the incline is reduced, do you expect the normal force to increase, decrease, or stay the same? Sample problem A gardener mows a lawn with an oldfashioned push mower. The handle of the mower makes an angle of 320 with the surface of the lawn. If a 249 N force is applied along the handle of the 21-kg mower, what is the normal force exerted by the lawn on the mower? Draw a free body diagram for the skier. 6/18/2017 19 Bertrand Sample problem Larry pushes a 200 kg block on a frictionless floor at a 45o angle below the horizontal with a force of 150 N while Moe pulls the same block horizontally with a force of 120 N. a) Draw a free body diagram. b) What is the acceleration of the block? c) What is the normal force exerted on the block 6/18/2017 20 Bertrand Physics B Classwork/Notes: Applications of Newton’s Laws Name:____________________ Friction Definition: What causes friction? How is friction useful? How does friction depend on normal force? Static Friction Definition: Equation: What are some implications of the fact that the static friction equation is an inequality? 6/18/2017 21 Bertrand Static friction and applied horizontal force Draw a force diagram representing the force of static friction on a level surface. Sample Problem A 10-kg box rests on a ramp that is laying flat. The coefficient of static friction is 0.50, and the coefficient of kinetic friction is 0.30. a) What is the maximum horizontal force that can be applied to the box before it begins to slide? Static friction on a ramp Draw a force diagram representing the force of static friction on a ramp. b) What force is necessary to keep the box sliding at constant velocity? Kinetic Friction Definition: Sample Problem A 10-kg wooden box rests on a ramp that is lying flat. The coefficient of static friction is 0.50, and the coefficient of kinetic friction is 0.30. What is the friction force between the box and ramp if a) no force horizontal force is applied to the box? Equation: How does the magnitude of kinetic friction compare to the magnitude of static friction? b) a 20 N horizontal force is applied to the box? 6/18/2017 22 Bertrand c) a 60 N horizontal force is applied to the box? Sample Problem A 10-kg wooden box rests on a wooden ramp. The coefficient of static friction is 0.50, and the coefficient of kinetic friction is 0.30. What is the friction force between the box and ramp if a) the ramp is at a 25o angle? Laboratory Determine the coefficients of static and kinetic friction between the wooden block (felt side) and the cart track. The only additional equipment you may use is a meter stick, a clamp, and a pole. Write a mini-lab report that includes only the following: procedure: one for determination of each kind of friction. analysis: include diagrams (free-body), calculations, and results for each kind of friction. It is necessary to type the procedure section, but the rest of the report may be hand-written. Tension Definition: What causes tension? b) the ramp is at a 45o angle? Sample problem a) A 1,500 kg crate hangs motionless from a crane cable. What is the tension in the cable? Ignore the mass of the cable. c) what is the acceleration of the box when the ramp is at 45o? 6/18/2017 23 b) Suppose the crane accelerates the crate upward at 1.2 m/s2. What is the tension in the cable now? Bertrand Hooke’s Law Definition: Connected objects What do connected objects subject to a force have in common? Sample problem A 5.0 kg object (m1) is connected to a 10.0 kg object (m2) by a string. If a pulling force F of 20 N is applied to the 5.0 kg object as shown, Equation: A) what is the acceleration of the system? (Assume no friction; draw the figure and proceed). What is meant by the term restoring force? Sample problem A 1.50 kg object hangs motionless from a spring with a force constant of k = 250 N/m. How far is the spring stretched from its equilibrium length? B) what is the tension in the string connecting the objects? (Assume no friction) Sample problem Mass 1 (10 kg) rests on a frictionless table connected by a string to Mass 2 (5 kg). Find a) the acceleration of each block. (Draw the figure and proceed). Sample problem A 1.80 kg object is connected to a spring of force constant 120 N/m. How far is the spring stretched if it is used to drag the object across a floor at constant velocity? Assume the coefficient of kinetic friction is 0.60. b) the tension in the connecting string. 6/18/2017 24 Bertrand Sample problem Mass 1 (10 kg) rests on a table connected by a string to Mass 2 (5 kg). Find the minimum coefficient of static friction for which the blocks remain stationary. Uniform Circular Motion Definition: Why is uniform accelerated? Sample problem Mass 1 (10 kg) rests on a table connected by a string to Mass 2 (5 kg). If s = 0.30 and k = 0.20, what is a) the acceleration of each block? (Draw the figure and proceed). circular motion What is centrifugal force? When a car turns, you feel as if you are flung to the outside? Why? As a general rule, when you feel flung in a certain direction, in what direction is the acceleration? b) the tension in the connecting string? Acceleration in Uniform Circular Motion In what direction does it point? Sample problem Two blocks are connected by a string as shown in the figure. What is the acceleration, assuming there is no friction? (Draw the figure and proceed). Centripetal Acceleration Definition: Equation 6/18/2017 25 Bertrand Centripetal Force Definition: Equation: Sample problem A 1200-kg car rounds a corner of radius r = 45 m. If the coefficient of static friction between tires and the road is 0.93 and the coefficient of kinetic friction between tires and the road is 0.75, what is the maximum velocity the car can have without skidding? Sample problem You whirl a 2.0 kg stone in a horizontal circle about your head. The rope attached to the stone is 1.5 m long. What is the tension in the rope? 6/18/2017 26 Bertrand Work Sample problem Jane uses a vine wrapped around a pulley to lift a 70-kg Tarzan to a tree house 9.0 meters above the ground. a) How much work does Jane do when she lifts Tarzan? Definition: Equation: Units of Work (SI System) b) How much work does gravity do when Jane lifts Tarzan? Question: If a man holds a 50 kg box at arms length for 2 hours as he stands still, how much work does he do on the box? Sample problem Joe pushes a 10-kg box and slides it across the floor at constant velocity of 3.0 m/s. The coefficient of kinetic friction between the box and floor is 0.50. a) How much work does Joe do if he pushes the box for 15 meters? Question: If a man holds a 50 kg box at arms length for 2 hours as he walks 1 km forward, how much work does he do on the box? Question: If a man lifts a 50 kg box 2.0 meters, how much work does he do on the box? b) How much work does friction do as Joe pushes the box? Work and Energy Work changes mechanical energy. Sample problem A father pulls his child in a little red wagon with constant speed. If the father pulls with a force of 16 N for 10.0 m, and the handle of the wagon is inclined at an angle of 60o above the horizontal, how much work does the father do on the wagon? Positive work ________________ mechanical energy. Negative work ________________ mechanical energy. The two forms of mechanical energy are called: 6/18/2017 27 Bertrand Kinetic Energy b) Did air resistance do positive, negative or zero work on the acorn? Why? Definition: Equation: c) How much work was done by air resistance? Units of Energy (SI System): Sample problem A 10.0 g bullet has a speed of 1.20 km/s. a) What is the kinetic energy of the bullet? d) What was the average force of air resistance? b) What is the bullet’s kinetic energy if the speed is halved? Constant force and work: draw and label graph c) What is the bullet’s kinetic energy if the speed is doubled? The Work-Energy Theorem Definition: Variable force and work: draw and label graph Equation: Sample problem An 8.0-g acorn falls from a tree and lands on the ground 10.0 m below with a speed of 11.0 m/s. a) What would the speed of the acorn have been if there had been no air resistance? 6/18/2017 28 Bertrand Springs Why is the work done by a spring when it is stretched or compressed negative? Sample problem It takes 1000 J of work to compress a certain spring 0.10 m. a) What is the force constant of the spring? Equation for work done by spring: b) To compress the spring an additional 0.10 m, does it take 1000 J, more than 1000 J, or less than 1000 J? Verify your answer with a calculation. Graph for stretching of spring F Sample Problem How much work is done by the force shown when it acts on an object and pushes it from x = 0.25 m to x = 0.75 m? x Sample problem A spring with force constant 250 N/m is initially at its equilibrium length. a) How much work must you do to stretch the spring 0.050 m? Figure from “Physics”, James S. Walker, Prentice-Hall 2002 Sample Problem How much work is done by the force shown when it acts on an object and pushes it from x = 2.0 m to x = 4.0 m? b) How much work must you do to compress it 0.050 m? Figure from “Physics”, James S. Walker, Prentice-Hall 2002 6/18/2017 29 Bertrand Power Unit of Power (SI system): Force types Conservative forces: Work is path independent. Work along a closed path is zero. Work changes potential energy. Examples: gravity, springs Non-conservative forces: Work is path dependent. Work along a closed path is NOT zero. Work changes mechanical energy. Examples: friction, drag Unit of Power (British system): Definition: Definition: Equation: Potential energy Conversion from horsepower to Watts: Equation (gravity): The kilowatt-hour Definition: Equation (spring): Sample problem A man runs up the 1600 steps of the Empire State Building in 20 minutes seconds. If the height gain of each step was 0.20 m, and the man’s mass was 80.0 kg, what was his average power output during the climb? Give your answer in both watts and horsepower. Conservative forces and Potential energy Equation: What happens to potential energy when a conservative force does positive work? What about when a conservative force does negative work? Sample problem Calculate the power output of a 0.10 g fly as it walks straight up a window pane at 2.0 cm/s. More on paths and conservative forces. Q: Assume a conservative force moves an object along the various paths. Which two works are equal? Q: Which two works, when added together, give a sum of zero? Figure from “Physics”, James S. Walker, Prentice-Hall 2002 6/18/2017 30 Bertrand Sample problem A box is moved in the closed path shown. a) How much work is done by gravity when the box is moved along the path A->B->C? Sample problem If 30.0 J of work are required to stretch a spring from a 2.00 cm elongation to a 4.00 cm elongation, how much work is needed to stretch it from a 4.00 cm elongation to a 6.00 cm elongation? Figure from “Physics”, James S. Walker, Prentice-Hall 2002 Law of Conservation of Energy Statement: b) How much work is done by gravity when the box is moved along the path A->B->C->D->A? Sample problem A box is moved in the closed path shown. a) How much work would be done by friction if the box were moved along the path A->B->C? Law of Conservation of Mechanical Energy Equations: Pendulums and Energy Conservation Q: Where in pendulum swing is mechanical energy all potential energy? Figure from “Physics”, James S. Walker, Prentice-Hall 2002 b) How much work is done by friction when the box is moved along the path A->B->C->D->A? Q. What kind of potential energy exists in pendulum? Give the equation. Sample problem A diver drops to the water from a height of 20.0 m, his gravitational potential energy decreases by 12,500 J. How much does the diver weigh? 6/18/2017 Q: Where in pendulum swing is mechanical energy all kinetic energy? 31 Bertrand Springs and Energy Conservation Q: Where in spring oscillatin is mechanical energy all potential energy? For 2.0 m Q. What kind of potential energy exists in a spring? Give the equation. For 0.0 m Q: Where in spring oscillation is mechanical energy all kinetic energy? Sample problem Problem from “Physics”, James S. Walker, Prentice-Hall 2002 A 1.60 kg block slides with a speed of 0.950 m/s on a frictionless, horizontal surface until it encounters a spring with a force constant of 902 N/m. The block comes to rest after compressing the spring 4.00 cm. Find the spring potential energy, U, the kinetic energy of the block, K, and the total mechanical energy of the system, E, for the following compressions: 0 cm, 2.00 cm, 4.00 Sample problem What is the speed of the pendulum bob at point B if it is released from rest at point A? 40o 1.5 m For 0 cm A B Sample problem Problem from “Physics”, James S. Walker, Prentice-Hall 2002 A 0.21 kg apple falls from a tree to the ground, 4.0 m below. Ignoring air resistance, determine the apple’s gravitational potential energy, U, kinetic energy, K, and total mechanical energy, E, when its height above the ground is each of the following: 4.0 m, 2.0 m, and 0.0 m. Take ground level to be the point of zero potential energy. For 2.00 cm For 4.0 m For 4.00 cm 6/18/2017 32 Bertrand Law of Conservation of Energy Equations: For 0.50 m Work done by non-conservative forces Wnet = Wc + Wnc Wc = -U Wnet = K K = -U + Wnc Wnc = U + K = E Sample problem Problem from “Physics”, James S. Walker, Prentice-Hall 2002 Catching a wave, a 72-kg surfer starts with a speed of 1.3 m/s, drops through a height of 1.75 m, and ends with a speed of 8.2 m/s. How much non-conservative work was done on the surfer? For 1.00 m Pendulum lab Figure out how to demonstrate conservation of energy with a pendulum using the equipment provided. The photogates must be set up in “gate” mode this time. The width of the pendulum bob is an important number. To get it accurately, use the caliper. Turn in just your calculations, which must clearly show the speed you predict for the pendulum bob from conservation of energy, the speed you measure using the caliper and photogate data, and a %difference for the two. Sample problem Problem from “Physics”, James S. Walker, Prentice-Hall 2002 A 1.75-kg rock is released from rest at the surface of a pond 1.00 m deep. As the rock falls, a constant upward force of 4.10 N is exerted on it by water resistance. Calculate the nonconservative work, Wnc, done by the water resistance on the rock, the gravitational potential energy of the system, U, the kinetic energy of the rock, K, and the total mechanical energy of the system, E, for the following depths below the water’s surface: d = 0.00 m, d = 0.500 m, d = 1.00 m. Let potential energy be zero at the bottom of the pond. For 0.00 m 6/18/2017 33 Bertrand Physics B Notes: Momentum Momentum (p) Name:_________________ Momentum change of Lazy Ball: Definition: Equation: one particle Momentum change of Bouncy Ball: Equation: multiple particles Units: Impulse (J) Definition: Sample Problem Calculate the momentum of a 65-kg sprinter running east at 10 m/s. What does impulse change? Equations: Sample Problem Calculate the momentum of a system composed of a 65-kg sprinter running east at 10 m/s and a 75-kg sprinter running north at 9.5 m/s. Units: N s or kg m/s (same as momentum) What characterizes impulsive forces? Impulse (J) on a graph Change in momentum Equation: 6/18/2017 34 Bertrand Sample Problem Suppose a 1.5-kg brick is dropped on a glass table top from a height of 20 cm. A) What is the magnitude and direction of the impulse necessary to stop the brick? Law of Conservation of Momentum Definition: Equation: Sample problem A 75-kg man sits in the back of a 120-kg canoe that is at rest in a still pond. If the man begins to move forward in the canoe at 0.50 m/s relative to the shore, what happens to the canoe? B) If the table top doesn’t shatter, and stops the brick in 0.01 s, what is the average force it exerts on the brick? C) What is the average force that the brick exerts on the table top during this period? External versus internal forces External forces are Internal forces are What can external forces do that internal forces cannot? Explosions What type of forces exist in an explosion (external or internal?) 6/18/2017 35 Bertrand What is conserved in an explosion? Collisions Definiton: What is not conserved in an explosion? What is conserved in all collisions? Recoil Definition: Collision Types Describe an elastic collision. Which of Newton’s three laws is most applicable to recoil? Describe a perfectly inelastic collision. What is conserved in both elastic and inelastic collisions? Sample problem Suppose a 5.0-kg projectile launcher shoots a 209 gram projectile at 350 m/s. What is the recoil velocity of the projectile launcher? What is conserved in an elastic collision but not conserved in an inelastic collision? Sample Problem An 80-kg roller skating grandma collides inelastically with a 40-kg kid. What is their velocity after the collision? What is the change in kinetic energy? Sample problem An exploding object breaks into three fragments. A 2.0 kg fragment travels north at 200 m/s. A 4.0 kg fragment travels east at 100 m/s. The third fragment has mass 3.0 kg. What is the magnitude and direction of its velocity? Sample Problem A fish moving at 2 m/s swallows a stationary fish which is 1/3 its mass. What is the velocity of the big fish and after dinner? 6/18/2017 36 Bertrand Sample Problem A car with a mass of 950 kg and a speed of 16 m/s to the east approaches an intersection. A 1300-kg minivan traveling north at 21 m/s approaches the same intersection. The vehicles collide and stick together. What is the resulting velocity of the vehicles after the collision? 2-Dimensional Collisions What key concept do you need to remember when you work 2-dimensional collisions problems, either elastic or inelastic? Sample problem Sample Problem – elastic collision A 500-g cart moving at 2.0 m/s on an air track elastically strikes a 1,000-g cart at rest. What are the resulting velocities of the two carts? Sample Problem Suppose three equally strong, equally massive astronauts decide to play a game as follows: The first astronaut throws the second astronaut towards the third astronaut and the game begins. Describe the motion of the astronauts as the game proceeds. Assume each toss results from the same-sized "push." How long will the game last? 6/18/2017 37 Bertrand Mechanical Wave Definition: Wave types: transverse Definition: Draw a wave; label the parts Examples Wave types: longitudinal Definition: Speed of a wave Equation 1: Equation 2: Examples Period of a wave Equation: Reflection of waves Definition: . Problem: Sound travels at approximately 340 m/s, and light travels at 3.0 x 108 m/s. How far away is a lightning strike if the sound of the thunder arrives at a location 2.0 seconds after the lightning is seen? Characteristics of Fixed-end reflection Characteristics of Open-end reflection Problem: The frequency of an oboe’s A is 440 Hz. What is the period of this note? What is the wavelength? Assume a speed of sound in air of 340 m/s. 6/18/2017 Refraction of waves Definition: 38 Bertrand Principle of Superposition Definition: What can change when a wave refracts? What never changes when a wave refracts? Sound What type of wave is sound? Constructive interference Definition: How does the oscilloscope display a pure tone? Picture of waveforms undergoing constructive interference: What does a Fourier transform look like for a pure tone? Destructive interference. Definition: How does the oscilloscope display a complex tone? Picture of waveforms undergoing destructive interference: What does a Fourier transform look like for a complex tone? Sample Problem: Draw the waveform from its two components. Doppler Effect Definition: Approaching sound has ________ pitch. Retreating sound has ________ pitch. 6/18/2017 39 Bertrand Standing Wave Open-end standing waves 1st harmonic Definition: What role does reflection play in formation of a standing wave? 2nd harmonic What role does superposition play in a standing wave? 3rd harmonic Fixed-end standing waves 1st harmonic Mixed standing waves 1st harmonic 2nd harmonic 2nd harmonic 3rd harmonic 3rd harmonic 6/18/2017 40 Bertrand Sample Problem How long do you need to make an organ pipe that produces a middle C (256 Hz)? The speed of sound in air is 340 m/s. A) Draw the first harmonic. Resonance Definition: Beats Definition: B) Calculate the pipe length. Drawing: C) What is the wavelength and frequency of the 2nd harmonic? Diffraction Definition: Double-slit or multi-slit diffraction Equation: Sample Problem How long do you need to make an organ pipe whose fundamental frequency is a Csharp (273 Hz)? The pipe is closed on one end, and the speed of sound in air is 340 m/s. A) Draw the fundamental. Single slit diffraction Equation: Sample Problem Light of wavelength 360 nm is passed through a diffraction grating that has 10,000 slits per cm. If the screen is 2.0 m from the grating, how far from the central bright band is the first order bright band? B) Calculate the pipe length. C) What is the wavelength and frequency of the 2nd harmonic? 6/18/2017 41 Bertrand Sample Problem Graph: Light of wavelength 560 nm is passed through two slits. It is found that, on a screen 1.0 m from the slits, a bright spot is formed at x = 0, and another is formed at x = 0.03 m? What is the spacing between the slits? x(m) t Definitions: Amplitude Sample Problem Light is passed through a single slit of width 2.1 x 10-6 m. How far from the central bright band do the first and second order dark bands appear if the screen is 3.0 meters away from the slit? Assume 560 nm light. Period Frequency Ideal Springs What makes springs ideal? Periodic Motion Definition: Hooke’s Law Equation: What are mechanical devices that undergo periodic motion called? Period of a spring Equation: Simple Harmonic Motion Definition: Sample Problem Calculate the period of a 300-g mass attached to an ideal spring with a force constant of 25 N/m. Simple Harmonic Oscillators Definition: Examples: 6/18/2017 42 Bertrand Sample Problem A 300-g mass attached to a spring undergoes simple harmonic motion with a frequency of 25 Hz. What is the force constant of the spring? Sample problem A spring of force constant k = 200 N/m is attached to a 700-g mass oscillating between x = 1.2 and x = 2.4 meters. Where is the mass moving fastest, and how fast is it moving at that location? Sample Problem An 80-g mass attached to a spring hung vertically causes it to stretch 30 cm from its unstretched position. If the mass is set into oscillation on the end of the spring, what will be the period? Sample problem A spring of force constant k = 200 N/m is attached to a 700-g mass oscillating between x = 1.2 and x = 2.4 meters. What is the speed of the mass when it is at the 1.5 meter point? Sample Problem You wish to double the force constant of a spring. You A. Double its length by connecting it to another one just like it. B. Cut it in half. C. Add twice as much mass. D. Take half of the mass off. Sample problem A 2.0-kg mass attached to a spring oscillates with an amplitude of 12.0 cm and a frequency of 3.0 Hz. What is its total energy? Conservation of Energy Where does maximum kinetic energy occur? Where does maximum potential energy occur? Where does maximum total energy occur? 6/18/2017 43 Bertrand Pendulums When is a pendulum a good approximation of a simple harmonic oscillator? Pendulum Forces Equation: Sample problem Predict the period of a pendulum consisting of a 500 gram mass attached to a 2.5-m long string. Sample problem Suppose you notice that a 5-kg weight tied to a string swings back and forth 5 times in 20 seconds. How long is the string? Sample problem The period of a pendulum is observed to be T. Suppose you want to make the period 2T. What do you do to the pendulum? 6/18/2017 44 Bertrand A hollow tube of adjustable length, open at both ends, is held in midair as shown. A tuning fork with frequency 320 Hz vibrates at one end of the tube and causes the air in the tube to vibrate at its fundamental frequency. The speed of sound in the laboratory is 343 m/s. a) Draw the fundamental standing wave inside the tube. b) Determine the length of the tube that will support this fundamental frequency. c) Determine the next higher frequency at which this air column would resonate. Draw the standing wave represented by this frequency. (Do not change the length of the tube.) The tube is now submerged in a large, graduated cylinder filled with water. The tube is slowly raised out of the water and the same tuning fork, vibrating with frequency 320 Hz, is held a fixed distance from the top of the tube. d) Determine the height h of the tube above the water when the air column resonates for the first time. 6/18/2017 45 Bertrand S116B4 (15 points) Your teacher gives you a slide with two closely spaced slits on it. She also gives you a laser with a wavelength λ = 632 nm. The laboratory task that you are assigned asks you to determine the spacing between the slits. These slits are so close together that you cannot measure their spacing with a typical measuring device. a. From the list below, select the additional equipment you will need to do your experiment by checking the line next to each item. _____Meterstick _____Ruler _____Tape measure _____Light-intensity meter _____Large screen _____Paper _____Slide holder _____Stopwatch d. Outline the procedure that you would use to make the needed measurements, including how you would use each piece of the additional equipment you checked in a. e. Using equations, show explicitly how you would use your measurements to calculate the slit spacing. b. Draw a labeled diagram of the experimental setup that you would use. On the diagram, use symbols to identify carefully what measurements you will need to make. c. On the axes below, sketch a graph of intensity versus position that would be produced by your setup, assuming that the slits are very narrow compared to their separation. 46 A172 B2. A block of mass M is resting on a horizontal, frictionless table and is attached as shown above to a relaxed spring of spring constant k. A second block of mass 2M and initial speed vo collides with and sticks to the first block Develop expressions for the following quantities in terms of M, k, and vo a. v, the speed of the blocks immediately after impact b. x, the maximum distance the spring is compressed c. T, the period of the subsequent simple harmonic motion 47 (e) Calculate the frequency of oscillation of the 8.0 kg block (NEW). S117 B1 (15 points) An ideal spring of unstretched length 0.20 m is placed horizontally on a frictionless table as shown above. One end of the spring is fixed and the other end is attached to a block of mass M = 8.0 kg. The 8.0 kg block is also attached to a massless string that passes over a small frictionless pulley. A block of mass m = 4.0 kg hangs from the other end of the string. When this spring-and-blocks system is in equilibrium, the length of the spring is 0.25 m and the 4.0 kg block is 0.70 m above the floor. (a) (f) Calculate the maximum speed attained by the 8.0 kg block (NEW). On the figures below, draw free-body diagrams showing and labeling the forces on each block when the system is in equilibrium. (b) Calculate the tension in the string (REVIEW). (c) Calculate the force constant of the spring (REVIEW). The string is now cut at point P. (d) Calculate the time taken by the 4.0 kg block to hit the floor (REVIEW).. 48