Work, Energy and Momentum Notes 1 Work and Energy

Transcription

1 Work, Energy and Momentum Notes 1 Work and Energy Work is defined as the transfer of energy from one body to another. Or more rigorously: We can calculate the work done on an object with: the units of work are Nm or Joules Note that these are the same units as torque yet their values are used to describe very different quantities. Example 1 - Work against Gravity How much work is required to lift a 2.0 kg textbook from the floor to a height of 1.5 m at a constant velocity? Note: W = Fd, but what force do we need to exert to lift the book at a constant velocity? Example 2 - Work on an object How much work is done on a 4.0 kg medicine ball that is held at a height of 1.8 m for 10 s? Note: Is energy being used to hold the ball in this position? Is work actually being done ON THE BALL? Since the velocity is constant what is the net force acting on the book? Example 3 Forces at an angle A plucky youngster is pulling his sled at a constant velocity of 1.2 m/s. He pulls the 15 kg sled with a force of 35 N at an angle of 40 o to the horizontal. How much work does he do in pulling the sled 20 m? Draw an FBD showing the forces at work on the sled. Break F boy into its vertical and horizontal components. Does the vertical component of the force do any work? Rule: When finding the work done on an object we only consider. in the direction of motion Fancy New Definition of Work

2 Example 4 F net vs. F app A biology student is pushing a rope 15 m across a flat surface. The student pushes the rope with a force of 220 N while the force of friction is 120 N. How much work is the student doing? Note: To find the amount of work done by the student should we used F net or F app? Rule: When finding the total work done on an object we always use: Example 5 To scalar or not to scalar? Work is the product of a scalar and a vector, but work is a. However work can be positive or negative. 1) Car accelerates from rest 2) Skater stops suddenly 3) Lifting a object above your head 4) Ball rolls down a ramp Types of Energy: There are many forms of energy: mechanical, thermal, electrical, nuclear, chemical etc. One form can be converted into another by doing work. In this chapter will be concerned mostly with potential and kinetic (and just a hint of thermal) energy. Potential Energy (E p ): Kinetic Energy (E k ): Cause math is fun! Cause math is fun! Deriving the E p formula Deriving the E k formula Remember: Potential energy is always E p = Fd (in this case F = F g = mg) E p = mgd (let s change letters just for fun, and call h the vertical displacement) E p = mgh HOORAY! v 2 = v o 2 + 2ad (take v o = 0) v 2 = 2ad (but a = F/m) v 2 = 2Fd m (but Fd=W= E k, when v o = 0) v 2 = 2E k m E k = ½mv 2

3 Example 6 - Work- Energy Theorem for Net Force It is worth noting that the work done by the net force on an object is equal to the change in its kinetic energy: Example: The graph shows a variable force working on a 15.0 kg mass on a level surface which is initially at rest. Find: a. The total amount of work done. b. The final speed of the object assuming friction is negligible. Example: A 1270 kg car accelerates from 15 m/s to 25 m/s over a distance of 75 m. Determine the average net force that was required to do this. To summarize our work energy relationships: The Law of Conservation of Energy Energy cannot be or, only from one form into another. Therefore if only conservative forces act in in a system the in energy is always. Total Initial Energy = Total Final Energy Total Change in Energy = 0 Example The first peak of a roller coaster is 55 m above the ground. The 1200kg car starts from rest and goes down the hill and up the second hill which is 30 m high. How fast is the car traveling at the top of the second hill?

4 Back in grade 11 it really was that easy When non- conservative forces (such as friction) act on an object, not all energy is transferred between kinetic and potential. This is what physicists have termed REALITY. Deal with it. The work done by friction does produce another form of energy known as This energy is quickly conducted, convected or radiated in all directions and is effectively dispersed. Consider a block of wood sliding down a ramp with a small amount of friction. How would the block s kinetic energy at the bottom compare to its potential energy at the top? Why? The fact that the amount of energy in the block decreases as it slides down the ramp doesn t change the fact that the in the system is CONSTANT. We need modify our earlier equation for The Law of Conservation of Energy only slightly: Total Energy Initial = Total Energy Final Example A 5.0 kg block of wood is now pushed down a ramp with a velocity of 6.0 m/s. At the bottom of the ramp it is traveling at 7.5 m/s. a. How much thermal energy is generated due to friction? 3.5 m 1.5 m b. Determine the force of friction.

5 Work, Energy and Momentum Notes 2 Power and Efficiency In everyday language we often use the words WORK, ENERGY and POWER synonymously. However this makes the physics gods extremely furious because we should all know that: POWER is Mathematically we define power as: = = The unit of power is J/s or Watts (W) (this is sometimes confusing because W is also the symbol for work) Example: A physics student is setting up a wicked body slam on a biology student. He lifts the 75 kg student clear over his head to a height of 2.2 m in s. How much power did the physics student generate? Example: While cruising along level ground in a one horse open sleigh at 4.0 m/s, Mr. Lawson cracks the whip and speeds up to 12.0 m/s in 4.5 s. If the sleigh has a mass of 850 kg, how much power did it generate? Ignore friction. Another useful equation for power can be derived: Example: A student pushes 14 kg of their physics homework up a 40 o ramp at a constant velocity of 3.2 m/s. The friction force is 26 N. How much power must the student exert? Whenever we use a machine to do work some of the energy we put into the machine is always lost, mainly due to friction. For example an electric heater is efficient a car is efficient a lightbulb is efficient We can define efficiency in two ways: Efficiency = = Notice that efficiency is a ratio expressed as a percentage and therefore has no units!

6 The most common source of confusion when calculating efficiency is in understanding which values applies to work/power IN and which applies to work/power OUT. Work/Power In: Work/Power Out: Remember that energy is always LOST somewhere in using the machine, so Work IN Work OUT and Efficiency Example: The Top Thrill Dragster is one of the tallest roller coaster in the world. The car is accelerated along a level track until they take a 90 o vertical turn and travel to the peak, 120 m high. A typical fully loaded car has a mass of 2800 kg. Example: On the Incredible Hulk roller coaster the car is initially launched up a hill 34 m high, traveling from 0 to 64 km/h in 2.0 s. A full car has a mass of 4500 kg. a) Find the power output of the ride. a) Calculate the minimum amount of work done on the car in order for it to reach the peak. b) In reality the roller coaster is accelerated from 0 to 193 km/h in 3.8 s. Find the actual power input of the ride. b) The power consumption during the initial launch is actually 1.45 MW. Determine the efficiency of the ride during the initial launch. c) If the car pulls in to the station at 8.0 m/s. How much heat has been generated from the first peak? c) Determine the efficiency of the ride from start to peak. d) The car is finally brought to rest over a distance of 2.0 m. How much force is required?

7 Work, Energy and Momentum Notes 3 Momentum and Collisions Momentum is a quantity of motion that depends on both the mass and velocity of the object in question. The units of momentum are: Remember: Momentum is a quantity, with the same sign as its velocity. As with any vector you can assign any direction as positive and the opposite as negative, but as convention we will refer to up or to the right as positive and down or to the left as negative. Example: A baseball pitcher hurls a ball at 32 m/s. The batter crushes it and the ball leaves the bat at 48 m/s. What was the ball s change in momentum? Remember that momentum is a VECTOR which means: Impulse: Derivation: Recall that momentum is the product of and. Since we will not be dealing with changing masses, we can define an object s change in momentum as: Whenever a net force acts on a body, an acceleration results and so its momentum must change. Let s try to understand how forces relate to changes in momentum with a few examples. A student jumps off a desk. When they land they bend their knees on impact. Why does this help prevent some serious damage to their knees? Coaches for many sports such as baseball, tennis and golf can often be heard telling their athletes to follow through with their swing. Why is this so important? Conventional wisdom suggests that cars should be made tough and rigid to prevent injury during a collision, however newer vehicles are all built with large crumple zones. Why? A beanbag and a high bounce ball of equal masses are dropped from the same height. The beanbag is brought to a stop in the same time that the ball is in contact with the floor. Which one exerts a greater average force on the floor?

8 Example A 115 kg fullback running at 4.0 m/s East is stopped in 0.75 s by a head- on tackle. Calculate a) the impulse felt by the fullback. b) the impulse felt by the tackler. c) the average net force exerted on the tackler. Example A 1250 kg car traveling east at 25 m/s turns due north and continues on at 15 m/s. What was the impulse of the car exerted while turning the corner? The Law of Conservation of Momentum Momentum is a useful quantity because in a closed system it is always conserved. This means that in any collision, the total momentum before the collision must equal the total momentum after the collision. There are two ways of thinking about the conservation of momentum: (1) (2) Collisions can be grouped into two categories, Elastic Collisions: Inelastic Collisions: In reality collisions are generally somewhere in between perfectly elastic and perfectly inelastic. As a matter of fact, it is impossible for a macroscopic collision to ever be perfectly elastic. Perfectly elastic collisions can only occur at the atomic or subatomic level. Why can t macroscopic collision ever be truly elastic?

9 Inelastic Two rugby players of equal mass collide head on while traveling at the same speed. What is their final speed? A 9500 kg caboose is at rest on some tracks. An kg engine moving east at 12.0 m/s collides with it and they stick together. What is the velocity of the train cars after the collision? Is momentum conserved? Is energy conserved? Is kinetic energy conserved? Elastic A proton traveling at 2x10 3 m/s collides with a stationary proton and comes to rest. An alpha particle has a mass approximately 4 times larger than a proton. A proton traveling to the right at 3200 m/s strikes a stationary alpha particle it rebounds at 1920 m/s. What is the final speed of the alpha particle? What is the final speed of the other proton? Is kinetic energy conserved? Explosions A firecracker is placed in a pumpkin which explodes in into exactly two pieces. The first piece has a mass of 2.2 kg and flies due east at 26 m/s. The second chunk heads due west at 34 m/s. What was the initial mass of the pumpkin?

10 Work, Energy and Momentum Notes 4 Collisions in 2- D When dealing with collisions in 2- dimensions it is important to remember that momentum is a vector with magnitude and direction. When finding the total momentum we have to do: Collisions at 90 o : A 750 kg Peugeot travelling at 21 m/s West collides with a 680 kg Fiat travelling at 18 m/s South. If the two cars become entwined what is their total final velocity? Remember that it is momentum that is conserved, so we need to add the NOT Collisions not at 90 o (because life is never that easy ): A 4.0 kg bowling ball is moving east at an unknown velocity when it collides with a 6.1 kg frozen cantaloupe at rest. After the collision, the bowling ball is traveling at a velocity of 2.8 m/s 32 o N of E and the cantaloupe is traveling at a velocity of 1.5 m/s 41 o S of E. What was the initial velocity of the bowling ball? Before After Component Method We need to break the final momenta of the two objects into x and y components:

11 We then add the individual x and the individual y components to find our total momentum. Σp x = p 1x + p 2x = Σp y = p 1y + p 2y = Notice that the total momentum is all in the! This should be no surprise since the bowling ball was initially only moving in the x direction. Don t forget to solve for the initial velocity (magnitude and direction): Vector Addition: Simply add the vectors and solve with the sine or cosine law. Notice that the is either the initial or the final because momentum is. First we need to use geometry to solve for the angle opposite the total momentum. And then, start hammering:

12 Worksheet Work 1. A 20.0 N pomegranate is lifted at a constant velocity from the floor to a height of 1.50 m. How much work is done on the object? 2. A 15.0 N potato is moved horizontally 3.00 m across a level floor using a horizontal force of 6.00 N. How much work is done on the potato? 3. A 2.20 N pear is held 2.20 m above the floor for 10.0 s. How much work is done on the pear? 4. A 10.0 kg pink grapefruit is accelerated horizontally from rest to a velocity of 11.0 m/s in 5.00 s by a horizontal force. How much work is done on the pink grapefruit assuming no friction? 5. A 90.0 N box of papayas is pulled 10.0 m along a level surface by a rope. If the rope makes an angle of 20 o with the surface, and the force in the rope is 75.0 N, how much work is done on the box? 6. A 60.0 kg student runs at a constant velocity up a flight of stairs. If the height of the stairs is 3.2 m, what is the work done against gravity? 7. A 20.0 kg passionfruit is pulled horizontally 9.0 m along a level frictionless surface at a constant velocity. How much work is done on the passionfruit? 8. An 80.0 kg pumpkin is pushed up at a constant velocity along a frictionless incline as shown in the diagram. How much work is done on the pumpkin in moving it up the incline? 10.0 m 7.0 m 9. A 25.0 kg pickle is accelerated from rest through a distance of 6.0 m in 4.0 s across a level floor. If the friction force between the pickle and the floor is 3.8 N, what is the work done to move the object? 10. A 1165 kg car traveling at 55 km/h is brought to a stop while skidding 38 m. Calculate the work done on the car by the friction forces. 1) 30.0 J 2) 18.0 J 3) 0J 4) 605 J 5) 705 J 6) 1900 J 7) 0 J 8) 5500 J 9) 140 J 10) -1.4x10 5 J Worksheet 4.1 Ep and Ek 1. A 25.0 N object is held 2.10 m above the ground. What is the potential energy with respect to the ground? 2. An uncopressed spring is 20.0 cm in length. What is the potential energy of the spring when an average force of 65.0 N compresses it to a length of 13.5 cm? 3. A 2.75 kg box is at the top of a frictionless incline as shown in the diagram. What is the potential energy with respect to the bottom of the incline? 7.00 m 10.0 m 4. The bob of a pendulum has a mass of 2.0 kg and hangs 0.50 m above the floor. The bob is pulled sideways so that it is 0.75 m above the floor. What is its potential energy with respect to its equilibrium position?

13 5. A 2.00 x10 3 kg object is pushed to the top of an incline as shown. If the force applied along the incline is 6.00 x 10 2 N, what is the potential energy of the object when it is at the top of the incline with respect to the bottom? 6.0 m 12.0 m 6. A 3.0 kg ewok is traveling at a constant speed of 7.5 m/s. What is its kinetic energy? 7. The kinetic energy of a 20.0 N droid is 5.00 x 10 2 J. What is the speed of the droid? 8. A 10.0 N lightsaber is accelerated from rest at a rate of 2.5 m/s 2. What is the kinetic energy of the lightsaber after it has accelerated over a distance of 15.0 m. 9. A N Wookie falls off a cliff on Earth. What is its kinetic energy after it falls for 4.50 s? 10. An 8.0 kg bantha poodoo is dropped from a height of 7.0 m. What is the kinetic energy of the poodoo just before it hits the ground? 11. A 9.00 kg object falls off of a 1.2 m high table. If all of the objects potential energy is converted into kinetic energy just before it hits the floor, how fast is it moving? 12. Solve #11 using kinematics this time. Is there any difference? 1) 52.5 J 2) 4.2 J 3) 189 J 4) 4.9 J 5) J 6) 84 J 7) 22.1 m/s 8) 38 J 9) 1.2x10 5 J 10) 550 J 11) 4.85 m/s 12) 4.85 m/s Worksheet 4.1 Law of Conservation of Energy Use the Law of Conservation of Energy to solve the following problems. 1. Physics student is dropped (don t ask why or you re next). If they reach the floor at a speed of 3.2 m/s, from what height did they fall? 2. A heavy object is dropped from a vertical height of 8.0 m. What is its speed when it hits the ground? 3. A bowling ball is dropped from the top of a building. If it hits the ground with a speed of 37.0 m/s, how tall was the building? 4. A safe is hurled down from the top of a 1.3 x 10 2 m building at a speed of 11.0 m/s. What is its velocity as it hits the ground? 5. A box slides down a frictionless ramp. If it starts at rest, what is its speed at the bottom? 4.0 m 9.0 m 6. A pendulum is dropped from the position shown, 0.25 m above its equilibrium position. What is the speed of the pendulum bob as it passes through its equilibrium position? 1.00 m 0.25 m 12.0 m 30 o 7. A box slides down a frictionless incline as shown. If the box starts from rest, what is its speed at the bottom?

14 8. A roller coaster car starts from rest at point A. What is its speed at point C if the track is frictionless? A 12.0 m B C 2.0 m 4.0 m 9. A 2.5 kg object is dropped from a height of 10.0 m above the ground. Calculate the speed of the object as it hits the ground. 10. An 80.0 kg student running at 3.5 m/s grabs a rope that is hanging vertically. How high will the student swing? 11. A pendulum is 1.20 m long. If the pendulum is pulled until it makes a 25.0 o angle to the vertical, what is the speed of the pendulum bob when it passes through its equilibrium position? HINT: Determine the vertical drop of the pendulum bob first ) 0.52 m 2) 13 m/s 3) 69.8 m 4) 52 m/s 5) 8.9 m/s 6) 2.2 m/s 7) 10.8 m/s 8) 13 m/s 9) 14 m/s 10) 0.63 m 11) 1.5 m/s Worksheet 4.2 1) A 45.0 kg student runs at a constant velocity up the incline shown. If the power output of the student is 1.50 x 10 3 W, how long does it take the student to run the 9.0 m along the incline? (1.8 s) 6.0 m 9.0 m 2) A 20.0 kg object is lifted vertically at a constant velocity 2.50 m in 2.00 s. Calculate the power output of the student. (245 W) 3) A 2.00 kg object is accelerated uniformly from rest to 3.00 m/s while moving 1.5 m across a level frictionless surface. Calculate the power output. (9.0 W) 4) An 8.5 x 102 kg elevator is pulled up at a constant velocity of 1.00 m/s by a 10.0 kw motor. Calculate the efficiency of the motor. (83%) 5) A 5.0 kg object is accelerated uniformly from rest to 6.0 m/s while moving 2.0 m across a level surface. If the force of friction is 4.0 N, calculate the power output. (1.5 x 10 2 W) 6) A 5.00 x 10 2 W electric motor lifts a 20.0 kg object 5.00 m in 3.50 s. What is the efficiency of the motor? (56%) 7) If a 1.00 x 10 2 kw motor has an efficiency of 82%, how long will it take to lift a 50.0 kg object to a height of 8.00 m? (0.048 s)

15 Worksheet 4.3 Law of Conservation of Momentum 1. A 1.0 kg ball hits the floor with a velocity of 2.0 m/s. If the ball bounces up with a velocity of 1.6 m/s, what is the ball s change in momentum? (3.6 kgm/s) 2. A kg baseball is pitched horizontally at + 38 m/s. The batter hits a horizontal line drive at 38 m/s (the opposite direction!). What is the ball s change in momentum? ( kgm/s) 3. The 800 kg physics dragster is traveling at 35 km/h east when it hits the gas and accelerates at 12.5 m/s 2 for 3.25 s. What is its change in momentum during this time? (32500 kgm/s) 4. A 30.0 kg object moving to the right at a velocity of 1.00 m/s collides with a 20.0 kg object moving to the left with a velocity of 5.00 m/s. If the 20.0 kg object continues to move to the left at a velocity of 1.25 m/s, what is the velocity of the 30.0 kg object? (1.50 m/s left) 5. A 4.50 x 10 3 kg railway car is moving east at a velocity of 5.0 m/s on a level frictionless track when it collides with a stationary 6.50 x 10 3 kg caboose. If the two cars lock together upon impact, how fast are they moving after collision? (2.0 m/s east) 6. A 925 kg car moving at a velocity of 18.0 m/s right collides with a stationary truck of unknown mass. The two vehicles lock together and move off at a velocity of 6.50 m/s. What is the mass of the truck? (1640 kg) 7. A 50.0 g bullet strikes a 7.00 kg wooden block. If the bullet becomes imbedded in the block and they both move off at a velocity of 5.00 m/s, what was the initial velocity of the bullet? (705 m/s) 8. A 40.0 g hot dog moving with a velocity of 9.00 m/s to the right collides with a 55.0 g hot dog bun with a velocity of 6.00 m/s to the left. If the two objects stick together upon collision, what is the velocity of the combined masses? (0.316 m/s right) 9. A 76 kg student, standing at rest on a frictionless surface throws a 0.20 kg cream pie horizontally at 22 m/s at Mr. Trask who is standing to the student s left. What was the velocity of the student after they throw the pie? (0.058 m/s right) 10. A 25 kg turkey is fired from a 1.1 x 10 3 kg turkey launcher. If the horizontal velocity of the turkey is 325 m/s east, what is the recoil velocity of the launcher? (7.4 m/s west) 11. A vehicle with a rocket engine is being tested on a smooth track. Starting from rest the engine is fired for a short period of time, releasing 4.5 x 10 2 kg of gases. It is estimated that the average velocity of the gases is 1.4 x 10 3 m/s to the right, and that the maximum velocity of the vehicle is 45 m/s left. What is the mass of the vehicle? (1.4x10 4 kg) Worksheet Impulse 1. A rocket at rest with a mass of 9.5 x 10 3 kg is acted on by an average net force of 1.5 x 10 5 N upwards for 15 s. What is the final velocity of the rocket? 2. A 26.3 kg object is traveling at 21.0 m/s north. What average net force is required to bring this object to a stop in 2.60 s? 3. An average force of 31.6 N south is used to accelerate a 15.0 kg object uniformly from rest to 10.0 m/s. What is the change in momentum? 4. An average net force of 25.0 N acts north on an object for 7.20 x 10-1 s. What is the change in momentum of the object?

16 5. A 5.00 kg object accelerates uniformly from rest to a velocity of 15.0 m/s east. What is the change in momentum on the object? 6. An average net force caused an 11.0 kg object to accelerate uniformly from rest. If this object travels 26.3 m west in 3.20 s, what is the change in momentum of the object? 7. A 1.30 kg object is dropped from a height of 6.5 m. How far did the object fall when its momentum is 6.0 kgm/s? 8. An average net force of 16.0 N acts on an object for 2.00 x 10-1 s causing it to accelerate from rest to 3.50 m/s. What is the mass of the object? 9. A kg object is thrown vertically upward with an average applied force of 8.20 N by a student. The force is applied through a displacement of 1.50 m. a. What is the average net force acting on the object? b. What is the velocity of the object when it leaves the student s hand? (Assume initial velocity is zero) 1) 237 m/s 2) 212 N south 3) 150 Ns south 4) 18 Ns north 5) 75 Ns 6) 181 Ns west 7) 1.1 m 8) 0.91 kg 9)a N b. 4.4 m/s Worksheet A 1.4 x 10 3 kg car is westbound at a velocity of 37.0 km/h when it collides with a 2.0 x 10 3 kg truck northbound at a velocity of 35 km/h. If these two vehicles lock together upon collision, what is the initial velocity of the vehicles after collision? (7.2 m/s 37 o W of N) 2. A 6.2 kg object heading north at 3.0 m/s collides with an 8.0 kg object heading west at 3.5 m/s. If these two masses stick together upon collision, what is their velocity after collision? (2.4 m/s 56 o W of N) 3. A 4.0 x 10 4 N Truck moving west at a velocity of 8.0 m/s collides with a 3.0x10 4 N truck heading south at a velocity of 5.0 m/s. If these two vehicles lock together upon impact, what is their velocity?(5.0 m/s 25 o S of W) 4. A 50.0 kg object is moving east at an unknown velocity when it collides with a 60.0 kg stationary object. After collision, the 50.0 kg object is traveling at a velocity of 6.0 m/s 50.0 o N of E and the 60.0 kg object is traveling at a velocity of 6.3 m/s 38 o S of E. a. What was the velocity of the 50.0 kg object before collision? (9.86 m/s due east) b. Determine whether this collision was elastic or inelastic. (Ek loss of 340 J, so inelastic) 5. A 15.0 kg penguin waddling east at a velocity of 7.0 m/s collides with a stationary 10.0 kg penguin. After the collision the 15.0 kg penguin is traveling at a velocity of 4.2 m/s 20.0 o S of E. a. What is the velocity of the 10.0 kg penguin after collision? (5.1 m/s 25 o N of E) b. is this collision elastic or inelastic? (Inelastic, E k loss of 110J) 6. A watermelon explodes into three equal masses. One mass moves east at 15.0 m/s. If a second mass moves at a velocity of 10.0 m/s 45.0 o S of E, what is the velocity of the third mass? (Hint: the total momentum is zero, so how will your vector arrows add up?) BRAIN-MELTER SPECIAL Mr. Trask s head explodes into three pieces after thinking too hard about this very problem (but HOW!?). As we all know, Mr. Trask s head has a mass of 15kg (that s a lot of physics up there). A 6.0 kg chunk flies off at 12.0 m/s 15 o N of W and a 5.0 kg chunk sails at 8.0 m/s 35 o E of S. What is the velocity of the final piece?