Example: a 10 kg book initially at rest, falls vertically for a distance of 2m onto a table. Gravitational Potential Energy. Nonconservative Forces

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1 Gravitational Potential Energy An object can possess energy by virtue of its position (potential energy), not just its motion (kinetic energy) You can store that energy there by doing work to change the object s position If you lift an object to a height you do positive work on it, gravity does the same amount of negative work A spring also has potential energy (more with more compression) Example: a 10 kg book initially at rest, falls vertically for a distance of 2m onto a table. 1. What work did you need to do to lift the book from the table to the 2m height? 2. And when it falls from there to the table what kinetic energy does it have just before impact? (example of mixed concept problem) 3. What is the energy when the object is at 1m high (before it impacts)? Conservative Force Work done by a conservative force is independent of path taken Example: gravity does work -mgh while you go up ski lift; +mgh while you ski down. Total work is zero (you get it all back) Conservative forces: Gravity Spring Force Integral formalism: work done by a conservative force is zero when particle moves around a closed loop Nonconservative Forces A force is non-conservative if the work done is not zero around a closed path Nonconservative Forces (you loose energy, can t get it back by going back to the starting point) Friction Air resistance, drag Potential Energy Which cat has a greater gravitational potential energy? When you ski down a hill, the work done by gravity decreases potential energy (potential energy is converted to kinetic energy). Define potential energy: The work done by a conservative force equals the decrease in potential energy Derive Gravitational potential energy Derive Spring potential energy Nova h Milo 1

2 How about now? Find the gravitational potential energy of each cat Spring potential Example 7.3 (work in groups) Deo, 13 lbs (~6kg) Nova, 6 lbs (~3kg) 1m 0.3m A system consists of a 110-kg basketball player, the rim of a basketball hoop, and Earth. Assume that the potential energy of this system is zero when the player is standing on the floor and the rim is horizontal. Find the total potential energy of this system when the player is hanging on the front of the rim. Also, assume that the center of mass of the player is 0.80 m above the floor when he is standing and 1.30m when he is hanging. The force constant of the rim is 7.2kN/m and the front of the rim is displaced downward a distance of 15cm. What is energy? It s a property of matter that can be transmitted to other matter; it can affect other objects It s a property of matter such that the total quantity of this property in the universe is constant (seems to be conserved!) Energy can be from Gravity, motion, springs, light, sound Example of transfer in energy: Where does our electricity come from? Energy can be transferred by doing work (workkinetic energy theorem -last class) Transferring energy by doing work Energy can be transferred by doing work (work-kinetic energy theorem), applying a force causes motion and motion is associated with kinetic energy Positive work is done on an object only to the extent that the motion is in the same direction as the net force (dot product takes care of this) Accounting with Mechanical Energy: the overall Bank Balance When we judge how much energy a system has, we must have two categories: Kinetic energy (K sys ), and potential energy (U sys ). The sum of these two forms of energy (like two sub accounts in a bank balance) is the Mechanical Energy, E mech = K sys + U sys A conservative force acting alone on an object simply exchanges one form of energy to another. So when measuring the change in an object s total mechanical energy, we need only compute what work has been done on it by non-conservative forces: W nc = ΔE mech The non-conservative force is like a bank robber, it takes from the total account of energy. In the problems to follow, we ONLY care about: 1. initial energy 2. final energy 3. the work done by nonconservative forces in between! 2

3 Example: A 5 kg block, initially moving horizontally at 4m/s across a frictionless surface, then slides down a frictionless ramp, losing 2 m of altitude in the process. At the bottom of the ramp, it encounters another horizontal surface, but this one exerts a kinetic friction force of 10 N on the block. How far across this surface will the block slide before coming to rest? Conservation of Energy If there is no non-conservative work (work done by a non-conservative force) and no external work (work done by a force outside the system), then energy is conserved: ΔE mech = 0 ΔE mech = ΔK sys + ΔU sys = 0 means that ΔK sys = - ΔU sys Usually, there are no external forces because you assume gravity is an internal force in your system (Earth-object system) Example 7-4: Standing near the edge of the roof of a 12m high building, you kick a ball with an initial speed of v i =16 m/s at an angle of 60º above the horizontal. Neglecting effects due to air resistance, find (a) how high above the height of the building the ball rises, and (b) its speed just before it hits the ground. All that we ve learned is related. How can work be used in a problem? Get kinetic energy from velocity Get velocity, position, time, etc. from acceleration (like kinematics problem) Get acceleration from net force Get work from kinetic energy Analyze forces with FBDs and Newton s 2nd Relate kinetic energy to potential energy if you have conservative forces Find spring constants or height of fall Use forces and distance to find work With work, you can get power Special 2-D motion case: Circular Motion When an object is moving in a circle, something must be keeping it from continuing in a straight path Centripetal acceleration! Anything holding you moving in a circle Example 3-4: Rounding a curve A car is traveling east at 20m/s. It rounds a curve and 5.0 seconds later it is traveling north at 20m/s (same speed as before!). Find the average acceleration of the car. 3

4 Uniform Circular Motion Defined: motion in a circle at constant speed (not constant velocity!) Key point: When an object is moving in a circle, something must be keeping it from continuing in a straight path. Centripetal acceleration! Centripetal acceleration is anything causing an object to move in circular motion. And it is a vector that is ALWAYS directed toward the center of the circle. Centripetal acceleration is not: an additional acceleration not already present in the system. An object in circular motion can experience two kinds of acceleration: 1. Centripetal Acceleration: keeps object moving in a circle 2. Tangential Acceleration: Measures the change in speed along the arc of the circle Dispel Misconceptions What is centrifugal acceleration? Circular motion definitions Centripetal acceleration Tangential acceleration Speed as it relates to period 4

5 Example: A satellite moves at a constant speed in a circular orbit around the center of the earth and near the surface (take radius to be Earth s, 6370km). If the magnitude of its acceleration is g = -9.8m/s 2, find a) its speed and b) the time it takes to complete one revolution. A ball is whirled in a horizontal circle of radius r and speed v. The radius is increased to 2r keeping the speed of the ball constant. The period of the ball changes by a factor of A. one half. B. one. C. two. D. three. E. four. Example 7-8: Imagine that you have time-traveled back to the late 1800s and are watching a Coney Island roller coaster with a circular loop-the-loop. The car they are in is about to enter the loop when a 100lb sack of sand falls from the construction site platform and lands in the back seat of the car. No one is hurt, but the impact causes the car to lose 25 percent of its speed. The car started from rest at a point 2 times as high as the top of the loop. Will their car make it over the top of the loop without falling off? What is momentum? Mass in motion Like energy, it s a property of matter that can be transmitted to other matter; it can affect other objects It s a property of matter such that the total quantity of this property in the universe is constant (seems to be conserved!) Example of transfer in momentum: bounce Mass times speed, larger speed for smaller mass Mathematical definition continued Solving Momentum conservation problems 1. To have momentum conservation, there must be no net force 2. Draw a picture showing the system before and after the event Include coordinate system (label axes) Label initial and final velocity vectors 3. Equate initial and final momentum - Pay attention to signs! 4. Use given information in this equation to solve for quantity of interest Use minus signs with velocities if necessary and check units of result Example 8-1 During repair of the Hubble Space Telescope, an astronaut replaces a damaged solar panel during a spacewalk. Pushing the detached panel away into space, she is propelled in the opposite direction. The astronaut s mass is 60 kg and the panel s mass is 80 kg. Both the astronaut and the panel initially are at rest relative to the telescope. After the push the panel is moving at 0.30m/s relative to the telescope. What is her velocity relative to the telescope? 5

6 Momentum is a VECTOR Example 8-2 A runaway 14,000-kg railroad car is rolling horizontally at 4.00m/s toward a switchyard. As it passed by a grain elevator, 2000kg of grain suddenly drops into the car. How long does it take the car to cover the 500m distance from the elevator to the switchyard? Assume the grain falls straight down and friction is negligible. Example 8-3 A 40.0-kg skateboarder on a 3.00-kg board is training with two 5.00-kg weights. Beginning from rest, she throws the weights horizontally, one at a time, from her board. The speed of each weight is 7.00 m/s relative to her after it is thrown. Assumer the board rolls without friction a) How fast is she moving in the opposite direction after throwing the first weight? b) After throwing the second weight? Want to be a pool shark? The condition necessary for the Conservation of Linear Momentum in a given system is that A. energy is conserved. B. one body is at rest. C. the net external force is zero. D. internal forces equal external forces. E. None of the above. A golf ball and a Ping-Pong ball are dropped in a vacuum chamber. When they have fallen halfway to the floor, they have the same A. speed. B. potential energy. C. kinetic energy. D. momentum. E. speed, potential energy, kinetic energy, and momentum. Two cars of equal mass travel in opposite directions at equal speeds. They collide in a perfectly inelastic collision. Just after the collision, their velocities are A. zero. B. equal to their original velocities. C. equal in magnitude but opposite in direction from their original velocities. D. less in magnitude and in the same direction as their original velocities. E. less in magnitude and opposite in direction from their original velocities. 6

7 Momentum continued: Collisions Inelastic Collision: Kinetic energy is not conserved. Examples: Explosions (internal force pushes apart), Things sticking together, like putty or car crash (kinetic energy goes into deformation and sticking instead of motion) Elastic Collision: Conserves Kinetic Energy Happens between rigid objects like pool balls (all kinetic energy in is returned) Momentum can be conserved while Energy is not conserved Impulse and Average Force When objects collide, they exert large forces on each other for a brief time Impulse is a measure of both the strength and duration of the force Impulse is a vector The force may vary during the time of contact (integral definition on impulse picture) The x component of impulse is the area under the curve of an x component force as a function of time Impulse-momentum theorem derivation Units What are the units of Momentum? What are the units of Impulse? Momentum is conserved in which of the following? A. elastic collisions B. inelastic collisions C. explosions D. collisions between automobiles when friction from the road is negligible E. All of the above. Example 8-5 With an expert karate blow, you shatter a concrete block. Consider your hand to have a mass of 0.70 kg, to be moving 5.0 m/s as it strikes the block and to stop 6.0 mm beyond the point of contact. a) What impulse does the block exert on your hand? b) What is the approximate collision time and average force the block exerts on your hand? Perfectly inelastic collisions in one dimension Example 8-9: An astronaut of mass 60 kg is on a space walk. You throw him a repair manual with a speed of 4 m/ s relative to the spacecraft. He is initially at rest relative to the spacecraft before catching the 3.0-kg book. Find: a) his velocity just after he catches the book b) the initial and final kinetic energies of the book- astronaut system c) the impulse exerted by the book on the astronaut 7

8 Example 8-10 Ballistic Pendulum In a feat of public marksmanship, you fire a bullet into a hanging wood block, which is a device known as a ballistic pendulum. The block, with the bullet embedded, swings upward. Noting the height reached at the top of the swing, you immediately inform the crowd of the bullet s speed. How fast was the bullet traveling? One dimensional Elastic collisions Example 8-13: A 4.0-kg block moving to the right at 6.0 m/s undergoes an elastic head-on collision with a 2.0-kg block moving to the right at 3.0 m/s. Find their final velocities. Collisions in 3 dimensions (inelastic example, sticking together) Example 8-15: You are at the wheel of a 1200-kg car traveling east through an intersection when a 3000-kg truck traveling north through the intersection crashes into your car. Your car and the truck remain stuck together after impact. The driver of the truck claims you were at fault because you were speeding. You look for evidence to disprove this clam. The speed limit for the road on which you were driving is 80 km/hr. The speedometer on the truck was smashed on impact, leaving the needle stuck at 50 km/hr. Lastly the wreck initially skidded from the impact zone at an angle of 59 degrees north of east. Does this evidence support or undermine the claim that you were speeding? 3-D elastic collisions Example 8-16: An object with mass m1 and with an initial speed of 20 m/s undergoes an off-center collision with a second object of mass m2. The second object is initially at rest. After the collision the first object is moving at 15 m/s at an angle of 25 degrees with the direction of the initial velocity of the first object. In what direction is the second object moving? 8