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1 Unit 3 Momentum and Energy In this unit we are going to be looking at the two most fundamental laws of nature – the law of conservation of energy and the law of conservation of momentum. Chapter 6 – Work, Power and Efficiency Every object you see has some form of energy. There are two types of energy an object can have – kinetic energy and potential energy. Kinetic energy is the energy of motion and potential energy is the energy that is stored in an object. Mechanical energy is the combination of both kinetic and potential energy. What is Work? Work is not energy itself but it is a transfer of mechanical energy. A force does work on an object if it causes the object to move. Note that work is always done on an object and results in a change in the object. Physics defines work very specifically as force acting on an object causing the object to move a certain distance. The amount of work depends directly on the magnitude of the force and the displacement of the object along the line of the force. Work is the product of force and displacement when the force and displacement vectors are parallel and pointing in the same direction. W = F║Δd (W is scalar so vector notations are not used.) What is Not Work? Because of the specific definition of what work is there are several situations that may appear to involve work but from a physics point of view, actually involve no work. 2 Case 1: Applying a force that does not cause motion. The most common example for this situation is looking at a student pushing on the classroom wall. He may appear to be doing work. He feels tired after a while and has been exerting a force (expending energy) upon the wall. But, because the wall didn’t move or experience a change in displacement, no work was done. Case 2: Uniform motion in the absence of a force. This ties into Newton’s first law – an object will continue in motion unless acted upon by an external force. If we look at a hockey puck sliding along a frictionless surface, there was work done to get the puck moving but there is no work being done to keep it moving. Since the puck is not changing its motion, no force is acting upon it. Case 3: Applying a force that is perpendicular to the motion. If you look at carrying your physics book down the hall, both you and the book are moving. The force acting on the book however is perpendicular to the line of motion since your arm is exerting a force up. Picking up the book is work forces are parallel. Once you and the book are moving at a constant velocity, you are no longer doing work on the book. Another example of perpendicular forces is swinging an object on a string. Calculating Work The amount of work depends directly on the magnitude of the force and the displacement of the object along the line of the force. Work is calculated using the equation: W Fll d W is the work done, d is the displacement of the object and F is the force exerted on the object. 3 The derived unit for work is the joule, J, and is named after nineteenth century physicist, James Prescott Joule. One joule is equivalent to a N m Practice Question A student is rearranging her room. She decided to move her desk across the room, a total distance of 3.00 m. She moves the desk at a constant velocity by exerting a horizontal force of 200 N. Calculate the amount of work the student did on the desk in moving it across the room. Page 221 #1, 2, 3; Page 225 #4 – 10 4 Constant Force at an Angle It is rare in everyday situations that the forces acting on an object will be precisely parallel or perpendicular to the motion of an object. If we look at a situation similar to that in figure 6.11 on page 230, the applied force is at an angle relative to the direction of the wagon’s displacement. To determine the work being done by the child pulling the wagon, you must first find the component of the force that is parallel to the direction of motion. This will be done in a very similar way to how we found vertical and horizontal components of forces in the last unit. We will use trigonometry to resolve force into it components. When the force and displacement vectors are not parallel and are pointing in the same direction, work can be calculated using the formula: W Fd cos . According to this equation, the work being done by gravity will be 0 J. Gravity will being acting at 90o to the direction of displacement. The cosine of 90o is 0 so the work being done will also be 0. Practice Question: How much work is being done on a wagon being pulled with 80 N of force at an angle of 23o to the direction of motion? 5 Positive and Negative Work Positive work adds energy to an object; negative work removes energy from an object. If we look at the common force of friction, we can see how our new equation for work, W Fd cos , will show that friction results in negative work. If we were to drag a block across a desk, friction is going to work in the direction opposite to the direction of displacement. Friction can be said to be acting at 180o to the direction of displacement. Because the cosine of 180o is -1, the resulting work being done by friction will be negative. Practice Problem: A weight lifter is bench-pressing a barbell weighing 650 N through a height of 0.55 m. There are two distinct motions: (1) when the barbell is lifted up and (2) when the barbell is lowered back down. Calculate the work that the weight lifer does on the barbell during each of the two motions. Page 235 #14 – 18 6 Kinetic Energy The energy of motion is called kinetic energy. If you think about a few simple situations, you can begin to understand what quantities will be involved in calculating kinetic energy. If there was a bowling ball and a golf ball rolling toward you at the same velocity, which would you try to avoid? If there were two golf balls of the same mass coming toward you, one slow and one fast, which would you try to avoid? Your answers to these questions should lead you to think that mass and velocity play a role in the amount of kinetic energy an object possesses. 1 2 2 The equation for kinetic energy is: E k mv Kinetic energy is measured in joules and is a scalar quantity. Practice Question: A 200 g hockey puck, initially at rest, has a final velocity of 27 m/s. Calculate the kinetic energy of the hockey puck a) at rest and b) in motion. Page 238 #19 – 21 7 Work and Kinetic Energy The special relationship between doing work on an object and the resulting kinetic energy of the object is called the work-kinetic energy theorem. Because doing work on an object results in the object having velocity, there is kinetic energy. Before we make a mathematical expression for this theorem, we must make one assumption. That is to assume that all work done on a system results in kinetic energy only. Page 240 shows the entire derivation. The equation for the work – kinetic energy theorem is W Ek or W Ek 2 Ek1 These representations describe how doing work on an object can change the object’s kinetic energy (energy of motion). The work-kinetic energy theorem is part of the broader work-energy theorem that includes the concept that work can change an object potential energy, thermal energy, or other forms of energy. Practice: A physic student does work on a 2.5 kg curling stone by exerting 40 N of force horizontally over a distance of 1.5 m. a) Calculate the work done by the student on the curling stone. b) Assuming that the stone started from rest, calculate the velocity of the stone at the point of release. (Consider the ice to be frictionless) 8 Practice 2: A 75 kg skateboarder (including the board), initially moving at 8.0 m/s, exerts an average force of 200 N by pushing on the ground, over a distance of 5.0 m. Find the new kinetic energy of the skateboarder if the trip is completely horizontal. Page 245 – 246 #22 – 26 9 Potential Energy and Work-Energy Theorem It is possible to do work on an object but the object does not gain kinetic energy. Doing work on an object can result in a change in potential energy rather than kinetic energy. Potential energy is the energy stored by an object due to its position or condition. By doing work against the force of gravity a special form of potential energy called gravitational potential energy is gained. This is only one of several forms of potential energy. Chemical potential energy is stored in the food we eat, elastic potential energy is stored in elastic bands, and batteries store electrical potential energy. Gravitational potential energy has been used for hundreds of years to produce energy from water sources starting with water wheels and now includes reservoirs and dams. There are several factors that play a role in calculating gravitational potential energy. If you think about a Ping Pong ball and a golf ball dropped from the same height, which would you try to catch? What about a golf ball dropped from 10 cm and one dropped from 10 m, which would hit the ground harder? Would a golf ball dropped from 1 m hit the surface of the Earth or moon harder? Your answers to these questions should tell you that gravitational potential energy is affected by mass, height and the acceleration due to gravity. An important characteristic of all forms of potential energy is that there is no absolute zero position. You must assign a reference position and compare the potential energy of an object to that position. Typically the reference position is the solid surface toward which an object is falling or might fall. 10 The equation for gravitational potential energy is E g mgh . In this equation, h is the change in height from the reference position. Practice Question: You are about to drop a 3.0 kg rock onto a tent peg. Calculate the gravitational potential energy of the rock after you lift it to a height of 0.68 m above the tent peg. Page 250 #27 – 29 11 Work and Gravitational Potential Energy Similarly to work and kinetic energy, the equation for the work-energy theorem in terms of gravitational potential energy can be given by: W Eg or W Eg 2 Eg1 . The derivation for this formula can be found on page 251. Practice Problem: A 65.0 kg rock climber did 1.6 x 104 J of work against gravity to reach a ledge. How high did the rock climber ascend? Page 254 #30 – 34 12 Elastic Potential Energy When an object can stretch, compress, bend, or change in shape in some way and then return to its original position, it is said to be elastic. The energy stored in an object with elastic properties has a form of stored energy called elastic potential energy. Hooke’s Law Hooke’s law says that the force applied to extend or compress a spring will be proportional to the amount of extension or compression of the spring. The restoring force, the force exerted by the spring to return it to its original position, will act in the opposite direction to the applied force. Hooke’s law is given by the equation F = -kx for the restoring force and Fa = kx for the applied force. In this equation, x is the amount of extension or compression of the spring and k is the spring constant. Each spring has a different spring constant that is measured in N/m. Practice Question: A typical archery bow requires a force of 133 N to hold an arrow at “full draw” (pulled back 71 cm). Assuming that the bow obeys Hooke’s law, what is its spring constant? Page 258 #35 – 37 13 Work and Elastic Potential Energy Elastic potential energy for a perfectly elastic material is given by the 1 2 2 equation: E kx It is important to note that there is no material that is perfectly elastic. For a material to be perfectly elastic, it will return to precisely to its original form after being deformed. All materials will reach an elastic limit where they will not return to their original shape if stretched to that limit. Practice Question: A spring with a spring constant of 75 N/m is resting on a table. a) If the spring is compressed a distance of 28 cm, what is the increase in its potential energy? b) What force must be applied to hold the spring in this position? Page 261 #38, 39, 40 END for Jan 2010 14 Power Power is the rate at which work is done. It is the rate at which energy is transferred. It is given by the equation P W E or P t t The unit for power is the watt, W. Remember that work is done on an object and results in a transfer of energy to that object. Practice Question: 1. A crane is capable of doing 1.50 x 105J of work in 10.0 s. What is the power of the crane? 2. A cyclist and her mountain bike have a combined mass of 60.0 kg. She is able to cycle up a hill that changes her altitude by 4.00 x 102 m in 1.00 min. (Assume that friction is negligible.) a. How much work does she do against gravity in climbing the hill? b. How much power is she able to generate? 15 Efficiency It is inevitable that some energy will be lost when transferring from one type to another, usually in the form of heat. The efficiency of a machine or device is a ratio of the useful energy to the total energy. It is E calculated using the equation: Efficiency o 100% or Efficiency Wo 100% EI WI Eo and Wo are the useful output energy and work, respectively and EI and WI are the energy and work inputted. Practice Question: A model rocket engine contains explosives storing 3.50 X 105 J of chemical energy. The stored chemical energy is transformed into gravitational potential energy at the top of the rocket’s flight path. Calculate how efficient the rocket transforms stored chemical energy into gravitational potential energy if the 500 g rocket is propelled to a height of 100 m. Page 266 #41, 42, 43 Page 270 – 271 #44 – 50 16 Work Done When Forces are Changing Much like in our study of kinematics, there are often situations where the forces are not uniform. To determine the work done when the force acting upon an object is not uniform, calculating the area under the curve of a force – position graph will give the work done. Often the graphs we examine will be fairly straightforward similar to the example given on page 227. If you happen to encounter a graph that is not so easy to work with (see figure 6.10 on page 226), you must count all of the squares under the curve and estimate for partial squares. Although this method is not completely accurate, the mathematical solution for such a problem will not be taught in this course. Complete the example on page 227. Page 229 #11, 12 17