Name: Review and Preview We have come a long way in our study of mechanics. We started with the concepts of displacement and time, and built up to the more complex quantities of velocity and acceleration. These four were used in a description of motion called kinematics. Then we introduced mass and force into the mix and our descriptions of motion including these additions came under the heading of dynamics. Now we will introduce a new quantity called energy. You will have 5 minutes to come to a consensus on the following questions before sharing with the group. a) With your group, consider the types of energy you know of and come up with a general statement to describe energy as a concept. b) Where do we see examples of energy in everyday life? c) One way of describing energy is "the ability to make change". Does that fit with your definition above? Working Instead of using F=ma for force in an instant, we can look at force over a distance and see how that changes a system. When you apply a force over a physical distance that means you are working, or doing work. In physics we use the term work to describe a process of energy transfer. The simplest case is that of a constant force F, pushing an object that moves a distance d in the direction of the force. If it was initially at rest, this force over a distance would transfer energy of motion to the object. In this case we define the work as the product of the magnitude of the force, F, and the displacement d: In a slightly more general case, the force may not be the only one acting on the object, so the motion may not be in the direction of the force. In that case we take the magnitude of the component of the force that is along the direction of motion multiplied by d: Sometimes the component of force actually points in the direction opposite to the displacement. In this case we say that the work done is negative, otherwise (as in the preceding cases) it's positive. An example of a force that (almost) always does negative work is: *Can you come up with an example where this force does positive work? What is it?
Units of Work The units of work are constructed from those of force and displacement. The unit of force: The unit of displacement: Work = Force(displacement) cos q where cos q is dimensionless. Units of Work: In the MKS system, work is defined as 1 Joule. One Joule is the amount of it takes to move a mass 1. Example A box of m = 2 kg moves over a frictional floor with f k = 0.3 and has an applied force of FA = 25 N applied to it at a 30 angle. The crate moves 16m in the horizontal. a) How much work does FA do? Set up the answer with variables only: W = Solve with the values given: W = b) How much work does the normal force do? W = F d cos q W = c) Explain your answer: Practice Problems You move a 20 kg box 1.5 m across a rough floor at constant speed by pulling with a 100 N force at a 37 angle to the horizontal. a) How much work did you do? b) Draw a free body diagram of this problem. c) What are the other forces acting on the box? d) Did any other force do work? e) If so, which force and how much work? 2
Work Done by the Net Force As you saw in the example above, many forces can be acting on an object, and each force can do positive work, negative work, or no work at all. We could ask how much work is done by the vector sum of all the forces on the object ( F ). In the problem above, we know that there is acceleration (since F = ma), so we know that the velocity changes. We can express the work done in terms of the velocity change. Let's consider the simple case when the force is in the same direction as the displacement d and there is no change in height. Then the work done is Fd, and from Newton's second law of F = ma: W = Fd = To express this in terms of velocity, we use the kinematic equation v f 2 = v i 2 + 2aΔx or v f 2 = v i 2 + 2ad if you substitute Δx for d. Solve the kinematic equation for ad: Use your definition for ad and substitute it into the equation for W = Fd: Simplify into one final term and one initial term: Now we see the effect that the net force has when it does work on an object: It changes the "½mv f 2 " of the object. This combination of mass and velocity is the energy of motion of an object and has a specific name: the kinetic energy of the object. Kinetic Energy = ½ mass (velocity 2 ) K = ½ mv 2 à Energy of motion Simplify the above in terms of K: Practice Problems 1. Calculate the kinetic energies of the following moving objects: A) A 180 kg football player running at 8 m/s B) A 4.2 g bullet moving at 950 m/s C) A bicycle (15 kg) and rider (60 kg) moving at 15 m/s D) A 1.2 x 10 5 kg airliner at cruising speed of 1000 km/hr E) The 9 x 10 4 kg Space Shuttle in orbit with a speed of 7.86 km/s 3
2. A box of mass 50 kg is pulled across a rough horizontal floor by a horizontal rope with a tension of 200 N. The coefficient of kinetic friction between the box and the floor is 0.3. After the box has moved 5 m from rest, find A) The work done by the person pulling the rope B) The work done by the kinetic frictional force C) The work done by the net force D) The final speed of the box 3. A block of mass m is launched up a frictionless ramp, as shown to the right, with an initial speed v i. The block travels up the ramp and continues on the level section. A) In the box at right, draw a free-body diagram of the block as it moves up the frictionless ramp. B) For each force on the free-body diagram, state whether that force does positive, negative, or zero work. ur v i θ h C) If the ramp were made steeper, but everything else were the same, how would that affect the net work done on the block? Explain using equations and relationships between variables. Stored Energy Where is energy stored? As an example, consider a battery, where energy is stored as chemical energy and when a current runs through the battery, that energy flows through the circuit as electricity. 1. Where else can you store energy? 2. Try holding a Slinky stretched out. Although you are at rest, is there energy in this system? If so, where? 3. What happens when you let go of the Slinky? 4
The Earth's gravity allows us to store energy in different amounts. Within this lesson, we often transfer that energy to either the energy of motion or to work due to friction (heat energy). Work Done by the Gravitational Force The familiar "mg" force of gravity near the earth's surface can do positive, negative, or zero work on something depending on how it moves. Since the force of gravity always points down, toward the center of the earth, if the displacement of the object has a net downward component then the gravitational force does work. If the displacement has a net upward component, then it does work. What would the work be if the object stays at the same horizontal level? Explain your answer: Consider a ball of mass m moving through space from a height h i to a height h f, as shown at right. To calculate the work done by the gravitational force we multiply the force (mg) by the component of the displacement in the direction of the force. If a mass m moves from an original height h i above the ground to a height h f, by any path at all, then the work done by the gravitational force depends only on the difference between the heights (h f - h i ). This is because the sideways components of the displacement don't contribute to the work, since they're perpendicular to the gravitational force, which points downward. Work due to gravity = mass(gravity)(change in height) Wg = mgdh Practice Problems An 8 kg bowling ball sitting on the ball rack falls onto a nearby table, a distance of 1 meter. A) What is the work done by the gravitational force? B) Since in this case the gravitational force is the net force, what is the change in the ball's kinetic energy? C) The work done by gravity transfers entirely into the change in K of the ball. What is the speed of the ball just before hitting the table? D) How much work would it take to put the ball back on the rack (transfer the energy of motion to stored energy)? 5
Gravitational Potential Energy We think of the bowling ball 1 meter above the ground as being more "dangerous" than one sitting on the floor, in terms of its potential to hurt your toe. The height above ground is indicative of the stored energy the ball has and we call this type of stored energy Gravitational Potential Energy. This is the energy something has by virtue of its position relative to the earth; the higher it is, the more gravity has the potential for doing work on it, and the greater the kinetic energy it might build up if it were dropped. Gravitational potential energy is a relative quantity, because it is measured relative to an arbitrary "zero" level. In the bowling ball example above, we used the floor as the "zero" level, but that floor may be above or even below the surface of the earth. What's important is the change in the potential energy, which is the same no matter what zero level we choose. With that we define gravitational potential energy (Ug) of a mass as the work needed to lift the mass to its current position from an arbitrarily chosen zero level: Gravitational Potential Energy = mass x gravity x height Ug = mgh Conservative and Non-conservative Forces For some forces, like gravitation, the work done in moving an object from one point to another doesn't depend on the path that is taken. In the bowling ball example from above, you could chose many paths other than the one shown, as long as you start at h i and end at h f the work done by the gravitational force is W = mg(h f - h i ) A force that is not dependent on the path like gravity is called a conservative force. Other forces, such as the frictional force, do not behave this way. If we slide a book around on a table top, for instance, friction does (negative) work the whole time we're sliding it, so sliding it directly on a line from one corner of the table to the other would involve less work than taking a longer path. The frictional force is an example of a non-conservative force. Energy is easily transferred between conservative forces: stored, internal energy U becomes energy of motion K and then returns to the exact same amount of potential energy in an ideal pendulum. Energy is less easily transferred when non-conservative forces are involved: when you heat a log the internal chemical stored energy becomes heat energy and you cannot recreate the original log. The Work-Energy Theorem We have seen energy as an agent of change in the two methods below: Energy of Motion: Kinetic Energy (K) = (1/2)mv 2 Stored Energy: Potential Energy (U) = mgh As well as energy as the mechanism for energy transfer: Work = Fd cos θ = ΔK when U is constant or Work = ΔU when K is constant Combining the two, we find that we can create an equation describing the constant state of energy within a closed system Work = ΔK + ΔU In other words, the difference in energy from the initial state to the final state is equal to the amount of work done. 6
As an example, consider the ball toss video we saw at the beginning of the projectile motion unit. Describe how K and U change over time in that video and include how work is done on the system in pulling energy out. One of the consequences of this theorem is that if there are no non-conservative forces acting (W nc = 0), then no mechanical energy is transferred into or out of the system. We say that the total mechanical energy is conserved. This means that if you add up all the energy in the initial condition it will equal the energy in the final condition. E i = E f K i + U i = K f + U f When W nc = 0 Another way of saying that energy is conserved is to note that Energy Here = Energy There or Energy at the Start = Energy at the End Energy Here Practice Problem 1. A 1 kg ball is tied to a string of length 1 meter, and held in a horizontal position 2 meters off the floor. The ball is released and swings down to a vertical position as seen in the image at right. A) What is the total energy of the system at the initial position (energy here)? B) What is the total energy of the system at the final position (energy there)? Energy There C) Neglecting air resistance (which is a non-conservative force), how fast is the ball going when it reaches the bottom? D) If the ball actually ends up moving at 3 m/s at the bottom, how much work has been done by the force of air resistance? E) How much energy is dissipated by friction forces when the ball comes to rest after a few swings? F) Once the ball is stopped in the "there" position, how much work would you have to do to move the ball back up to the original height? 7
Additional Practice Problems 1. A 20 kg sled carrying a 40 kg girl is sliding at 12 m/s on smooth, level ice, when it encounters a rough patch of snow. A) What is the initial kinetic energy of the girl and the sled? B) If the rough ice exerts an average opposing force of 540 N, in what distance does the sled stop? C) What work is done by the rough ice in stopping the sled? 2. A frictionless roller coaster of mass m is given an initial speed of 5 m/s at the top of the first hill, which is 25 meters high. A) Find the K of the roller coaster at point A in terms of m. A B C D B) Find the speed of the roller coaster at point B, a height of 10 meters. C) Find the speed of the roller coaster at point C, at ground level. D) At point D, the roller coaster just barely makes it over the top. How high is point D? 3. Going back to the Toss.dv video from the projectile motion unit, sketch out the graphs that would display the potential, kinetic and total energy of the ball during the toss. Ignore air resistance and assume the ball was caught at the same height from which it was thrown. Potential Energy of Ball Kinetic Energy of Ball Total Mechanical Energy of Ball Time Time Time 8
How would you expect the total mechanical energy graph to look if you did not ignore air resistance? Sketch it below. Total Mechanical Energy of Ball e Time Power It takes 9.8 Joules of work to raise 1 kg of mass by 1 meter, no matter how quickly it is done. That is, the work done doesn't depend on how fast it happens. The term power takes account not only of how much work is done, but how quickly it is done. Power means the rate at which work is done. Average power P is defined as the ratio of the work done to the time it takes. Write out the equation for power based on that definition: In the MKS system, power is measured in Joules per second, and 1 Joule/s is given the name Watt in honor of James Watt, who invented the steam engine. In the British system, power is in foot pounds per second (ft lb/s), but more commonly in horsepower, which was intended to mean the power capacity of an average horse. One horsepower (hp) is defined as 550 ft lb/s. In other words, an average horse should take one second to lift 550 pounds up one foot. Recall the definition of work: W = Fd cos θ. If the force points in the same direction as the displacement, then this simplifies to W = Fd. Divide both sides of the equation by the time it takes to do the work: Power can be defined as work over time and average velocity can be defined as distance over time. Write another expression for average power: Practice Problems 1. In the 2008 Tantalus Time Trial, an altitude gain of about 500 meters, Mr. Clarke was timed at about 26 minutes, Mr. Adams was timed at about 34 minutes, and Mr. Gearen was timed at about 35 minutes. Mr. Clarke and Mr. Gearen each have a mass (including bicycle) of about 100 kg, and the mass of Mr. Adams and his bicycle is about 85 kg. Neglect friction. Assume the work required is only that needed to lift the racer and bike vertically up the mountain, and that the racer has constant velocity throughout. A) How much work did they each do? 9
B) Determine their power outputs in watts for the time trial. 2. How long must a 60 Watt light bulb be on in order to do 2.16 x 10 5 J of work? 3. One kilocalorie of food energy is converted to 4186 joules. If the average person consumes 2000 kcal of food energy a day (within 24 hours), what is the power from that food energy to the adult? 4. The diameter of Earth is about 12,800 km and its mass is 5.97 x 10 24 kg. a. What is the kinetic energy of one point on the Earth's equator due to the daily rotation of the planet? b. What is the power due to the Earth's daily motion at that point? 10