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Chapter 6 So far, we’ve been obsessed with motion and how it relates to Newton’s Laws and forces. Now we will be concerned with two scalar quantities, work & energy and how they are always conserved or remain constant. ENERGY WORK the ability to do work a transfer of energy Wait, what?! 1. Work Done by a Constant Force 2. Work Done by a Varying Force 3. Kinetic Energy, and the Work-Energy Principle 4. Potential Energy 5. Conservative and Nonconservative Forces 6. Mechanical Energy and Its Conservation 7. Problem Solving Using Conservation of Mechanical Energy 8. Other Forms of Energy; Energy Transformations and the Law of Conservation of Energy 9. Energy Conservation with Dissipative Forces: Solving Problems 10. Power DEFINITION IMPORTANT INFO Scalar Quantity a force acting through a distance the product of the magnitudes of the displacement times the component of force parallel to the displacement. Symbol: W Units: Joule (J) Magnitude only 1 J = 1 Nm *W = Fd W – Work (J) F – Force (N) d – displacement (m) - For work to be does done: • Recap As long as this person 1. force must be applied. notAlift or lower the bag of 2. The object must move. groceries, he is doing no work 3. on The it. object must move in the • direction of the force. The force he exerts has no Iscomponent the force ininthe thedirection directionofof motion? motion – it is perpendicular No – So, is work being done? to the direction of motion. What about a force acting at an angle to the displacement? The work done by a constant force is defined as the distance the object moved multiplied by the component of the force in the direction of displacement: θ is the angle between the direction of the F and d 1. Draw a free-body diagram. 2. Choose a coordinate system. 3. Apply Newton’s laws to determine any unknown forces. 4. Find the work done by a specific force. 5. To find the net work, either: find the net force and then find the work it does, or find the work done by each force and add. A 50 kg crate is pulled 40 m along a horizontal floor by a constant force exerted by a person, Fp = 100 N, which acts at a 37° angle as shown. The floor is rough and exerts a friction force Ff = 50 N. Determine the work done by each force acting on the crate, and the net work done on the crate. mg 1. Draw a FBD. FN Fp Ff Fg 1. Choose a coordinate system. Let’s choose +x to be in the direction of motion. FN Fp d = 40 m Ff Fpx Fg Find any unknown forces using Newton’s 2nd. 3. 4. All forces are known (or can be determined easily). Determine the work done by each force. FN Fp = 100 N θ = 37° Ff = 50 N d = 40 m Fg Determine the work done by each force. 4. WN = perpendicular to direction of motion = zero Wg = perpendicular to direction of motion = zero Wf = Fdcosθ = (50 N) (40 m) (cos 180) = -2000 J WpII=Fdcosθ = (100 N) (40 m) (cos 37) = 3200 J Determine the net work done on the crate. 5. Wnet = 0 + 0 + (-2000 J) + (3200 J) = 1200 J a) Determine the work a hiker must do on a 15.0 kg backpack to carry it up a hill of height h = 10.0 m. a) Determine the work done by gravity on the backpack. a) Determine the net work done on the backpack. For simplicity, assume the motion is smooth and the velocity is constant. Concept Questions p. 160 #3-6 Practice Problems p. 162 #2-5, 8 + 10 xkcd.com What if the applied force is not constant - it keeps changing? How can we determine work done by a varying force? We could do my favorite thing! ▪ Let’s graph it! ▪ Plot Fcosθ vs. d. ▪ What represents the work done on this type of graph? The work done between any two points on the curve is equal to the area under the curve between those two points. Try #12 (a & b) on p.162. Kinetic Energy – the energy of motion. KE is measured in Joules. Translational Kinetic Energy: A moving object can do work on another moving object that it strikes - it can exert a force on an object and move it a distance. (Therefore, W is done.) 1 2 KE ( mv ) 2 KE is a scalar quantity that is always either zero or greater than zero. Why? The net work done on an object is equal to the change in the kinetic energy of the object. Wnet KE Wnet 1 2 1 2 mv f mvi 2 2 • If the net work is positive, the kinetic energy increases. • If the net work is negative, the kinetic energy decreases. Potential Energy is the energy of position. Examples of potential energy: • An object at some height above the ground • A stretched or compressed spring • A stretched elastic band or string In raising a mass m to a height h, the work done by the external force is Therefore, we define the gravitational potential energy: Ug = mgy The energy an object has because of its position at some height above the ground. Ug = mgh Notice, the higher an object is, the more gravitational PE it has. This potential energy can become kinetic energy if the object is dropped. Potential energy is a property of a system as a whole, not just of the object (because it depends on external forces). If Ug = mgy, where do we measure y from? It doesn’t matter, as long as we are consistent about where we choose y = 0. Only changes in potential energy can be measured. Potential energy can also be stored in a spring when it is compressed; the figure below shows potential energy yielding kinetic energy. The force required to compress or stretch a spring is: Where: k is called the spring constant, and needs to be measured for each spring. x is the displacement from the unstretched position. Also known as the spring equation or Hooke’s Law. The force increases as the spring is stretched or compressed. We find that the potential energy of the compressed or stretched spring, measured from its equilibrium position, can be written: Elastic PE, or Us = ½ kx2 Note: For a spring, we choose the reference point for zero PE at the spring’s natural position. GRAVITATIONAL PE ELASTIC PE Concept Question p.161 #9 Concept Question p.161 #7 Practice Problems Review Example 6-7 on p.146 Practice Problems Do #26, 29 + 32 on p.163 Do #27, 28, 30, 31 on p.163 CONSERVATIVE NONCONSERVATIVE Forces for which the work done does not depend on the path taken but only on the initial and final positions. Forces for which the work done depends not only on the starting and ending points but also on the path taken. Example: Gravity Example: Friction Although potential energy is always associated with a force with a force, not all forces have potential energy. Potential energy can only be defined for conservative forces. Review Examples 6-4, 6-5, and 6-6 on page 143-144. Complete the Following Problems on p.162: #15-20 Therefore, we must distinguish between the work done by conservative forces and the work done by nonconservative forces. We find that the work done by nonconservative forces is equal to the total change in kinetic and potential energies: WNC = ΔKE + ΔPE If there are no nonconservative forces, the sum of the changes in the kinetic energy and in the potential energy is zero – the kinetic and potential energy changes are equal but opposite in sign. This allows us to define the total mechanical energy (E): E = KE + PE And its conservation: E1 = E2 = constant In the image on the left, the total mechanical energy is: If there is no friction, the speed of a roller coaster will depend only on its height compared to its starting height. For an elastic force, conservation of energy tells us: Law of Conservation of Energy: Energy cannot be created nor destroyed. (It can change form.) Though there are several types of energy, the total energy of a closed system remains constant. Changes are NOT perfect – Energy is a scalar quantity expressing the capacity to do work. There are many types of energy. Most fit into two broad categories: 1. Potential Energy 2. Kinetic Energy When energy changes form, some energy is lost as heat (thermal energy). Friction also usually accompanies these changes – still lost as heat. Some other forms of energy: Work is done when energy is transferred from one object to another. Accounting for all forms of energy, we find that the total energy neither increases nor decreases. Energy as a whole is conserved. electric energy nuclear energy thermal energy chemical energy If there is a nonconservative force such as friction, where do the kinetic and potential energies go? They become heat; the actual temperature rise of the materials involved can be calculated. Problem Solving: See p. 157 1. Draw a picture. 2. Determine the system for which energy will be conserved. 3. Figure out what you are looking for, and decide on the initial and final positions. 4. Choose a logical reference frame. 5. Apply conservation of energy. 6. Solve. Power is the rate at which work is done. In the SI system, the units of power are watts. 1 W = 1 J/s = 1 N∙m/s 1 horsepower = 746 W So, the difference between walking and running up stairs is power – the work done and change in gravitational potential energy is the same. Power is also needed for acceleration and for moving against the force of gravity. The average power can be written in terms of the force and the average velocity: • Work: • Kinetic energy is energy of motion: • Potential energy is energy associated with forces that depend on the position or configuration of objects. • The net work done on an object equals the change in its kinetic energy – the work-energy principle. • If only conservative forces are acting, mechanical energy is conserved. • Power is the rate at which work is done. Conservation of Mechanical Energy p.163 #33, 34, 39, 42 Law of Conservation of Energy p.164 # 47 + 48 Power p.165 # 58, 59, 60