In this section you will: Solve problems using the law of conservation of energy. Section 11.2

Conservation of Energy Consider a ball near the surface of Earth. The sum of gravitational potential energy and kinetic energy in that system is constant. As the height of the ball changes, energy is converted from kinetic energy to potential energy, but the total amount of energy stays the same. In our everyday world, it may not seem as if energy is conserved. Section 11.2

Law of Conservation of Energy The law of conservation of energy states that in a closed, isolated system, energy can neither be created nor destroyed; rather, energy is conserved. Under these conditions, energy changes from one form to another while the total energy of the system remains constant. Section 11.2

Mechanical Energy The sum of the kinetic energy and gravitational potential energy of a system is called mechanical energy. In any given system, if no other forms of energy are present, mechanical energy is represented by the following equation. Mechanical Energy of a System E = KE + PE The mechanical energy of a system is equal to the sum of the kinetic energy and potential energy if no other forms of energy are present. Section 11.2

Roller Coasters In the case of a roller coaster that is nearly at rest at the top of the first hill, the total mechanical energy in the system is the coaster’s gravitational potential energy at that point. If another hill along the track is higher than the first one. The roller coaster would not be able to climb the higher hill because the energy required to do so would be greater than the total mechanical energy of the system. Section 11.2

Pendulums The simple oscillation of a pendulum also demonstrates conservation of energy. The system is the pendulum bob and Earth. Usually, the reference level is chosen to be the height of the bob at the lowest point, when it is at rest. If an external force pulls the bob to one side, the force does work that gives the system mechanical energy. Section 11.2

Pendulums At the instant the bob is released, all the energy is in the form of potential energy, but as the bob swings downward, the energy is converted to kinetic energy. The figure shows a graph of the changing potential and kinetic energies of a pendulum. Section 11.2

Pendulums When the bob is at the lowest point, its gravitational potential energy is zero, and its kinetic energy is equal to the total mechanical energy in the system. Note that the total mechanical energy of the system is constant if we assume that there is no friction. Section 11.2

Loss of Mechanical Energy The oscillations of a pendulum eventually come to a stop, a bouncing ball comes to rest, and the heights of roller coaster hills get lower and lower. Where does the mechanical energy in such systems go? Any object moving through the air experiences the forces of air resistance. In a roller coaster, there are frictional forces between the wheels and the tracks. Section 11.2

Loss of Mechanical Energy When a ball bounces off of a surface, all of the elastic potential energy that is stored in the deformed ball is not converted back into kinetic energy after the bounce. Some of the energy is converted into thermal energy and sound energy. Section 11.2

Loss of Mechanical Energy The motion of the molecules that make up an object is called thermal energy. Much of the energy used to overcome the force of friction is converted into thermal energy. This can cause a slight increase in the temperature of an object. Section 11.2

Conservation of Mechanical Energy During a hurricane, a large tree limb, with a mass of 22.0 kg and at a height of 13.3 m above the ground, falls on a roof that is 6.0 m above the ground. A. Ignoring air resistance, find the kinetic energy of the limb when it reaches the roof. B. What is the speed of the limb when it reaches the roof? Section 11.2

Step 1: Analyze and Sketch the Problem Conservation of Mechanical Energy Step 1: Analyze and Sketch the Problem Section 11.2

Conservation of Mechanical Energy Sketch the initial and final conditions. Choose a reference level. Section 11.2

Conservation of Mechanical Energy Draw a bar graph. Section 11.2

Conservation of Mechanical Energy Identify the known and unknown variables. Known: m = 22.0 kg g = 9.80 m/s2 hlimb = 13.3 m vi = 0.0 m/s hroof = 6.0 m KEi = 0.0 J Unknown: PEi = ? KEf = ? PEf = ? vf = ? Section 11.2

Step 2: Solve for the Unknown Conservation of Mechanical Energy Step 2: Solve for the Unknown Section 11.2

Conservation of Mechanical Energy A. Set the reference level as the height of the roof. Solve for the initial height of the limb relative to the roof. h = hlimb – hroof Section 11.2

Conservation of Mechanical Energy Substitute hlimb = 13.3 m, hroof = 6.0 m h = 13.3 m – 6.0 m = 7.3 m Section 11.2

Conservation of Mechanical Energy Solve for the initial potential energy of the limb. PEi = mgh Substitute m = 22.0 kg, g = 9.80 m/s2, h = 7.3 m PEi = (22.0 kg) (9.80 m/s2) (7.3 m) = 1.6×103 J Section 11.2

Conservation of Mechanical Energy Identify the initial kinetic energy of the limb. The tree limb is initially at rest. KEi = 0.0 J Section 11.2

Conservation of Mechanical Energy The kinetic energy of the limb when it reaches the roof is equal to its initial potential energy because energy is conserved. KEf = PEi PEf = 0.0 J because h = 0.0 m at the reference level. KEf = 1.6×103 J Section 11.2

Conservation of Mechanical Energy B. Solve for the speed of the limb. Section 11.2

Conservation of Mechanical Energy Substitute KEf = 1.6×103 J, m = 22.0 kg Section 11.2

Step 3: Evaluate the Answer Conservation of Mechanical Energy Step 3: Evaluate the Answer Section 11.2

Conservation of Mechanical Energy Are the units correct? Velocity is measured in m/s and energy is measured in kg·m2/s2 = J. Do the signs make sense? KE and the magnitude of velocity are always positive. Section 11.2

Conservation of Mechanical Energy The steps covered were: Step 1: Analyze and Sketch the Problem Sketch the initial and final conditions. Choose a reference level. Draw a bar graph. Section 11.2

Conservation of Mechanical Energy The steps covered were: Step 2: Solve for the Unknown Set the reference level as the height of the roof. Solve for the initial height of the limb relative to the roof. Solve for the speed of the limb. Step 3: Evaluate the Answer Section 11.2

Question 1 Write the law of conservation of energy and state the formula for the mechanical energy of the system. Section 11.2

Answer 1 Answer: The law of conservation of energy states that in a closed, isolated system, energy can neither be created nor destroyed; rather, energy is conserved. Under these conditions, energy changes from one form to another, while the total energy of the system remains constant. Since energy is conserved, if no other forms of energy are present, kinetic energy and potential energy are inter-convertible. Section 11.2

Answer 1 Answer: Mechanical energy of a system is the sum of the kinetic energy and potential energy if no other forms of energy are present. That is, mechanical energy of a system E = KE + PE. Section 11.2

Question 2 Two brothers, Jason and Jeff, of equal masses, jump from a house 3 m high. If Jason jumps to the ground and Jeff jumps to a platform 2 m high, what can you say about their kinetic energy? Section 11.2

Question 2 A. The kinetic energy of Jason when he reaches the ground is greater than the kinetic energy of Jeff when he lands on the platform. B. The kinetic energy of Jason when he reaches the ground is less than the kinetic energy of Jeff when he lands on the platform. C. The kinetic energy of Jason when he reaches the ground is equal to the kinetic energy of Jeff when he lands on the platform. D. Neither Jason nor Jeff possesses kinetic energy. Section 11.2

Answer 2 Reason: The kinetic energy at the ground level is equal to the potential energy at the top. Since potential energy is proportional to height, and since the brothers have equal masses, Jason will have greater kinetic energy since his height above the ground decreases more than Jeff’s. Section 11.2