Chapter 6 Conservation of Energy

MFMcGraw-PHY 1401Ch06b - Energy - Revised: 6/25/102 Conservation of Energy Work by a Constant Force Kinetic Energy Potential Energy Work by a Variable Force Springs and Hooke’s Law Conservation of Energy Power

MFMcGraw-PHY 1401Ch06b - Energy - Revised: 6/25/103 The Law of Conservation of Energy The total energy of the Universe is unchanged by any physical process. The three kinds of energy are: kinetic energy, potential energy, and rest energy. Energy may be converted from one form to another or transferred between bodies.

MFMcGraw-PHY 1401Ch06b - Energy - Revised: 6/25/104

MFMcGraw-PHY 1401Ch06b - Energy - Revised: 6/25/105 Work by a Constant Force Work is an energy transfer by the application of a force. For work to be done there must be a nonzero displacement. The unit of work and energy is the joule (J). 1 J = 1 Nm = 1 kg m 2 /s 2.

MFMcGraw-PHY 1401Ch06b - Energy - Revised: 6/25/106 Only the force in the direction of the displacement that does work. An FBD for the box at left: x y F w N  The work done by the force F is: rxrx  F rxrx Work - Example

MFMcGraw-PHY 1401Ch06b - Energy - Revised: 6/25/107 The work done by the normal force N is: The normal force is perpendicular to the displacement. The work done by gravity (w) is: The force of gravity is perpendicular to the displacement. Work - Example

MFMcGraw-PHY 1401Ch06b - Energy - Revised: 6/25/108 The net work done on the box is: Work - Example

MFMcGraw-PHY 1401Ch06b - Energy - Revised: 6/25/109 In general, the work done by a force F is defined as where F is the magnitude of the force,  r is the magnitude of the object’s displacement, and  is the angle between F and  r (drawn tail-to-tail). Work Done

MFMcGraw-PHY 1401Ch06b - Energy - Revised: 6/25/1010 Example: A ball is tossed straight up. What is the work done by the force of gravity on the ball as it rises? FBD for rising ball: x y w rr Work - Example

MFMcGraw-PHY 1401Ch06b - Energy - Revised: 6/25/1011 A box of mass m is towed up a frictionless incline at constant speed. The applied force F is parallel to the incline. Question: What is the net work done on the box?  F w N F x y  Apply Newton’s 2 nd Law: An FBD for the box: Inclined Plane-V Constant

MFMcGraw-PHY 1401Ch06b - Energy - Revised: 6/25/1012 The magnitude of F is: If the box travels along the ramp a distance of  x the work by the force F is The work by gravity is Example continued: Inclined Plane-V Constant

MFMcGraw-PHY 1401Ch06b - Energy - Revised: 6/25/1013 Example continued: The work by the normal force is: The net work done on the box is: Inclined Plane-V Constant

MFMcGraw-PHY 1401Ch06b - Energy - Revised: 6/25/1014 Example: What is the net work done on the box in the previous example if the box is not pulled at constant speed? Proceeding as before: Inclined Plane-Acceleration New Term

MFMcGraw-PHY 1401Ch06b - Energy - Revised: 6/25/1015 Kinetic Energy is an object’s translational kinetic energy. This is the energy an object has because of its state of motion. It can be shown that, in general Net Work = Change in K

MFMcGraw-PHY 1401Ch06b - Energy - Revised: 6/25/1016 Example: The extinction of the dinosaurs and the majority of species on Earth in the Cretaceous Period (65 Myr ago) is thought to have been caused by an asteroid striking the Earth near the Yucatan Peninsula. The resulting ejecta caused widespread global climate change. If the mass of the asteroid was kg (diameter in the range of 4- 9 miles) and had a speed of 30.0 km/sec, what was the asteroid’s kinetic energy? This is equivalent to ~10 9 Megatons of TNT. Kinetic Energy

MFMcGraw-PHY 1401Ch06b - Energy - Revised: 6/25/1017 Gravitational Potential Energy Part 1- Close to Earth’s Surface Potential energy is an energy of position. There are potential energies associated with different forces. Forces that have a potential energy associated with them are called conservative forces. Not all forces are conservative, i.e. Friction.

MFMcGraw-PHY 1401Ch06b - Energy - Revised: 6/25/1018 Gravitational Potential Energy Every conservative force has an associated potential function. F is equal in magnitude and opposite in direction of mg. The object is raised a distance Δx doing work against the gravitational field. The change in U is positive

MFMcGraw-PHY 1401Ch06b - Energy - Revised: 6/25/1019 The change in gravitational potential energy (only near the surface of the Earth) is where  y is the change in the object’s vertical position with respect to some reference point. You are free to choose to location of this reference point where ever it is convenient. Gravitational Potential Energy

MFMcGraw-PHY 1401Ch06b - Energy - Revised: 6/25/1020 The table is 1.0 m tall and the mass of the box is 1.0 kg. Ques: What is the change in gravitational potential energy of the box if it is placed on the table? First: Choose the reference level at the floor. U = 0 here. GPE Problem U = 0

MFMcGraw-PHY 1401Ch06b - Energy - Revised: 6/25/1021 Example continued: Now take the reference level (U = 0) to be on top of the table so that y i =  1.0 m and y f = 0.0 m. The results do not depend on the location of U = 0. GPE Problem

MFMcGraw-PHY 1401Ch06b - Energy - Revised: 6/25/1022 Mechanical energy is The total mechanical energy of a system is conserved whenever nonconservative forces do no work. That is E i = E f or  K =  U. Total Mechanical Energy Then if  K increases  U decreases and vice versa

MFMcGraw-PHY 1401Ch06b - Energy - Revised: 6/25/1023 A cart starts from position 4 with v = 0.0 m/s to the left. Find the speed of the cart at positions 1, 2, and 3. Ignore friction. Mechanical Energy Problem

MFMcGraw-PHY 1401Ch06b - Energy - Revised: 6/25/1024 Or use E 3 =E 2 Or use E 3 =E 1 E 2 =E 1 Mechanical Energy Problem

MFMcGraw-PHY 1401Ch06b - Energy - Revised: 6/25/1025 A roller coaster car is about to roll down a track. Ignore friction and air resistance. 40 m 20 m y=0 m = 988 kg (a) At what speed does the car reach the top of the loop? Roller Coaster Problem

MFMcGraw-PHY 1401Ch06b - Energy - Revised: 6/25/1026 (b) What is the force exerted on the car by the track at the top of the loop? Nw y x Example continued: FBD for the car: Apply Newton’s Second Law: Roller Coaster Problem

MFMcGraw-PHY 1401Ch06b - Energy - Revised: 6/25/1027 (c) From what minimum height above the bottom of the track can the car be released so that it does not lose contact with the track at the top of the loop? Example continued: Using conservation of mechanical energy: Solve for the starting height Roller Coaster Problem

MFMcGraw-PHY 1401Ch06b - Energy - Revised: 6/25/1028 Example continued: v = v min when N = 0. This means that What is v min ? The initial height must be Roller Coaster Problem

MFMcGraw-PHY 1401Ch06b - Energy - Revised: 6/25/1029 What do you do when there are nonconservative forces? For example, if friction is present The work done by friction. Nonconservative Forces In the presence of non-conservative forces Total Mechanical Energy is not conserved.

MFMcGraw-PHY 1401Ch06b - Energy - Revised: 6/25/1030 Gravitational Potential Energy Part 2 - Away from Earth’s Surface The general expression for gravitational potential energy is: It is convenient in this case to assign the zero of PE to a point an infinite distance away.

MFMcGraw-PHY 1401Ch06b - Energy - Revised: 6/25/1031 Gravitational Potential Energy The potential energy can be viewed as the work required to bring a second mass from infinity to its present position r. Since we can a conservative force that has a potential we don’t have to do the force x distance calculation.

MFMcGraw-PHY 1401Ch06b - Energy - Revised: 6/25/1032 Example: What is the gravitational potential energy of a body of mass m on the surface of the Earth? Gravitational Potential Energy The negative sign is useful in describing bound states, i.e. A moon bound to an planet, as having a negative energy. A positve energy state would represent a free state, i.e. an unbound moon that had somehow bee freed from its planet..

MFMcGraw-PHY 1401Ch06b - Energy - Revised: 6/25/1033 A planet of mass m has an elliptical orbit around the Sun. The elliptical nature of the orbit means that the distance between the planet and Sun varies as the planet follows its orbital path. Take the planet to move counterclockwise from its initial location. QUES: How does the speed of a planet vary as it orbits the Sun once? The mechanical energy of the planet-sun system is: r B A Planetary Motion

MFMcGraw-PHY 1401Ch06b - Energy - Revised: 6/25/1034 Starting from point “B” (where the planet moves the slowest), as the planet moves in its orbit r begins to decrease. As it decreases the planet moves faster. At point “A” the planet reaches its fastest speed. As the planet moves past point A in its orbit, r begins to increase and the planet moves slower. r B A Planetary Motion At point “B” the planet is the farthest from the Sun. At point “A” the planet is at its closest approach to the sun.

MFMcGraw-PHY 1401Ch06b - Energy - Revised: 6/25/1035 Work by a Variable Force Work can be calculated by finding the area underneath a plot of the applied force in the direction of the displacement versus the displacement.

MFMcGraw-PHY 1401Ch06b - Energy - Revised: 6/25/1036 x (m) F x (N) F3F3 F2F2 F1F1 x3x3 x2x2 x1x1 The work done by F 1 is Example: What is the work done by the variable force shown below? The net work is then W 1 +W 2 +W 3. The work done by F 2 is The work done by F 3 is

MFMcGraw-PHY 1401Ch06b - Energy - Revised: 6/25/1037 By hanging masses on a spring we find that stretch  applied force. This is Hooke’s law. For an ideal spring: F x =  kx F x is the magnitude of the force exerted by the free end of the spring, x is the measured stretch of the spring, and k is the spring constant (a constant of proportionality; its units are N/m). A larger value of k implies a stiffer spring. Spring Force

MFMcGraw-PHY 1401Ch06b - Energy - Revised: 6/25/1038 (a) A force of 5.0 N applied to the end of a spring cause the spring to stretch 3.5 cm from its relaxed length. Ques: How far does a force of 7.0 N cause the same spring to stretch? For springs F  x. This allows us to write Solving for x 2 : Spring Force

MFMcGraw-PHY 1401Ch06b - Energy - Revised: 6/25/1039 Example continued: (b) What is the spring constant of this spring? Or Spring Force

MFMcGraw-PHY 1401Ch06b - Energy - Revised: 6/25/1040 An ideal spring has k = 20.0 N/m. What is the amount of work done (by an external agent) to stretch the spring 0.40 m from its relaxed length? Spring Force

MFMcGraw-PHY 1401Ch06b - Energy - Revised: 6/25/1041 The work done in stretching or compressing a spring transfers energy to the spring. Below is the equation of the spring potential energy. Elastic Potential Energy The spring is considered the system

MFMcGraw-PHY 1401Ch06b - Energy - Revised: 6/25/1042 A box of mass 0.25 kg slides along a horizontal, frictionless surface with a speed of 3.0 m/s. The box encounters a spring with k = 200 N/m. Ques: How far is the spring compressed when the box is brought to rest? Elastic Potential Energy

MFMcGraw-PHY 1401Ch06b - Energy - Revised: 6/25/1043 Power Average Power The unit of power is the watt. 1 watt = 1 J/s = 1 W. 1 Horsepower (hp) = 550 ft-lbs/s = W Instantaneous Power Power is the rate of energy transfer.

MFMcGraw-PHY 1401Ch06b - Energy - Revised: 6/25/1044 A race car with a mass of kg completes a quarter-mile (402 m) race in a time of 4.2 s starting from rest. The car’s final speed is 125 m/s. (Neglect friction and air resistance.) Ques: What is the engine’s average power output? Power - Car Example

MFMcGraw-PHY 1401Ch06b - Energy - Revised: 6/25/1045 Summary Conservation of Energy Calculation of Work Done by a Constant or Variable Force Kinetic Energy Potential Energy (gravitational, elastic) Power