Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 1 Chapter 6: 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

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 2 § 6.1 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.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 3

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 4 § 6.2 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.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 5 It is 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

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 6 The work done by the 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.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 7 The net work done on the box is:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 8 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).

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 9 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

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 10 Example: A box of mass m is towed up a frictionless incline at constant speed. The applied force F is parallel to the incline. What is the net work done on the box?  F w N F x y  Apply Newton’s 2 nd Law:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 11 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:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 12 Example continued: The work by the normal force is: The net work done on the box is:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 13 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:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 14 § 6.3 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

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 15 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.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 16 § 6.4 Gravitational Potential Energy Part 1 Objects have potential energy because of their location (or configuration). There are potential energies associated with different (but not all!) forces. Such a force is called a conservative force. In general

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 17 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 that you are free to choose.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 18 Example: What is the change in gravitational potential energy of the box if it is placed on the table? The table is 1.0 m tall and the mass of the box is 1.0 kg. First: Choose the reference level at the floor. U=0 here.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 19 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.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 20 Mechanical energy is Whenever nonconservative forces do no work, the mechanical energy of a system is conserved. That is E i =E f or  K= -  U.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 21 Example (text problem 6.27): A cart starts from position 4 with v = 15.0 m/s to the left. Find the speed of the cart at positions 1, 2, and 3. Ignore friction.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 22 Or use E 3 =E 2 Or use E 3 =E 1 E 2 =E 1 Example continued:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 23 Example (text problem 6.82): 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?

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 24 (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:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 25 (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

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 26 Example continued: v=v min when N=0. This means that What is v min ? The initial height must be

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 27 What do you do when there are nonconservative forces? For example, if friction is present The work done by friction.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 28 § 6.5 Gravitational Potential Energy Part 2 The general expression for gravitational potential energy is:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 29 Example: What is the gravitational potential energy of a body of mass m on the surface of the Earth?

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 30 § 6.6 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.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 31 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

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 32 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.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 33 Example (text problem 6.48): (a) If forces of 5.0 N applied to each end of a spring cause the spring to stretch 3.5 cm from its relaxed length, 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 :

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 34 Example continued: (b) What is the spring constant of this spring? Or

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 35 Example (text problem 6.46): 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? F x (N) x (m) x 1 =0.4 m kx 1

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 36 The work done in stretching/compressing a spring transfers energy to the spring. § 6.7 Elastic potential energy

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 37 Example: 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. How far is the spring compressed when the box is brought to rest?

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 38 § 6.8 Power Average Power The unit of power is the watt. 1 watt = 1 J/s = 1 W. Instantaneous Power Power is the rate of energy transfer.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 39 Example (text problem 6.73): 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. What is the engine’s average power output? Neglect friction and air resistance.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 40 Summary Conservation of Energy Calculation of Work Done by a Constant or Variable Force Kinetic Energy Potential Energy (gravitational, elastic) Power