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Work & Energy

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Explain work, energy and power. The Law of Conservation of Energy is utilized as well as conservative and non conservative systems.
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Work & Energy

2. 2. WORK & ENERGY • Work, in a physics sense, has a precise definition, unlike the common use of the word. When you do your home “work” you probably, from a physics stand point, did no work at all ! • Work is defined as force applied in the direction of the motion multiplied times the distance moved. • When work is done by moving an object in a horizontal direction, work equals the applied force times the cosine of the angle of the applied force times the distance the object is moved. • W = F (cos ) x s, (s stands for distance) • Work is a scalar quantity (it has no direction). The sign of a work quantity (positive or negative) indicates the direction of energy flow as into or out of a system but does not give it a direction as in a vector quantity. 2
3. 3. 3
4. 4. WORK & ENERGY • The terms work and energy are interchangeable. Energy is defined as the ability to do work. • Kinds of work and energy • (1) mechanical work – work done by applying a force over a distance • (2) work of friction – work required to overcome friction • (3) gravitational potential energy – energy needed to lift an object against the force of gravity • (4) elastic potential energy – the energy stored in a compressed or stretched spring • (5) kinetic energy – energy an object has because of its motion (velocity) 4
5. 5. Distance moved Applied force Vertical component Horizontal component  Vertical component = Applied force x sin  Horizontal component = Applied force x cos   Work = force in direction of motion x distance moved The horizontal force component is in the direction of the motion 5
6. 6. FORCE OF FRICTION = 0 WORK DONE = 0 FORCE OF FRICTION > 0 WORK DONE = FFRICTION x DISTANCE Recall: Ffriction = coefficient of friction x Fnormal and on a horizontal surface: Fnormal = weight of object = mass x gravity 6
7. 7. GRAVITATIONAL POTENTIAL ENERGY • When an object is lifted, work is done against the force of gravity (the weight of the object). • Since weight is a force and the height to which an object is lifted is a distance, then force times distance equals work done. • Weight of an object can be calculated using mass time gravity. When objects are lifted near the surface of the earth, gravity is assumed to be constant at 9.8 m/s2 (32 ft/s2). • If object are lifted well beyond the earth’s surface gravity diminishes to progressively smaller values and the work done in the lifting becomes less and less. 7
8. 8. GRAVITATIONAL POTENTIAL ENERGY • Potential energy change equals weight times change in height. • Weight equals mass times gravity • Potential energy change equals mass times gravity times height (distance lifted)   8
9. 9. MEASUREMENT OF POTENTIAL ENERGY IS RELATIVE Boy Two! You’re at a High potential I sure am ! Who are they kidding ?? One Two Three Two is at a higher potential energy than One but lower than Three. Two’s potential energy is negative relative to Three’s and positive relative to One’s. If this point was used as reference, One, Two and Three would all have negative potential energies. 9
10. 10. Radius of Earth = 4000 miles scale150 lbs Two Radius of Earth = 8000 miles scale37.5 lbs Three Radius of Earth = 12000 miles scale16.7 lbs ¼ wt 1/9 wt Normal wt 10
11. 11. Calculating Work in Different Gravitational Fields • Potential energy changes are different in different gravitational field because the value of g changes. • As seen in the previous slide, at an altitude of one earth radii above the earth (4000 miles) gravity is ¼ of normal gravity (1/4 x 9.8 m/s2 = 2.45 m/s2). At two earth radii altitude, gravity is 1.09 m/s2. • An object of mass 10 kg is lifted 5 meters on earth. The work done (potential energy increase) is (P.E. = mgh) 10 kg x 9.8 m/s2 x 5 m = 490 joules. • At one earth radii, work done is 10 kg x 2.45 m/s2 x 5 m = 122.5 joules (1/4 of the work done in lifting the same object on earth) • At two earth radii above the earth (8000 miles altitude) the work done on the same object is 10 kg x 1.09 m/s2 x 5 m = 54.4 joules (1/9 of the work required to lift the object on earth) 11
12. 12. KINETIC ENERGY • Kinetic energy is the energy of motion. In order to possess kinetic energy an object must be moving. • As the speed (velocity) of an object increases its kinetic energy increases. The kinetic energy content of a body is also related to its mass. The most massive objects at the same speed contain the most kinetic energy. • Work = force x distance (W = F x s ) • Recall that F = mass x acceleration (F = m x a) • Therefore: Work = m x a x s • Also, for an object initially at rest, recall that acceleration equals the final velocity squared divided by twice the distance traveled: a = v2 / (2 s) • Work = m (v2 / (2 s)) s, canceling out the distance term (s) gives, Work = (m v2 ) / 2 or 1/2 m v2 • Since the object is in motion, the work content is called kinetic energy and therefore: K.E. = 1/2 m v2 12
13. 13. High kinetic energy. High velocity ! Kinetic energy = 0 No motion ! 13
14. 14. ELASTIC POTENTIAL ENERGY • Elastic potential energy refers to the energy which is stored in stretched of compressed items such as springs or rubber bands. • The elongation or compression of elastic bodies is described by Hooke’s Law. This law relates the force applied to the elongation or compression experienced by the body. • In plain words, Hooke’s Law says, “the harder you pull on a spring, the more it stretches”. This relationship is given by the equation: F = k x. • F is the applied force, k is a constant called the spring constant or Hooke’s constant and x is the elongation of the spring. • Springs with large k values are hard to stretch or compress such as a car spring. Those with small constants are easy of stretch or compress such as a slinky spring. 14
15. 15. 400 grams 200 grams F O R C E (N) ELONGATION (M) Slope = spring constant 600 grams Elongation of spring 15
16. 16. F o r c e (N) Distance (M) x Constant force Work = force x distance Constant force Distance moved Force x distance equals area under the graph Work = area under a force versus distance graph 16
17. 17. F O R C E (N) ELONGATION (M) X1 X2 F1 F1 Area under the graph gives the work to stretch the spring. Work needed to stretch the spring to x2 is ½ F2 times x2 Work needed to stretch the spring to x1 is ½ F1 times x1 Work needed to stretch the spring from x1 to x2 is (½ F2 times x2) – ( ½ F1 times x1) Since F = kx and W = ½ Fx, W = ½ (kx) x or W = ½ kx2 and work from x1 to x2 is given by: W = ½ k (x2 2 – x1 2) 17
18. 18.  18
19. 19. F O R C E (N) DISPLACEMENT (M) X1 X2 WORK = AREA UNDER THE CURVE W =  F  X (SUM OF THE BOXES) WIDTH OF EACH BOX =  X AREA MISSED - INCREASING THE NUMBER BOXES WILL REDUCE THIS ERROR! AS THE NUMBER OF BOXES INCREASES, THE ERROR DECREASES! BOX METHOD 19
20. 20. Finding Area Under Curves Mathematically • Areas under force versus distance graphs (work) can be found mathematically. The process requires that the equation for the graph be known and integral calculus be used. • Recall that integration is also referred to as finding the antiderivative of a function. • The next slide reviews the steps in finding the integral of the basic function, y = kxn. 20