Work and Energy

1. F F F F F cos q F cos q F F x x F cos 90 = F0 W = F cos qx W = Fx x W = 0 Work and Energy • Energy ~ an ability to accomplish change • Work: a measure of the change produced by a force • Work = “a force through the displacement” • portion of the force along displacement × displacement • W = Fxx =F cos qx Units: 1Newton . 1 meter = 1 joule = 1J 1 ft-lb =1.356 J

2. Example: A child pulls a toy 2.00 m across the floor by a string, applying a force of constant magnitude 0.800 N. During the first meter, the string is parallel to the floor. During the second meter the string makes a 30º angle with the horizontal direction. What is the total work done by the child on the toy?

3. F x x1 x2 • Work and Energy with varying forces • Take average force, small sub-intervals Dxi F3 F F2 F1 … Dx3 DxN x Dx2 Dx4 Constant Force – near trivial example W = F.(x2-x1)

4. F x • Varying Force Example: Force of a Spring (GET OUT SOME SPRINGS!) • |F| = kx (Hooke’s “Law”) • k is the spring constant or force constant of that spring • From un-stretched to stretched/compressed by x: • area under curve = area of right triangle • W = ½ “height”. “width” = ½ kxx = ½ kx2 • Example: How much work is required to extend an exercise spring by 45 cm if the spring constant k is 310 n/m? What force is required to stretch it this far?

5. Energy: the capacity to do work • Kinetic Energy: energy associated with motion • Potential Energy: energy associated with position • Rest Energy, Thermal Energy, chemical energy... • Kinetic Energy for an object under a constant force • from motion in a straight line the work-energy theorem

6. Example: A baseball-player throws a 0.170 kg baseball at a speed of 36.0 m/s. What is its kinetic energy? Example: How much work is done to move a 1840 kg Jaguar XJ6 automobile from rest to a speed of 27.0 m/s on a level road? If this takes place of a distance of 117 m, what is the average force?

7. Potential Energy • energy associated with position • example: gravitational potential energy • Work done to raise an object a height h: W = mgh • = Work done by gravity on object if the object descends a height h. • identify source of work as Potential Energy • PE = mgh • other types of potential energy • electrical, magnetic, gravitational, compression of spring ... Example: How much potential energy does a 7.5 kg ceiling fan have with respect to the floor when it is 3.00 m above it?

8. Example: A 500 kg mass of a pile driver is dropped from a height of 3m onto a piling in the ground. The impact drives the piling 1.00 cm deeper into the ground. If the original potential energy of the mass is converted into work in driving the piling into the ground, what is the frictional force acting on the piling?

9. Elastic Potential Energy • energy stored in stretching or compressing a spring • Work done compressing: W = ½ k x2 • = work that can be extracted by releasing the spring • PE = ½ k x2 Example: A 1550 kg Pontiac Gran Prix is supported by 4 coil springs, each with a spring constant of 7.00E4 N/m. How much are the springs compressed by the weight of the car? How much energy is stored in this compression?

10. Conservation of Energy • Conservation Principle: For an isolated system, a conserved quantity keeps the same value no matter what changes the system undergoes. • Conservative Forces: Work done can be written as a change in potential energy. • Conservation of Mechanical Energy: The total amount of energy in an isolated system always remains constant, even though energy transformations from one form to another may occur. • Usually consider initial and final times: • KEi +PEi = KEf +PEf • or • Ei = Ef

11. Example: a plant is knocked off a window sill, where it falls from rest to the ground 5.27 m below. How fast is it going when it hits the ground? demo: rollercoaster and 4 track race

12. Example: A block of mass m is released from rest and slides down a frictionless track of height h. At the bottom of the track is a spring with a spring constant k attached to a wall. How far will the spring be compressed at the maximum point of compression?

13. (simplified) Example: A roller coaster starts at the top of a 48 m tall hill at an initial speed 0f 0.50 m/s before it plunges to its low point 3m above the ground. From there it climbs a smaller hill only 16 m high. • What is the speed of the train as it crests this hill? • The curvature of the crest of the hill has a curvature of radius 20 m. • What is the centripetal acceleration at the top of the hill? • What force must be exerted on a 1.5 kg video camera being held by one of the riders?

14. Nonconservative forces • Wc +Wnc = DKE • so • Wnc = DE = DKE+DPE • Friction is loss of energy • Einitial = Efinal + |Wfriction|

15. Example: A 55kg carton with an initial speed pf 0.45 m/s slides down a ramp inclined at an angle of 23 degrees. If the coefficient of friction is 0.24, how fast will the carton be moving after it has traveled a distance of 2.1 m down the ramp?

16. Power: the rate at which work is done

17. Example: A 70 kg person runs up a staircase 3.0 m high in 3.5 s. How much power does he develop climbing the stairs?