WORK AND ENERGY Work done by a constant
WORK AND ENERGY Work done by a constant force Kinetic Energy Gravitational Potential Energy Simple Machines
WORK F x W = Fx SI unit of work = Newton-meter = Joule
EXAMPLE FN Accelerating a crate on a truck m = 150 kg f a = 2 m/s 2 mg f = ma = (150)(2) = 300 N If the truck accelerates for x = 50 m, the work done on the crate is: W = (f)x = 300(50) = 15000 J
KINETIC ENERGY • • • The work done on the crate is W = max Use x = 1/2 at 2 W = 1/2 m(at)2 = 1/2 mv 2 Kinetic Energy = KE = 1/2 mv 2 SI unit of kinetic energy = Joule Work-Energy Theorem: W = KEf - KEi
Crate Example Backwards W = 15000 J. What is v? 1/2 mv 2 = 15000, so v 2 = 30000/m = 30000/150 = 200(m/s)2 v = 14. 1 m/s
Example: Space Ship • m = 50000 kg, v 0 = 10, 000 m/s • Engine force = 500, 000 N, x = 3, 000 m. What is final speed? • W = (5*105 N)(3*106 m) = 1. 5*1012 J • KEf = KEi + W = 2. 5*1012 + 1. 5*1012 = 4*1012 J • vf = (2 KEf/m)1/2 = 12, 600 m/s
Gravitational Potential Energy • The gravity force can do positive or negative work on an object. • W = mg(h 0 - h) • All that counts is the vertical height change. • PE = mgh
EXAMPLE: PILE DRIVER Mass is dropped on a nail from a height h. Wg = mgh = 1/2 mv 2 It exerts force F on nail, pushing it into the wood a distance d, and coming to a stop. Wn = -Fd = -1/2 mv 2 F = mg(h/d) M
f d l THE LEVER L Work done on one end = work done by the other end. fd = FD f/F = D/d = L/l D F
WORK-ENERGY THEOREM: GRAVITY DOING THE WORK W = 1/2 mvf 2 - 1/2 mvi 2 = KE = - PE W = - PE KE + PE = 0 Mechanical Energy = E = KE + PE = CONSTANT When friction can be ignored
Principle of Conservation of Mechanical Energy • E remains constant as an object moves provided that no work is done on it by external friction forces.
EXAMPLE: PENDULUM Forces: Gravity Tension (does no work) E = KE + PE remains constant as pendulum swings
BOUNCING BALL E = PE = mgh h -vf vf E = KE = 1/2 mv 2 Initial Before bounce After bounce
DOUBLE BALL BOUNCE A problem in relative motion b B -v -v -v v f Just after big ball hits floor, vb. B = -2 v Just after little ball hits big ball, vb. B = 2 v and vbf = vb. B+ v. Bf= 3 v. How high will it rise? h = vbf 2/2 g = 9 v 2/2 g = 9 h 0
Using the Conservation of Mechanical Energy • Identify important forces. Friction forces must be absent or small. • Choose height where gravitational PE is zero. • Set initial and final KE + PE equal to each other
Roller Coaster • After a vertical drop of 60 m, how fast are the riders going? • Neglecting friction, mechanical energy will be conserved. • Ei = mgh Ef = 1/2 mv 2 • v = (2*9. 8*60)1/2 = 34. 3 m/s (76 mph)
Roller Coaster Again • If the final speed is 32 m/s, how much work was done by friction on a 60 kg rider? • Wnc = Ef - Ei = 1/2 mv 2 - mgh • = 1/2*60*(32)2 - 60*9. 8*60 • = 30700 - 35300 = - 4600 J
Power • • P = Work/Time = W/t SI unit = J/s = watt (W) 1 horsepower (hp) = 746 W If a force F is needed to move an object with average speed vav, then the power required is Pav = Fvav
Accelerating a Car • A 1500 kg car accelerates with a = 5 m/s 2 for 6 s. What power is needed? • F = ma = 7500 N • vf = at = 30 m/s so vav = 15 m/s • Pav = Fvav = 1. 1*105 W (151 hp)
Car at constant speed • Car going 60 mph (27 m/s) requires F = 200 N to overcome friction. • What power is required from the engine? • P = Fv = 200*27 = 5400 W = 7. 2 hp
TOUR DE FRANCE What power does cyclist need? Air friction force = f = kv 2 P = fv = kv 3 P = 1 k. W for v = 25 mph What power does Superman need to go 50 mph? P = 1 k. W(v 2/v 1)3 = 8 k. W
Principle of Energy Conservation • Energy can be neither created nor destroyed, but only converted from one form to another.
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