Work done by a constant force Kinetic Energy Gravitational Potential Energy Simple Machines WORK AND ENERGY

WORK x F W = Fx SI unit of work = Newton-meter = Joule

Accelerating a crate on a truck f mg FNFN f = ma = (150)(2) = 300N a = 2 m/s 2 m = 150 kg If the truck accelerates for x = 50 m, the work done on the crate is: W = (f)x = 300(50) = J EXAMPLE

KINETIC ENERGY The work done on the crate is W = max Use x = 1/2at 2 W = 1/2m(at) 2 = 1/2mv 2 Kinetic Energy = KE = 1/2mv 2 SI unit of kinetic energy = Joule Work-Energy Theorem: W = KE f - KE i

Crate Example Backwards W = 15000J. What is v? 1/2mv 2 = 15000, so v 2 = 30000/m = 30000/150 = 200(m/s) 2 v = 14.1 m/s

Example: Space Ship m = 50000kg, v 0 = 10,000 m/s Engine force = 500,000 N, x = 3,000,000m. What is final speed? W = (5*10 5 N)(3*10 6 m) = 1.5*10 12 J KE f = KE i + W = 2.5* *10 12 = 4*10 12 J v f = (2KE f /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

M Mass is dropped on a nail from a height h. W g = mgh = 1/2mv 2 F = mg(h/d) W n = -Fd = -1/2mv 2 It exerts force F on nail, pushing it into the wood a distance d, and coming to a stop. EXAMPLE: PILE DRIVER

l L Work done on one end = work done by the other end. d D f F fd = FD f/F = D/d = L/l THE LEVER

W = 1/2mv f 2 - 1/2mv i 2 =  KE = -  PE  KE +  PE = 0 W = -  PE Mechanical Energy = E = KE + PE = CONSTANT WORK-ENERGY THEOREM: GRAVITY DOING THE WORK 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.

Forces: Gravity E = KE + PE remains constant as pendulum swings Tension (does no work) EXAMPLE: PENDULUM

h Initial -v f Before bounce vfvf After bounce BOUNCING BALL E = PE = mgh E = KE = 1/2mv 2

-v v Just after big ball hits floor, v bB = -2v f b B h = v bf 2 /2g = 9v 2 /2g = 9h 0 and v bf = v bB + v Bf = 3v. How high will it rise? Just after little ball hits big ball, v bB = 2v DOUBLE BALL BOUNCE A problem in relative motion

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. E i = mgh E f = 1/2mv 2 v = (2*9.8*60) 1/2 = 34.3 m/s (76 mph)

Roller Coaster Again If the final speed is 32m/s, how much work was done by friction on a 60 kg rider? W nc = E f - E i = 1/2mv 2 - mgh = 1/2*60*(32) *9.8*60 = = 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 v av, then the power required is P av = Fv av

Accelerating a Car A 1500 kg car accelerates with a = 5m/s 2 for 6 s. What power is needed? F = ma = 7500 N v f = at = 30 m/s so v av = 15m/s P av = Fv av = 1.1*10 5 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

Air friction force = f = kv 2 P = fv = kv 3 P = 1 kW for v = 25 mph What power does Superman need to go 50 mph? P = 1 kW(v 2 /v 1 ) 3 = 8 kW TOUR DE FRANCE What power does cyclist need?

Principle of Energy Conservation Energy can be neither created nor destroyed, but only converted from one form to another.