work done by a constant force kinetic energy gravitational potential energy simple machines work and...
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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
FN
f = ma = (150)(2) = 300N
a = 2 m/s2
m = 150 kg
If the truck accelerates for x = 50 m, the work done on the crate is:
W = (f)x = 300(50) = 15000 J
EXAMPLE
KINETIC ENERGY
• The work done on the crate is W = max
• Use x = 1/2at2
• W = 1/2m(at)2 = 1/2mv2
• Kinetic Energy = KE = 1/2mv2
• SI unit of kinetic energy = Joule
• Work-Energy Theorem: W = KEf - KEi
Crate Example Backwards
W = 15000J. What is v?
1/2mv2 = 15000, sov2 = 30000/m = 30000/150 = 200(m/s)2
v = 14.1 m/s
Example: Space Ship
• m = 50000kg, v0 = 10,000 m/s
• Engine force = 500,000 N, x = 3,000,000m. What is final speed?
• W = (5*105N)(3*106m) = 1.5*1012 J
• KEf = KEi + W = 2.5*1012 + 1.5*1012 = 4*1012 J
• vf = (2KEf/m)1/2 = 12,600 m/s
Gravitational Potential Energy
• The gravity force can do positive or negative work on an object.
• W = mg(h0 - h)
• All that counts is the vertical height change.
• PE = mgh
M
Mass is dropped on a nail from a height h.Wg = mgh = 1/2mv2
F = mg(h/d)
Wn = -Fd = -1/2mv2
It exerts force F on nail, pushing itinto the wood a distance d, and coming to a stop.
EXAMPLE: PILE DRIVER
l
L
Work done on one end = work doneby the other end.
d
D
f
F
fd = FD
f/F = D/d = L/l
THE LEVER
W = 1/2mvf2 - 1/2mvi
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
-vf
Beforebounce
vf
Afterbounce
BOUNCING BALLE = PE = mgh
E = KE = 1/2mv2
-v
-v
-v
v
Just after big ball hits floor, vbB = -2v
f
b
B
h = vbf2/2g = 9v2/2g = 9h0
and vbf = vbB+ vBf= 3v. How high will it rise?
Just after little ball hits big ball, vbB = 2v
DOUBLE BALL BOUNCEA 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.
• Ei = mgh Ef = 1/2mv2
• 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?
• Wnc = Ef - Ei = 1/2mv2 - 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 = 5m/s2 for 6 s. What power is needed?
• F = ma = 7500 N
• vf = at = 30 m/s so vav = 15m/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
Air friction force = f = kv2
P = fv = kv3
P = 1 kW for v = 25 mph
What power does Superman need to go50 mph?
P = 1 kW(v2/v1)3 = 8 kW
TOUR DE FRANCEWhat power does cyclist need?
Principle of Energy Conservation
• Energy can be neither created nor destroyed, but only converted from one form to another.
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