Unformatted text preview: W=
F (x power
dx by a
dU = a
Mechanical)dx = − delivered = − force on Ui − Uf
dx
i
i
i moving particle: Work = W = Kf − Ki = −Uf + Ui
Thus : Ki the i = and the f
P = power, depends on+ UforceKf + Uvelocity.
A simple formula: (W = work)
E = K + U is conserved r
dW
F · d
v
P=
=
= F ·
dt
dt
Example: lift 100 kg a height of 1 meter in 1 second:
P = (100 kg ) (g) (1m/s) = 31000 Watts = 10 light bulbs
Demo: bike generator Q: How many horses does it take to replace a light bulb?
A: 0.13 horses
1 horsepower = 746 Watt.
1 lightbulb/1horse = 100 W/746W = 0.13 Automotive horsepower...
How much power should a car engine provide in order for the
car to be capable of going up a 30 degree hill at a speed of 100
km/hr? (=28 m/s). Assume the car’s mass is 1000kg.
v 30° Fg = mg sin 30° P= mg sin30°v = (1000)(9.8) sin 30°(28)/746 = 184 hp Hydropower — Gravitational Potential Energy Hoover Dam — Colorado River (Nevada/Arizona) Hydropower — Gravitational Potential Energy Niagara Falls — NY/Canada 57 m drop
2800 m3/s flow rate Niagara Falls Horseshoe Falls, Canada Robert Moses
Hydroelectric Power Plant
& Reservoir
Lewiston, NY Hydropower In time Δt: — Gravitational Potential Energy ΔV H (V= volume) ΔV ΔUg = Δm g H = ΔV ρwater g H
Available power: P = g ΔU g Δt = Δm
Δt g H = ΔV
Δt ρ water gH P = (flow rate) (density of water) g H Ex. Niagara Falls H = 57 m
flow rate P = g ΔV
Δt ρ water ΔV
Δt ~ 2800 m3 / s gH ~ (2.8 x 103m3/s)(1.0 x 103 kg/m3)(9.8 N/kg)(57 m) = 1600 megaWatts Aside: ¿What’s wrong with this picture‽ No “adhesion”, i.e. no glue, only static friction what’s wrong with this picture...... Recall if held by static friction, tan θmax = μs
But based on the picture θmax > 80°
Thus, μs > 5.7 !
But I said, “for all practical purposes” μs < 1
hmmmm........ Gecko inspired superfriction:
Cartoon of friction synthetic nanofiber
(what is nano? A: the diameter of the fiber, 600nm) Gecko’s feet Center of mass and
Momentum * So far: we’ve treated objects as point particles. * We now consider co!ections of particles or extended objects.
* We wi! see: there is a special point ca!ed the CENTER OF
MASS that behaves like a point particle in response to external
forces.
* This leads to MOMENTUM CONSERV
ATION
* A! these new concepts are derived "om Newton’s Laws. Center of Mass
Motivating demo: projectile motion of extended objects.
Observe: there is a special point on the object that has a perfect
parabolic trajectory as for projectile motion.
The object does not have to be rigid: De...
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This note was uploaded on 09/29/2012 for the course MATH 1920 taught by Professor Pantano during the Spring '06 term at Cornell.
 Spring '06
 PANTANO
 Multivariable Calculus

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