Transcript Document

Work and Kinetic Energy
Work: a measure of the change produced by a force
Work = force through the displacement
W=Fs
(assuming force is constant!)
Units: 1 Newton . 1 meter = 1 joule = 1J = 1 N . m
F
F
x
W=Fx
Phys211C6 p1
Ex: A car is pushed by a constant 200 N force across a level street for a distance of 20 m. How much work is done by the
person pushing on the car?
Phys211C6 p2
Work: portion of the force along displacement * displacement
W = F cos f x
F
F
F cos f
x
W = F cos f |x|
F
F
F cos f
F
F
F cos 90 = F0
x
W=0
x
W = F |x|
Total Work = Fnet cos f x where f is the angle between net force and displacement
What is net work done on an object moving at constant velocity?
Phys211C6 p3
Ex: An object with weight 15,000 N is dragged for a distance of 20.0 m across level ground. The force is exerted
through a cable with a tension of 5,000 N which makes an angle of 36.9º above the horizontal. There is a 3,500 N
frictional force opposing the motion.
What is the work done by gravity, the force applied by the tension and friction?
Phys211C6 p4
Kinetic Energy
motion in a straight line from constant acceleration (and force)
W  F ( x2  x1 )  Fx  ma x
2 nd Law
v2  v1  2ax
from 1 - d motion
2
2
W  max   m
v2  v1 1
1
2
2
1

 mv2  mv1   mv 2 
2
2
2
2

2
2
define Kinetic Energy
Work  Energy Theorem
1 2
K  mv
2
W  K  K 2  K1
Phys211C6 p5
Ex: Take previous example of the 15,000 N dragged across level ground. Determine the final speed via the WorkEnergy Theorem and via Newton’s Laws. Take the initial speed to be 3.00 m/s.
Example: A pile driver uses a 200 kg hammer is dropped from a height of 3m above a beam that is being driven
into the ground. The rails which guide the hammer exert a 60.0 N frictional force on the falling hammer-head.
The beam is driven 7.4 cm further into the ground with each impact.
What is the speed of the hammer as it hits the beam?
What is the average force of the hammer on the pile when struck?
Phys211C6 p6
Interpreting Kinetic Energy
accelerating an object from rest
Wnet = K – 0 = K
Kinetic Energy is the total work necessary to accelerate an
object from rest to its final speed.
conversely
Kinetic Energy is the total work an object can do in the
process of being brought to rest.
Phys211C6 p7
Example: Consider two objects of mass m and 2m are accelerated from rest.
Compare the work done on them, their final kinetic energies and their final speeds if they are under the influence
of identical forces acting over the same distances.
Compare the work done on them, their final kinetic energies and the distances they must have traveled if they are
under the influence of identical forces and end up with the same final speed.
Phys211C6 p8
Work and Energy with varying forces
Take average force, small sub-intervals xi
W  F1x1  F2 x2    FN x N
F3
F
F2
F1
areas of rectangles
W  lim  Fi xi
x 0
x2
  Fdx
x2
x3
x4
…
xN
x1
area under curve!
Constant Force – near trivial example
W = F.(x2x1)
F
x1
x2
x
Phys211C6 p9
x
Varying Force Example: Force of a Spring
|F| = kx (Hooke’s “Law”)
k is spring constant or force constant
F
note that force and displacement are in opposite directions!
From origin to x:
area under curve = area of right triangle
x
W = ½ “height”. “width” = ½ kx x = ½ kx2
From x1 to x2:
area is difference between two triangles
F
W = ½ kx22  ½ kx12
these results are for the work done on the spring
x1
x2
Phys211C6 p10
Example: A 600 N individual steps upon a spring scale which is compressed by 1.00 cm under her weight.
What is the force constant?
How much work is done on the spring?
Phys211C6 p11
Work-Energy for a varying Force
dv dv dx dv
note : a 

 v
dt dx dt dx
mv
x2
Wnet   Fdx
x1
x2
x2
x2
x1
x1
x1
 Fdx   ma dx   m
dv
v dx
dx
v
v2
  mv dv area under curve of straight line!
v1
x2
Wnet
1
1
2
2
  Fdx  mv2  mv1  K
2
2
x1
Phys211C6 p12
Example: A .100 kg mass is attached to the end of a spring which has a force constant of 20.0 N/m. The spring is
initially unstretched. The mass is given an initial speed of 1.50 m/s to the right. Find the maximum distance the mass
moves to the right if
the surface is frictionless
the coefficient of kinetic friction between the mass and the surface is .47
Phys211C6 p13
Work and energy along a curve:
small increment of work along a small displacement
dW = F cos f dl = F.dl
Add up increments of work along all such displacements
W   dW
P2
P2
  P2
 F  dl   F cos f dl   F|| dl   F dl||
P2
P1
P1
P1
P1
Phys211C6 p14
Example: A small child of weight w is push slowly by a horizontal force until the swing chain makes an angle qf with
respect to the vertical. Determine the work done by
the tension in the chain (near trivial),
the “push” force (needs calculus), and
qf
the force of gravity
Phys211C6 p15
Power: the rate at which work is done
work done
Average Power 
time interval
W
Joules  J 
Pav =
units : Watts (W ) 
 
t
second  s 
Instantane ous Power  work rate
W dW
P = lim

t 0 t
dt
in terms of velocity
Δs
Pav=F||
 F||vav
Δt
 
P  F v
Phys211C6 p16
Example: A 50.0 kg marathon runner is to run up the stairs of the 443 m tall Sears Tower in Chicago in 15 minutes.
What is the runner’s average power exerted in this effort?
Phys211C6 p17