Download Work and Energy

Work and Energy
• Work (W) is done on an object by an force
when the object moves through a distance
(displacement). Since Force and
displacement are vectors, work has to be a
scalar.
We use the
r scalar product:
r
W = F s = Fs cos θ [Joule J = Nm]
Energy [J] is defined as the ability to do work.
Or “Work is the Energy supplied to on object
to make it move”.
Work is an energy transfer by the application of a force.
For work to be done there must be a nonzero displacement.
How much work is 1 Joule? – Let’s compare
Annual U.S. energy use
8 x 1019
Mt. St. Helens eruption
1018
Burning one gallon of gas
108
Human food intake/day
Melting an ice cube
107
104
Lighting a 100-W bulb for 1 6000
minute
Heartbeat
0.5
Turning page of a book
10–3
Hop of a flea
10–7
Breaking a bond in DNA
10–20
The Law of
Conservation of Energy
The total energy of the Universe is unchanged
by any physical process.
The three kinds of energy are: kinetic
energy, potential energy, and rest energy.
Energy may be converted from one form to
another or transferred between bodies.
Work done by a force – an example
Consider you are pushing a box, then:
ÆMore work is required to exert a greater force for a finite
distance
Æ More work is required to exert a finite force over a greater
distance
ÆTherefore: “Work equals force times displacement”
rr
W = F s = Fs cos θ [Joule J = Nm]
θ = 0 ; cos0 = 1 (angle between the two vectors)
⇒W = F s
Work W can be………….
+W
0
-W
Example 1: A person pulls a suitcase If the person
would pull
horizontal little
force would be
necessary to do
the same
work!!!
Example 2: Work done by gravity
Force needed to lift up
the box F1 = mg,
W1=Fs=(mg)h =mgh
cos(0)=mgh
Work done by gravity
Wg= mgh cos(180)= -mgh
It is only the force in the direction of
the displacement that does work.
Free Body Diagram for the box
F
θ
Δrx
y
Δrx
N
θ
w
x
F
WF = Fx Δrx = (F cos θ )Δx
The work done by the force N is:
WN = 0
The normal force is perpendicular to the displacement.
The work done by gravity (w) is: Wg = 0
The force of gravity is perpendicular to the
displacement.
Wnet = WF + WN + Wg
= (F cos θ )Δx + 0 + 0
= (F cos θ )Δx
Example: A ball is tossed straight up. What
is the work done by the force of gravity on the
ball as it rises?
y
Δr
FBD for
rising
ball:
x
w
r r
Wg = WΔy cos 180°
= − mgΔy
Conceptual Checkpoint
Which way is more work?
Total Work??
When more then a force acts on an object the
total work……………….
Force F1 (e.g. Friction) does work W1, Force F2
does work W2, etc.
Wtotal = W1 + W2 + W3 + ...... = ∑ W i
OR
Calculate the net force (or total force)
r r
W = Ftotal s = Ftotal s cos θ
Example: A box of mass m is towed up a
frictionless incline at constant speed. The
applied force F is parallel to the incline. What
is the net work done on the box?
y
F
N
F
x
θ
θ
w
Apply Newton’s
2nd Law:
Fx = F − w sin θ = 0
∑
∑F
y
= N − w cos θ = 0
The magnitude of F is:
F = mgsinθ
If the box travels along the ramp a distance of
Δx the work by the force F is
WF = FΔx cos0° = mgΔx sinθ
The work by gravity is
Wg = wΔx cos(θ + 90°) = − mgΔx sin θ
Example continued:
The work by the normal force is:
WN = NΔx cos 90° = 0
The net work done on the box is:
Wnet = WF + Wg + WN
= mgΔx sin θ − mgΔx sin θ + 0
=0
Graphical Representation of
Work
Plot force vs position and the area
under the curve represents the Work
W= F d
(The area of a rectangle with length
a and with b: Area = ab)
What if Force isn´t constant?
Split the curve in several
intervals/ rectangles and add
them up OR with calculus
Are work and speed are related?
Of course: When the total work done on a
object is:
Positive, its speed increases (W >0, vf>vi)
Negative, ist speed decreases (W<0, vf<vi)
Conclusion: There is a connection between
work and change in speed
Work-Energy Theorem
Kinetic Energy (K or Ek) is energy associated
with the state of motion of an object. The
faster the object moves, the greater is its
kinetic energy. When the object is stationary,
its kinetic energy is zero.
Kinetic Energy:
KE=Ekin= ½ mv2 linear
KE=Erot= ½ Jω2 rotational
• Net work that is done on body (by
net force) equals change in kinetic
energy
• Wnet = ΔKE=Ekin = ½ mv22 - ½ mv12
• all moving bodies (v ≠ 0 if linear, ω
≠ 0 if rotational)
If we see something with
mass moving, we know is
has kinetic energy!!!!!
Remember Springs?
What about the work done on a spring?
r
r
Hooke´s Law F = −k Δx (Hooke´s Law)
F is the magnitude of the
force exerted by the free
end of the spring, x is the
measured stretch of the
spring, and k is the spring
constant (a constant of
proportionality; its units
are N/m).
Remember:
F~x
Area: W=Fd
W=(kx)x/2
W= kx2/2
Power
Power is a measure how quickly work is
done
P=W/t or P= Fs/t=F(s/t)=Fv
SI Unit J/s= W (watt)
Other common unit
Horsepower hp: 1hp = 746W
Source
Approximate power
(W)
Hoover Dam
1.34 x 109
Car moving at 40 mph
7 x 104
Home stove
1.2 x 104
Sunlight falling on one square
meter
1380
Refrigerator
615
Television
200
Person walking up stairs
150
Human brain
20
Example
You want to accelerate your car from
13.4m/s (30mph) to 17.9m/s (40mph).
What is the minimum power your car
need to do this procedure?
Calculate the power output of a 1.8-g
beetle as it walks up a window pane at 2.3
cm/s. The beetle walks on a path that is at
25° to the vertical, as illustrated in Figure 7–
18.
Two springs, with force constants and are
connected in parallel, as shown in Figure .
How much work is needed to stretch this
system a distance x?