Work, Energy and Power
Work in 1-D
• Work in 1-D with a constant force is
W  Fx x
• Units - N·m or J (joules)
• Ex. While mowing the lawn, I press
directly into the handle with a force of 80N.
The handle makes an angle of 40° wrt.
horizontal. How much work is done on the
mower after it is pushed 25m?
Work of Friction
• On the previous problem, the lawn exerts a
frictional force of 50N. How much work
does it do on the lawnmower?
q
F
Work in 3-D
• Work in 3-D with a constant force is
W  Fr cosq
• Only want component of force in the
direction of the displacement.
• Work is a scalar.
F q
• Scalar Product (Dot Product)
 
W  F  r  F  r cosq
r
Dot Product
• With unit vectors
iˆ iˆ  1
iˆ  ˆj  0

ˆj  ˆj  1
iˆ  kˆ  0

kˆ  kˆ  1
ˆj  kˆ  0
W  Fxiˆ  Fy ˆj  Fz kˆ  rxiˆ  ry ˆj  rz kˆ
• Therefore,
W  Fx rx  Fy ry  Fz rz

Kite Flying
• During a strong gust of
wind a kite rises vertically
r
2.5m and moves to the
west 5.4 m. If the force
Fwind
from the gust of wind is
q
15N directed 20° upward
from due east, what work
was done on the kite by
the gust of wind?
1-D varying Force
• Work inx 1-D with a varying force is
W   F ( x)dx
x
x0
• Work on a Spring
F
Assume x0 = 0
Force you exert on the spring.
F  kx
Work you do on the spring
x
W   kxdx  kx
0
1
2
2
Note: During the
same motion the
spring does
negative work.
Compressed Spring
• How much work is done on a spring with a
spring constant of 35 N/m if it is
compressed 0.50m?
x
F
3-D Varying Force
• Most general form of the definition of work
r
W  
r0
 
F dr
• This is called a line integral.
• Ex. How much work is done on a 5.0kg
block by gravity as it slides down a
1m
parabola from x=1m to x=0m?
• For the parabola

yx
2
dr  dx iˆ  2 xdx ˆj
-1m
0m
1m
Kinetic Energy
• Kinetic Energy - Energy of Motion
Wnet
dv
  Fnet dx   m dx   mvdv
dt
v
f
Wnet   mvdv  mv
1
2
vi
K  mv
1
2
2 vf
vi
 mv  mv
2
• Work-Energy Theorem
Wnet  K f  K i  K
1
2
2
f
1
2
2
i
Braking Car
• A 600 kg car is traveling at 24 m/s. The
driver slams on the brakes and reduces the
car’s speed to 13 m/s before hitting a stalled
car. What work was done on the car while
braking and what work was done on the car
by the collision?
v
Power
• Power - Rate at which
energy is used or produced.
– Average Power
E
P
t
– Instantaneous Power
dE
P
dt
– Units - watt (W)
At 157 kg
• Ex. How much power does and lifting
Alexeyev have if lifting the 246 kg
weight 2m in 3 s?
Energy & Energy
Transfer
• System/Environment
– Kinetic energy
– Internal energy
– Potential energy
• Energy transfer
–
–
–
–
–
–
+Work (E → S)
Mechanical Waves
Heat
Matter transfer
Electrical transmission
EM radiation
System
Environment