Download Work

Chapter 7: Work; Energy of a System
Reading assignment: Chapter 7
Homework : CQ1, CQ5, QQ1, QQ2, AE2, 1, 9, 11, 12, 14, 15, 17,
25, 27, 31, 33, 42, 43
Due dates:
Tu/Th, and MWF section: Monday, Feb. 28
Remember HW 6 due Monday, Feb. 21
• The concept of energy (and the conservation of energy – chapter 8)
is one of the most important topics in physics.
• Work
• Kinetic energy
• Energy approach (“system model”) is often simpler than Newton’s
second law (“particle model”).
• Scalar product (dot product); work is scalar product of force and
displacement.
• Hooke’s law, springs
• Power
Work
(as defined by a physicist)
Definition:
The work done on an object by an external force is:
the product of the component of the force in the direction
of the displacement and the magnitude of the displacement.
W  F  d  cos
How much work is done
when just holding up an
object?
W  F  d  cos
W
How much work is done
when the displacement is
perpendicular to the
force?
W  F  d  cos
W
What is the work done when lifting?
(at constant speed)
- By the applied force?
- By the gravitational force?
W  F  d  cos
W
Sign convention:
W positive:
How much work is the strongest man
doing when he lifts a 100 kg boulder by 1 m.
If F and d are parallel.
A.
0J
If energy is transferred into the system.
B.
9800 J
C.
9.8 J
D.
100 J
E.
None of the above
W negative:
If F and d are antiparallel.
If energy is transferred out of the system.
Work is a scalar quantity (not a vector).
Work has units of
Newton·meter (N·m) = the Joule (J)
Born: Dec. 24 1818, Salford, Lancashire, England
Died: Oct. 11, 1889
Joule studied the nature of heat and discovered its relationship
to mechanical work. This led to the theory of conservation of
energy, which led to the development of the first law of
thermodynamics.
(from Wikipedia)
Black board example 7.1
A donkey is pulling a
cart with a force of
magnitude F = 500 N at an
angle of 30º with the
horizontal. The cart is
moving at constant
velocity.
500 N
1. Complete the free body
diagram.
2. Calculate the work done (a) by the donkey (applied force) (b)
gravity, (c) normal force, (d) frictional force, as the cart is pulled
for one mile (1609 m).
Definition of dot product and work
Work is the scalar product (or dot product)
of the force F and the displacement d.
 
W  F  d  F  d  cos
F and d are vectors
W is a scalar quantity
Scalar product between
vector A and B
Definition:
 
A  B  AB  cos
Scalar product is commutative:
   
A B  B  A
Distributive law of multiplication:
  
   
A  (B  C)  A  B  A  C
Scalar Product using unit vectors:
We have the vectors A and B:




A  Ax i  Ay j  Az k
Then:
 
A  B  Ax Bx  Ay By  Az Bz
 
A  A  Ax Ax  Ay Ay  Az Az  A2




B  Bx i  By j  Bz k
Black board example 7.2
A particle moving in the x-y plane undergoes a
displacement d = (2.0i + 3.0j) m as a constant force
F = (5.0i + 2.0j) N acts on the particle. Calculate
(a) The magnitude of the displacement and the force.
(b) The work done by F.
(c) The angle between F and d.
What if the force varies? We have to integrate the force along x
xf
Work done by a varying force:
W   F  x dx
xi
Thus, the work is equal to the area under the F(x) vs. x curve.
Black board example 7.3
i-clicker
A force acting on a particle
varies as shown in the
Figure.
What is the work done on the particle as it is moved from x = 0 to
x = 6 m? (Hint: It is the area under the curve.)
A. 5 J
B. 10 J
C. 20 J
D. 25 J
E. 30 J
Consider a spring
Hooke’s law:
(Force required to stretch or
compress a spring by x):
Fs  k  x
k is the spring constant of a spring.
Stiff springs have a large k value.
Work done by a spring
xi
1
2
2
W  k ( xi  x f )
2
xf
Black board example 7.4
A 0.500 kg mass is hung
from a spring extending
the spring by a distance
x = 0.2 m
(a) What is the spring constant of the spring?
(b) How much work was done on the spring?
(c) How much work was done by gravity on the spring?
A) -0.49J
B) 0J
C) 0.49J
D) 0.98J
E) 24.5J
Work-Kinetic Energy Theorem
Fnet , x  m  a x
v f  vi  2a x x
2
2
1
1
2
2
mv f  mvi  Fx x
2
2
K  K f  Ki  W
Change in the kinetic energy of a particle = net work done on the particle.
The kinetic energy of a particle is:
1
2
K  mv
2
Black board example 7.5
A cannon ball of mass m = 1 kg moves at 500 m/s.
A truck of mass m = 10,000 kg moves at 5 m/s
Which has more kinetic energy?
A. The truck
B. The cannon ball
C. Same
D. Need more information
Work due to friction
If friction is involved in moving objects, work has to be
done against the kinetic frictional force.
This work is:
Wf  fk  d
Black board example 7.6
Angus is pulling a 10,000 kg truck with all his might
(2000N) on a frictionless surface for 10.0 m.
(a) How much work is the man doing?
(b) What is the speed of the truck after 10 m.
(c) What is the speed of the truck after 10 m if there is
friction? (friction coefficient: 0.0153)
Black board example 7.7
A man loads a refrigerator onto a truck using a ramp. He thinks about the Physics of
lifting it straight up versus rolling it up the ramp. (Ignore friction).
Which requires
a larger force?
a longer distance?
more work?
A. Lifting.
A. Lifting
A. Lifting
B. Same.
B. Same
B. Same
C. Rolling
C. Rolling
C. Rolling
Power
dW
Power is the rate at which work is done: P 
dt
Average power (work done per time interval t):
W
P
t
The power can also be expressed as:
 


dW
ds
P
F
 F v
dt
dt
Dot product
The units of power are joule/sec (J/s) = Watt (W)
James Watt (1736-1819); Scottish inventor and engineer whose
improvements to the steam engine were fundamental to the changes
wrought by the Industrial Revolution.
(from Wikipedia)
Black board example 7.8
An elevator having a total mass of
1800 kg moves upward against
a frictional force of 4000N at a
constant speed of 3 m/s.
(a) What is the power delivered by
the motor?