1. Fundamentals of Physics School of Physical Science and Technology
Mechanics
(Bilingual Teaching)
张昆实
School of Physical Science and Technology
Yangtze University

2. Chapter 7 Kinetic Energy and work
7-3 Work and Kinetic Energy
7-4 Work Done by a Gravitational Force
7-5 Work Done by a Spring Force
7-6 Work Done by a General Variable
Force
7-7 Power

3. 7-1 Energy
★Newton’s laws of motion allow us to analyze many kinds of motion. However, the analysis is often complicated, requiring details about the motion that we simply do not know.
There is another technique for analyzing motion , which involves energy.
Kinetic energy is energy associated with the state of motion of an object.
SI unit: joule(J),
(7-1)
(7-2)

4. 7-2 Work
work is energy transferred to or from an object by means of a force acting on the object. Energy transferred to the object is positive work, and energy transferred from the object is negative work.
You acceletate an object its kinatic energy increaced the work done by your force is positive;
You deceletate an object its kinatic energy decreaced the work done by your force is negative;

5. 7-3 Work and Kinetic Energy
Finding an Expression for Work
A bead can slide along a frictionless horizontal wire ( axis), A constant force
directed at an angle to the wire, accelerates the bead along the wire.
(7-3)
Bead
wire
start
end
(7-4)
(7-5)
(7-6)

6. 7-3 Work and Kinetic Energy
To calculate the work done on an object
by a force during a displacement, we use
only the force component along the object’s
displacement. The force component perpen-
dicular to the displacement does zero work.
Since
General form
(7-7)
Scalar (dot) product
(7-8)
Cautions: (1) constant force (magnitude and direction) (2) particle-like object.

7. 7-3 Work and Kinetic Energy
Signs for work
positive work
negative work
does’t do work
A force does positive work when it has a vector component in the same direction as the displacement, and it does negative work when it has a vector component in the opposite direction. It does zero work when it has no such vector component.

8. 7-3 Work and Kinetic Energy
Units for work
The unit for work is the same as the unit for energy
(7-9)
net work
When two or more forces act on an object, their net work is the sum of the individual workes by the forces, which is also equal to the work that would be done on the object by the net force of those forces.

9. 7-3 Work and Kinetic Energy
Work-kinetic Energy Theorem
(7-5)
(7-10)
change in the kinetic
energy of a particle
net work done on
the particle =
(7-11)
kinetic energy
Before the net work
kinetic energy after
The net work is done =
“Work-kinetic Energy Theorem”
P120 bottom; Problem 7-2 +
the net
Work done

10. 7-4 Work Done by a Gravitational Force
A tomato is thrown upward, the work done by the
grivatational force :
(7-12) d F g
In the rising prosses
(7-13) d F g
minus sign: the grivatational force transfers energy (mgd) from the object’s kinetic energy, consistant with the slowing of the object.
In the falling down prosses
(7-14)
plus sign: the grivatational force transfers energy (mgd) to the object’s kinetic energy, consistant with the speeding up of the object.

11. 7-4 Work Done by a Gravitational Force
Work done in lifting and lowering an object
Lifting an object d F g
Displacement: upward
Lifting force: positive work;
transfers energy to the object;
Gravitational force: negative
work; transfes energy from
the object. d F g
Lowering an object. Displacement: downward
Lifting force: negative work; transfers energy
from the object;
Gravitational force: positive work; transfes
energy to the object.

12. 7-4 Work Done by a Gravitational Force
Work done in lifting and lowering an object
The change in the kinetic
energy due to these two
energy transfers is d F g d F g
(7-15) If
(7-16)
(7-17)
(Work in lifting and lowering; )
Up:
Down:
The angle between and .

13. 7-5 Work Done by a Spring Force
X axis along the spring
The spring force
relaxed state
Fig(a): A block is attached to a
spring in equilibrium (neither
compessed nor stretched).
stretched
Fig(b): when stretched to right
The spring pulls the block to the
Left (restoring force)
compressed
Fig(c): when compressed to left
The spring pulls the block to the
right (restoring force)
The spring force
(Hooke’s law) (7-20)
(variable force)
K: spring constant
(Hooke’s law) (7-21)

14. 7-5 Work Done by a Spring Force
The work done by a spring force
Simplifying assumptions:springmassless; ideal
spring(obeys Hooke’s law); contact frictionless.
Calculus metheod
(7-22)
Sumintegration x F dx dW xi xf O
segments
(7-23)
(7-24)
(7-25)
( work by a spring force )

15. 7-5 Work Done by a Spring Force
The work done by a spring force
Work is positive if the block ends up closer to
the relaxed position (x=0) than it was initially. It is
negative if the block ends up father away from x=0.
It is zero if the block ends up at the same distance
from x=0.
(7-25)
If and then
( work by a spring force )
(7-26)

16. 7-5 Work Done by a Spring Force
The work done by an applied force
Suppose we keep applying a force on the
block, our force does work , the spring force
does work on the block.
The change in
the kinetic energy
of the block
(7-27)
If the block is stationary before and after the
displacement, then
(7-28)

17. 7-5 Work Done by a Spring Force
The work done by an applied force
(7-28)
If a block that is attached to a spring is stationary
before and after a displacement, then the work
done on it by the applied force displacing it is the
negative of the work done on it by the spring
force.
Sample problem 7-8 : P128

18. 7-6 Work Done by a General Variable Force
One-dimensional Analysis
the same method as the calculation of the
work done by a spring force (calculus).
(7-29)
(7-30)
(7-31)
(7-32)
( Work:
variable force )

19. 7-6 Work Done by a General Variable Force
Three-dimensional Analysis
A three-dimensional force acts on a body
(7-33)
Simplifications: only depends on and so on
Incremental displacement
(7-34)
The work done by during is
(7-35)
(7-36)

20. 7-6 Work Done by a General Variable Force
work-kinetic energy theorem with a variable force Let us prove:
(7-32)
( Work: variable force )
(7-37)
(7-38)
(7-40)
From the “chain rule” of calculus
(7-39)

21. 7-6 Work Done by a General Variable Force
work-kinetic energy theorem with a variable force Let us prove:
(7-37)
(7-40)
Substituting Eq into Eq. 7-37:
(7-41)
work-kinetic energy theorem

22. 7-7 Power Power : The power due to a force is the rate
at which that force does work on an object.
If the force does work during a time interval
, the average power due to the force over
that time interval is
(7-42)
Instantaneous power is the
instantaneous rate of doing
work
(7-43)
If the direction of a force is
at an angle to the direction
of travel of the object, the
Instantaneous power is
(7-44)