# 7.4 Work Done by a Varying Force - PowerPoint PPT Presentation

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## 7.4 Work Done by a Varying Force

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### A moving hammer strikes a nail and comes to rest. ... done on the hammer is negative: Wh = Kh = Fd = 0 mhvh2 0. Example 7.11 Moving Hammer can do Work on ... – PowerPoint PPT presentation

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Title: 7.4 Work Done by a Varying Force

1
7.4 Work Done by a Varying Force
2
Work Done by a Varying Force
• Assume that during a very small displacement, Dx,
F is constant
• For that displacement, W F Dx
• For all of the intervals,

3
Work Done by a Varying Force, cont
• Sum approaches a definite value
• Therefore
• (7.7)
• The work done is equal to the area under the
curve!!

4
Example 7.7 Total Work Done from a Graph (Example
7.4 Text Book)
• The net work done by this force is the area under
the curve
• W Area under the Curve
• W AR AT
• W (B)(h) (B)(h)/2 (4m)(5N) (2m)(5N)/2
• W 20J 5J 25 J

5
Work Done By Multiple Forces
• If more than one force acts on a system and the
system can be modeled as a particle, the total
work done ON the system is the work done by the
net force
• (7.8)

6
Work Done by Multiple Forces, cont.
• If the system cannot be modeled as a particle,
then the total work is equal to the algebraic sum
of the work done by the individual forces

7
Hookes Law
• The force exerted BY the spring is
• Fs kx (7.9)
• x is the position of the block with respect to
the equilibrium position (x 0)
• k is called the spring constant or force constant
and measures the stiffness of the spring (Units
N/m)
• This is called Hookes Law

8
Hookes Law, cont.
• When x is positive (spring is stretched), Fs is
negative
• When x is 0 (at the equilibrium position), Fs is
0
• When x is negative (spring is compressed), Fs is
positive

9
Hookes Law, final
• The force exerted by the spring (Fs ) is always
directed opposite to the displacement from
equilibrium
• Fs is called the restoring force
• If the block is released it will oscillate back
and forth between x and x

10
Work Done by a Spring
• Identify the block as the system
• The work as the block moves from
• xi xmax to xf 0

(7.10)
11
Work Done by a Spring, cont
• The work as the block moves from
• xi 0 to xf xmax

(7.10) (a)
12
Work Done by a Spring, final
• Therefore
• Net Work done by the spring force as the block
moves from xmax to xmax is ZERO!!!!
• For any arbitrary displacement xi to xf

(7.11)
13
Spring with an Applied Force
• Suppose an external agent, Fapp, stretches the
spring
• The applied force is equal and opposite to the
spring force
• Fapp Fs (kx) ? Fapp kx

14
Spring with an Applied Force, final
• Work done by Fapp
• when xi 0 to xf xmax is
• WFapp ½kx2max
• For any arbitrary displacement xi to xf

(7.12)
15
Active Figure 7.10
16
7.5 Kinetic Energy And the Work-Kinetic Energy
Theorem
• Kinetic Energy is the energy of a particle due to
its motion
• K ½ mv2 (7.15)
• K is the kinetic energy
• m is the mass of the particle
• v is the speed of the particle
• Units of K Joules (J)
• 1 J Nm (kgm/s2)m kgm2/s2 kg(m/s)2
• A change in kinetic energy is one possible result
of doing work to transfer energy into a system

17
Kinetic Energy, cont
• Calculating the work
• Knowing that
• F ma mdv/dt m(dv/dt)(dx/dx) ?
• Fdx m(dv/dx)(dx/dt)dx mvdv
• (7.14)

18
Work-Kinetic Energy Theorem
• The Work-Kinetic Energy Principle states
• SW Kf Ki DK (7.16)
• In the case in which work is done on a system and
the only change in the system is in its speed,
the work done by the net force equals the change
in kinetic energy of the system.
• We can also define the kinetic energy
• K ½ mv2 (7.15)

19
Work-Kinetic Energy Theorem, cont
• Summary Net work done by a constant force in
accelerating an object of mass m from v1 to v2
is
• Wnet ½mv22 ½mv12 ? DK
• Net work on an object Change in Kinetic
Energy
• Its been shown for a one-dimension constant
force. However, this is valid in general!!!

20
Work-Kinetic Energy REMARKS!!
• Wnet work done by the net (total) force.
• Wnet is a scalar.
• Wnet can be positive or negative since ?K can be
both or
• K ? ½mv2 is always positive. Mass and v2 are
both positive. (Question 10 Homework)
• Units are Joules for both work kinetic energy.
• The work-kinetic theorem relates work to a
change in speed of an object, not to a change in
its velocity.

21
Example 7.8 Question 14
• (a). Ki ? ½m v2 0
• K depends on v2 0 m gt 0
• If v ?2v
• Kf ½m (2v)2 4(½mv2 ) 4Ki
• Then Doubling the speed makes an objects
kinetic energy four times larger
• (b). If SW 0 ? v must be the same at the final
point as it was at the initial point

22
Example 7.9 Work-Kinetic Energy Theorem (Example
7.7 Text Book)
• m 6.0kg first at rest is pulled to the right
with a force F 12N (frictionless).
• Find v after m moves 3.0m
• Solution
• The normal and gravitational forces do no work
since they are perpendicular to the direction of
the displacement
• W F Dx (12)(3)J 36J
• W DK ½ mvf2 0 ?
• 36J ½(6.0kg)vf2 (3kg)vf2
• Vf (36J/3kg)½ 3.5m/s

23
Example 7.10 Work to Stop a Car
• Wnet Fdcos180 Fd Fd
• Wnet ?K ½mv22 ½mv12 Fd ?
• -Fd 0 ½m v12 ? d ? v12
• If the cars initial speed doubled, the stopping
distance is 4 times greater. Then d 80 m

24
Example 7.11 Moving Hammer can do Work on Nail
• A moving hammer strikes a nail and comes to rest.
The hammer exerts a force F on the nail, the nail
exerts a force F on the hammer (Newton's 3rd
Law) m
• Work done on the nail is positive
• Wn ?Kn Fd ½mnvn2 0 gt 0
• Work done on the hammer is negative
• Wh ?Kh Fd 0 ½mhvh2 lt 0

25
Example 7.12 Work on a Car to Increase Kinetic
Energy
• Find Wnet to accelerate the 1000 kg car.
• Wnet ?K K2 K1 ½m v22 ½m v12
• Wnet ½(103kg)(30m/s)2 ½(103kg)(20m/s)2 ?
• Wnet 450,000J 200,000J 2.50x105J

26
Example 7.13 Work and Kinetic Energy on a Baseball
• A 145-g baseball is thrown so that acquires a
speed of 25m/s. ( Remember v1 0)
• Find (a). Its K.
• (b). Wnet on the ball by the pitcher.
• (a). K ? ½mv2 ½(0.145kg)(25m/s)2 ?
• K ? 45.0 J
• (b). Wnet ?K K2 K1 45.0J 0J ?
• Wnet 45.0 J

27
7.6 Non-isolated System
• A nonisolated system is one that interacts with
or is influenced by its environment
• An isolated system would not interact with its
environment
• The Work-Kinetic Energy Theorem can be applied to
nonisolated systems

28
Internal Energy
• The energy associated with an objects
temperature is called its internal energy, Eint
• In this example, the surface is the system
• The friction does work and increases the internal
energy of the surface

29
Active Figure 7.16
30
Potential Energy
• Potential energy is energy related to the
configuration of a system in which the components
of the system interact by forces
• Examples include
• elastic potential energy stored in a spring
• gravitational potential energy
• electrical potential energy

31
Ways to Transfer Energy Into or Out of A System
• Work transfers by applying a force and causing
a displacement of the point of application of the
force
• Mechanical Waves allow a disturbance to
propagate through a medium
• Heat is driven by a temperature difference
between two regions in space

32
More Ways to Transfer Energy Into or Out of A
System
• Matter Transfer matter physically crosses the
boundary of the system, carrying energy with it
• Electrical Transmission transfer is by electric
current
• Electromagnetic Radiation energy is transferred
by electromagnetic waves

33
Examples of Ways to Transfer Energy
• d) Matter transfer
• e) Electrical Transmission
• a) Work
• b) Mechanical Waves
• c) Heat

34
Conservation of Energy
• Energy is conserved
• This means that energy cannot be created or
destroyed
• If the total amount of energy in a system
changes, it can only be due to the fact that
energy has crossed the boundary of the system by
some method of energy transfer
• Mathematically SEsystem ST (7.17)
• Esystem is the total energy of the system
• T is the energy transferred across the system
boundary
• Established symbols Twork W and Theat Q
• The Work-Kinetic Energy theorem is a special case
of Conservation of Energy

35
Material for the Final