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7.4 Work Done by a Varying Force


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

7.4 Work Done by a Varying Force
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,

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

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

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)

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

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
  • This is called Hookes Law

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

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

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

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

(7.10) (a)
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

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

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

Active Figure 7.10
7.5 Kinetic Energy And the Work-Kinetic Energy
  • 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

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)

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)

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

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.

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

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

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

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

Example 7.12 Work on a Car to Increase Kinetic
  • 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

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

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
  • The Work-Kinetic Energy Theorem can be applied to
    nonisolated systems

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

Active Figure 7.16
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

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
  • Mechanical Waves allow a disturbance to
    propagate through a medium
  • Heat is driven by a temperature difference
    between two regions in space

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

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

Conservation of Energy
  • Energy is conserved
  • This means that energy cannot be created or
  • 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
  • Established symbols Twork W and Theat Q
  • The Work-Kinetic Energy theorem is a special case
    of Conservation of Energy

Material for the Final
  • Examples to Read!!!
  • Example 7.7 (page 195)
  • Example 7.9 (page 201)
  • Example 7.12 (page 204)
  • Homework to be solved in Class!!!
  • Problems 11, 26
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