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WorkWork
• Energy has the ability to do work; it can move matter.
• Work may be useful or destructive.
IntroductionIntroduction
• Work is defined as the product of the force component that is parallel to an object’s motion and the distance that the object is moved.
WorkWork
• Mechanical work is done by a force on a system.
• W ≡ Fd cos θ • Work is done by a force F
through a displacement d.
WorkWork
• W ≡ Fd cos θ • θ is the smallest angle
(≤180°) between the force and displacement vectors when they are placed tail-to-tail.
WorkWork
• W ≡ Fd cos θ • Work is a scalar.• Work can be positive,
negative, or zero, depending on the angle θ.
WorkWork
• θ < 90°: Work is positive. • 90° < θ < 180°: Work is
negative.• θ = 90°: Work is zero.• Units: Joules (J)• 1 J ≡ 1 N × 1 m
WorkWork
• This is the unit used for both work and energy.
• It must not be confused with the N · m, used for torque; joules are never used for torque.
Joule (J)Joule (J)
• Any kind of force can do work.
• No work is done if no object moves (since d = 0).
• Example 9-1: Why is the angle 0°?
Calculating WorkCalculating Work
• Force-distance graph• The area “under the curve”
of a force-distance graph approximates the work done on a system by the force.
Determining Work Graphically
Determining Work Graphically
• For a constant force, the “area” is rectangular and simple to calculate.
• Be sure to select the appropriate units for your result (typically N × m = J).
Determining Work Graphically
Determining Work Graphically
• An external force to stretch a spring is an example of a varying force.
Determining Work Graphically
Determining Work Graphically
• Equilibrium position: the normal or relaxed length of the spring
• Fex: an external force• d = Δx = x2 – x1 • x1 is equilibrium position.
SpringsSprings
• Fex = kd• k is a proportionality
constant called the spring constant.
• Work done on a spring by an external force is positive.
Hooke’s LawHooke’s Law
• no mass• value of k is truly constant
throughout its range of displacements
• exemplifies a Hooke’s Law force
Ideal SpringsIdeal Springs
• Wex = ½k(Δx)².• This is consistent with its
force-distance graph.
Ideal SpringsIdeal SpringsHow much work is done to
stretch a spring from its equilibrium position by Δx?
• How much work is done by the spring?
• According to Newton’s 3rd Law:
Ideal SpringsIdeal Springs
Fs = -Fex
Fs = -kd
• Work done by the spring is negative because the displacement is opposite the spring’s force.
• This is true whether the spring is stretched or compressed.
Ideal SpringsIdeal Springs
• The force-distance graph of the work done by the spring is below the x-axis.
• In Example 9-3, the two forces are opposites of each other.
Ideal SpringsIdeal Springs
• Defined: the time-rate of work done on a system
• Average power: the work accomplished during a time interval divided by the time interval
PowerPower
• Average power:
PowerPower
P =WΔt
P = Fv cos θ
Fd cos θΔt
=
• Power is a scalar quantity.
• The unit of power is the Watt (W).
• 1 W = 1 J/s
PowerPower
EnergyEnergy
Kinetic EnergyKinetic Energy• mechanical energy
associated with motion• positive scalar quantity
measured in joules
Work-Energy TheoremWork-Energy Theorem
• states that the total energy done on a system by all the external forces acting on it is equal to the change in the system’s kinetic energy
Wtotal = ΔK = K2 – K1
Kinetic EnergyKinetic Energy• can be defined as:
K = ½mv²
• Note that kinetic energy must mathematically be a positive quantity.
Potential EnergyPotential Energy• energy due to an object’s
condition or position relative to some reference point assumed to have zero potential energy
• measured in joules
Potential EnergyPotential Energy• takes various forms:
• gravitational• elastic• electrical
• results from work done against a force
Conservative ForcesConservative Forces
• One of the following things must be true:• The net work done by the
force on a system as it moves between any two points is independent of the path followed by the system.
Conservative ForcesConservative Forces
• One of the following things must be true:• The net work done by the
force on a system that follows a closed path (begins and ends at the same point in space) is zero.
Conservative ForcesConservative Forces
• Examples of conservative forces:• gravitational force• any central force• any Hooke’s law force
Conservative ForcesConservative Forces
• energy expended when doing work against them is stored as potential energy and can be regained as kinetic energy
• if not, it is called a nonconservative force
Conservative ForcesConservative Forces
• Examples of nonconservative forces:• kinetic frictional force• internal resistance forces• fluid drag
Conservative ForcesConservative Forces
• When work is done against nonconservative forces, the energy is not stored as potential energy but is converted into other forms of mechanically unusuable energy.
• work required to move masses apart against the force of gravity
• near earth’s surface, work done lifting against gravity:
Gravitational Potential Energy
Gravitational Potential Energy
Wlift = |mg|Δh
• Work must be done against a force in order to increase the potential energy of a system with respect to that force.
Gravitational Potential Energy
Gravitational Potential Energy
Wg = -ΔUg
• requires a well-defined reference point for height
• The Ug = |mg|h formula is still in effect, where h is the distance the object can fall.
Relative Potential Energy
Relative Potential Energy
• defined as the potential energy per kilogram at a specified distance r from a zero reference distance
• near the earth’s surface:
Gravitational PotentialGravitational Potential
Ug(r) = |g|h
• for any object of mass m at any distance r from mass M:
Gravitational PotentialGravitational Potential
The units are J/kg
Ug(r) = -GMr
• Gravitational potential will always be negative, but when the objects are moved farther apart, it is a positive change in potential energy.
• Gravity can do work!
Gravitational PotentialGravitational Potential
• Work must be done against a force in order to increase the potential energy of a system with respect to that force.
Elastic Potential Energy
Elastic Potential Energy
• ΔUs = change in spring’s potential energy
Elastic Potential Energy
Elastic Potential Energy
ΔUs = ½k(d2x2 – d1x
2)
Total Mechanical
Energy
Total Mechanical
Energy
All mechanical work on a system can be subdivided
into the work done by conservative forces (Wcf) and
the work done by nonconservative forces
(Wncf).
Wtotal = Wcf + Wncf = ΔK
The work done by nonconservative forces is equal to the change of the
system’s total energy.Total mechanical energy is
the sum of a system’s kinetic and potential energies.
E ≡ K + U
We can also say that the work accomplished by all
nonconservative forces on a system during a certain process is equal to the
change of total mechanical energy of a system.
Wncf = ΔE
If mechanical energy is conserved, we obtain:
ΔK = -ΔU
K1 + U1 = K2 + U2
If mechanical energy is not conserved, we obtain:
K1 + U1 = K2 + U2 + Wncf