Work. Energy has the ability to do work; it can move matter. Work may be useful or destructive....

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