Work and Energy

WorkO Work is defined as the force parallel to the

direction of motion times the distance.

W = F(parallel) d = F d cos θ

O If the direction of the force is opposite of the motion, the work is considered to be negative.

O Unit of measurement if the Joule.1 J = 1 N m = 1 kg m2/s2

Examples of work.

(a) The work done by the force F on this lawn mower is Fd cos θ . Note that F cos θ is the component of the force in the direction of motion.

(b) A person holding a briefcase does no work on it, because there is no motion. No energy is transferred to or from the briefcase.

(c) The person moving the briefcase horizontally at a constant speed does no work on it, and transfers no energy to it.

(d) Work is done on the briefcase by carrying it up stairs at constant speed, because there is necessarily a component of force F in the direction of the motion. Energy is transferred to the briefcase and could in turn be used to do work.

(e) When the briefcase is lowered, energy is transferred out of the briefcase and into an electric generator. Here the work done on the briefcase by the generator is negative, removing energy from the briefcase, because F and d are in opposite directions.

Example Work Problem

A rope is used to pull a metal box 18.0 m across the floor. The rope is held at a 48º angle with the floor, and a force of 540 N is used. How much work does the force on the rope do?

O Gravitational Potential EnergyO due to the height of an object

with massO Elastic Potential Energy

O The energy in springy thingsO Kinetic Energy

O due to motionO Energy is measured in Joules (J)

Forms of Mechanical Energy

Kinetic EnergyThe energy of an object that is due to the object’s motion is called kinetic energy.

Kinetic energy depends on speed and mass.

2

2

1

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kinetic energy = mass speed2

KE mv

Kinetic Energy – It’s moving

Kinetic EnergyO Kinetic energy can be translational or

rotational.O Translational means the object shifts

position from one point in space to anotherO Example: You walk from one class to

another.O Rotational mean some type of

circular motion is involved.O Example: wheels

Potential Energy

• Potential Energy is stored energy due to position, shape, or condition, or configuration (arrangement).

• Some forms of potential energy are NOT mechanical.

Example: electrical potential energy

Gravitational PE (UG)

• Gravitational potential energy is the potential energy stored when work is done against gravity.

• UG depends on height from a zero level.

O UG = mgh

O gravitational PE = mass free-fall acceleration height

PE in a SpringO Energy is stored when a spring is either

compressed or stretched.O The potential energy of a spring

depends of displacement of the spring and the spring constant, k.

Us = ½ k x2

O The spring constant is a factor that represents the stiffness of the spring. It can be calculated with Hooke’s Law.

Hooke’s LawO The force to

displace a spring is equal to a constant multiplied by the displacement.

O

Work-Energy TheoremO The work done on a system transfers

energy into the system. O Net work is the work done by an

external net force.O The net work done is equal to the

change in kinetic energy.

Wnet = ½ m vf2 – ½ m vi

2

Conservative vs. Nonconservative Forces

Conservative Nonconservative

O The path taken does not matter.

O The amount of energy used depends only upon the starting and ending points.

O Energy is added or removed by the system.

O Friction is an example of a nonconservative force.

Figure 7.14

O The amount of the happy face erased depends on the path taken by the eraser between points A and B, as does the work done against friction. Less work is done and less of the face is erased for the path in (a) than for the path in (b). The force here is friction, and most of the work goes into thermal energy that subsequently leaves the system (the happy face plus the eraser). The energy expended cannot be fully recovered.

Conservation of Mechanical Energy

O Mechanical Energy is conserved when only conservative forces are involved. (No friction)

O In this special situation, the following relationships can be used to solve problems:

K + Ug = constant

ORKbefore + Ug before = Kafter + Ug after

Conservation of Energy with Conservative Forces

O A toy car is pushed by a compressed spring and coasts up a slope. Assuming negligible friction, the potential energy in the spring is first completely converted to kinetic energy, and then to a combination of kinetic and gravitational potential energy as the car rises. The details of the path are unimportant because all forces are conservative—the car would have the same final speed if it took the alternate path shown.

1. Identify the relevant points in the problemAt one point is you know enough to calculate the energy, at the other point you want to know something.

2. Write down the conservation of energy formula

3. Write down the forms of energy at each point.

4. Substitute in the formulas for each type of energy.

5. Substitute numbers with units.6. Solve for the answer – check units.

Solving conservation of energy problems

Example Conservation Problem

O Maria is changing the tire of her car on a steep hill 15.0 m high. She trips and drops the 10.0-kg spare tire, which rolls down the hill with an initial speed of 2.00 m/s. What is the speed of the tire at the top of the next hill, which is 5.00 m high? (Ignore the effects of rotational KE and friction.)

O The total amount of energy in a system is constant in a closed, isolated system.

O The energy can change form.O Some of the energy may be work

done by nonconservative forces (Wnc).

O Kbefore + Ug before + Wnc = Kafter + Ug after

Conservation of Energy

Conservation of EnergyO Other types of energy may need to

be considered in analyzing conservation of energy problems.

O Other types of energy include: electrical, chemical, radiant, nuclear, thermal.

Ki + Ug i + Wnc + OEi = Kf + Ug f + OEf

PowerO Power is the rate at which work is

done.O Power = work done/time intervalO P = W/tO Units: J/s = Watt = W

O Energy is equal to power multiplied by time. (E = Pt)

Example Power Problem

A box that weighs 525 N is lifted a distance of 15.0 m straight up by a rope. The job is done in 8.0 seconds. What power is developed in watts and kilowatts?