Ch. 8: Conservation of Energy. Energy Review. Kinetic Energy : K = ( ½) mv 2 Associated with movement of members of a system Potential Energy Determined by the configuration of the system (location of the masses in space). Gravitational PE : U g = mgy

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Ch. 8: Conservation of Energy

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Determined by the configuration of the system(location of the masses in space).

Gravitational PE: Ug = mgy

Elastic PE(ideal spring): Ue = (½)kx2

Internal Energy

Related to the temperature of the system

Types of Systems

Nonisolated systems

Energy can cross the system boundary in a variety of ways

Total energy of the system changes

Isolated systems

Energy does not cross the boundary of the system

The total energy of the system is a constant

(CONSERVED!)

Ways to Transfer Energy Into or Out of A System

Work– transfers energy by applying a force & causing a displacement of the point of application of the force

Mechanical Waves– allows 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 System

Matter Transfer– matter physically crosses the boundary of the system, carrying energy with it

Electrical Transmission– transfer by electric current

Electromagnetic Radiation– energy is transferred by electromagnetic waves

Examples of Energy Transfer

a) Work b) Mechanical Waves c) Heat

d) Matter Transfer e) Electrical Transmission

f) Electromagnetic Radiation

Conservation of Energy

TOTAL Energy is conserved

“Total”means the sum of all possible kinds of energy.

“Conserved”means that it remains constant in any process. In other words, Total Energy can be neither created nor destroyed, but only can be transformed from one form to another or transferred across a system boundary.

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

The total change in the energy of a system = the

total energy transferred across a system boundary.

DEsystem = ST

Esystem= total energy of the system

T = energy transferred across the system boundary

Established symbols: Twork = W & Theat = Q

Others just use subscripts

The Work-Kinetic Energy Theorem is a special case of Conservation of Energy

A full expansion of the above equation gives:

D K + D U + DEint = W + Q + TMW + TMT + TET + TER

Isolated System

For an isolated system,ΔEmech = 0

Remember Emech = K + U

This is conservation of energy for an isolated system with no nonconservative forces acting

If nonconservative forces are acting, some energy is transformed into internal energy

Conservation of Energy becomes: DEsystem = 0

Esystem is all kinetic, potential, & internal energies

The most general statement of the isolated system model

For an isolated system, the changes in energy can be written out and rearranged

Kf + Uf = Ki + Ui

This applies only to a system in which only conservative forces act!

Example 8.1 – Free Fall

Calculate the speed of the ball at a distance y above the ground

Use energy

System is isolated so the only force is gravitational which is conservative

So, we can use conservation of mechanical energy!

Conservation of Mechanical Energy

Kf + Ugf = Ki + Ugi

Ki = 0, the ball is dropped

Solve for vf:

The equation for vfis consistent with the results obtained from kinematics

Example 8.2 – Grand Entrance

An actor, mass mactor = 65 kg, in a play is to “fly” down to stage during performance. Harness attached by steel cable, over 2 frictionless pulleys, to sandbag, mass mbag = 130 kg, as in figure. Need length R = 3 m of cable between nearest pulley & actor so pulley can be hidden behind stage. For this to work, sandbag can never lift above floor as actor swings to floor. Let initial angle cable makes with vertical beθ. Calculate the maximum value θcan have such that sandbag lifts off floor.

Free Body Diagrams

Step 1: To find actor’s speed at bottom, let yi = initial height above floor & use

Conservationof Mechanical Energy

Ki + Ui = Kf + Uf

or 0 + mactorgyi = (½)mactor(vf)2 + 0(1)

mass cancels. From diagram,

yi = R(1 – cos θ)

So, (1) becomes:(vf)2 = 2gR(1 – cosθ) (2)

Step 2: Use N’s 2nd Law for actor at bottom of path (T = cable tension).

Actor Sandbag

at bottom

Actor:∑Fy = T – mactorg = mactor[(vf)2/R]

or T = mactorg + mactor[(vf)2/R] (3)

Step 3: Want sandbag to not move. N’s 2nd Law for sandbag: