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Energy

- Analyzing the motion of an object can often get

to be very complicated and tedious requiring

detailed knowledge of the path, frictional

forces, etc. - There has to be an easier way
- It turns out that there is it is done by

analyzing the objects energy.

Energy

- The something that enables an object to do work

is energy. - Energy is measured in Joules (J).
- Forms of Energy
- Mechanical (kinetic and potential)
- Thermal (heat)
- Electromagnetic (light)
- Nuclear
- Chemical

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Mechanical Energy

- Mechanical energy is the form of energy due to

the position or the movement of a mass.

Kinetic Energy

- Kinetic energy is the energy of motion it is

associated with the state of motion of an object. - The faster an object is moving, the greater its

kinetic energy an object at rest has zero

kinetic energy. - For an object of mass m, we will define kinetic

energy as - The SI unit of kinetic energy is the Joule (J).

Kinetic Energy

- If we do positive work on an object by pushing on

it with some force, we can increase the objects

kinetic energy (and thereby increasing its

speed). - We can account for the change in kinetic energy

by saying that the force transferred energy from

you to the object. - If we do negative work on an object by pushing on

it with some force in the direction opposite to

the direction of motion, we can decrease the

objects kinetic energy (and decrease its

speed). - We can account for the change in kinetic energy

by saying that the force transferred energy from

the object to you.

Kinetic Energy

- Whenever we have a transfer of energy via a

force, we say that work is done on the object by

the force. - Work W is energy transferred to or from an object

by means of a force acting on that object. - Energy transferred to the object is positive

work. - Energy transferred from the object is negative

work. - Work is nothing more than transferred energy it

therefore has the same units as energy and is

also a scalar quantity. - Note that nothing material is transferred.
- Think of it like the balance in two bank

accounts when money is transferred the number

for one account goes down by some amount and the

number for the other account goes up by the same

amount.

Work-Energy Theorem

- Suppose we have an bead which is constrained to

move only along the length of a frictionless

wire. - We then supply a constant force F on the bead at

some angle ? to the wire. - Because the force is constant, we know that the

acceleration will also be constant.

Work-Energy Theorem

- But because of the constraint, only the force in

the x direction matters, thus Fx max where m

is the beads mass. - We can relate the beads velocity at some

distance down the wire to the acceleration using

Work-Energy Theorem

- Solving for ax, substituting into the Fx

equation, multiplying both sides by d, and

distributing the ½m throughout the equation

Work-Energy Theorem

- But we can see that the right side of the

equation is no more than the kinetic energy after

the force has been applied minus the kinetic

energy before the force was appliedand that

by definition is the work done

Work-Energy Theorem

- When calculating the work done on an object by a

force during a displacement, use only the

component of the force that is parallel to the

objects displacement. - where ? is the angle between the force F and the

horizontal. - The force component perpendicular to the

displacement does no work.

Work-Energy Theorem

- Work-Energy Theorem the net work done on an

object is equal to the change in kinetic energy

of the object. - W Kf Ki Fd 0.5m(vf2 - vi2)
- A net force causes an object to change its

kinetic energy because a net force causes an

object to accelerate, and acceleration means a

change in velocity, and if velocity changes,

kinetic energy changes.

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Gravitational Potential Energy

- Potential energy (U) an object may store energy

because of its position. Energy that is stored

is called potential energy because in the stored

state it has the potential to do work. - Work is required to lift objects against Earths

gravity. - Potential energy due to elevated positions is

gravitational potential energy. - The amount of gravitational potential energy

possessed by an elevated object is equal to the

work done against gravity in lifting it. - Ug Fwh mgh

Work and Potential Energy

- When we throw a tomato up in the air, negative

work is being done on the tomato which causes it

to slow down during its ascent. - As a result, the kinetic energy of the tomato is

reduced eventually to zero at the highest

point. - But where did that energy go???

Work and Potential Energy

- Where it went was into an increase in the

gravitational potential energy of the tomato. - The reverse happens when the tomato begins to

fall down. - Now the positive work done by the gravitational

force causes the gravitational potential energy

to be reduced and the tomatos kinetic energy to

increase.

Work and Potential Energy

- From this we can see that for either the rise or

fall of the tomato, the change ?U in the

gravitational potential energy is the negative of

the work done on the tomato by the gravitational

force - In equation form we get

- Only changes in potential energy have meaning it

is important that all heights be measured from

the same origin. - In many problems, the ground is chosen as the

zero level for the determination of the height. - As the ball falls from A to B, the potential

energy at A is converted to kinetic energy at B.

- The amount of potential energy of the ball at

point A will equal the amount of kinetic energy

of the ball at point B.

Elastic Potential Energy

- Stretching or compressing an elastic object

requires energy and this energy is stored in the

elastic object as elastic potential energy. - The work required to stretch or compress a spring

is dependent on the force constant k. - The force constant will not change for a

particular spring as long as the spring is not

permanently distorted (which occurs when the

elastic limit of the spring is exceeded). - F kx

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Elastic Potential Energy

- The force required to stretch/compress an elastic

object is not a constant force. The work needed

varies with the amount of stretch/compression. - W 0.5kx2
- The potential energy of an elastic object is

equal to the work done on the elastic object. - Ue 0.5kx2
- In actual practice, a small fraction of the work

in stretching/compressing an elastic object is

converted into heat energy in the spring.

Work and Elastic Potential Energy

- If we give the block a shove to the right, the

kinetic energy of the block is transferred into

elastic potential energy as the spring

compresses. - The work done in compressing the spring is the

negative of the change in the blocks kinetic

energy. - And of course the reverse happens when the spring

stretches back out potential energy gets

transformed back into kinetic energy.

Conservation of Mechanical Energy

- Conservative forces all the work done is stored

as energy and is available to do work later.

Example gravitational forces, elastic forces. - Nonconservative (dissipative) forces the force

generally produces a form of energy that is not

mechanical. - Friction is a nonconservative (dissipative) force

because it produces heat (thermal energy, not

mechanical). - The total amount of energy in any closed system

remains constant.

Conservation of Mechanical Energy

- The sum of the potential and kinetic energy of a

system remains constant when no dissipative

forces (like friction) act on the system. - Law of Conservation of Energy energy cannot be

created or destroyed it may be changed from one

form to another or transferred from one object to

another, but the total amount of energy never

changes.

Conservation of Mechanical Energy

- In a closed system in which gravitational

potential energy and kinetic energy are involved,

the potential energy at the highest point is

equal to the kinetic energy at the lowest point. - mgh 0.5mv2

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- Notice that the sum of the potential energy (PE)

and kinetic energy (KE) at every point is 40000

J. Energy is conserved.

- The velocity at the lowest point can be

determined by the height

Conservation of Mechanical Energy

- The change in velocity due to a change in height

can also be determined - The height can be determined by the initial

velocity (vf 0 m/s)

Conservation of Mechanical Energy

- In a closed system in which elastic potential

energy and kinetic energy are involved, the

potential energy at the maximum distance of

stretch/compression is equal to the kinetic

energy at the equilibrium (rest) position.

Conservation of Mechanical Energy

- Conservation of Energy Equation
- W done (by applied force) Ugravitational before

Uelastic before K before Ugravitational

after Uelastic after K after W done

(usually by friction) - Fappliedd m g hi 0.5 k xi2 0.5 m vi2

m g hf 0.5 k xf2 0.5 m vf2 FFd

- If there is no change in height m g hi and m

g hf drop out of the equation. - If there is no spring or elastic object 0.5

k xi2 and 0.5 k xf2 drop out of the equation. - If there is no change in velocity 0.5 m vi2

and 0.5 m vf2 drop out of the equation. - If there is no applied force (a push/pull that

you supply) Fappliedd drops out of the

equation. - If there is no friction FFd drops out of the

equation.

Helpful Online Links

- Work Energy Theorem
- The Work-Energy Theorem
- Elastic Constant k
- Hookes Law Applet