Ch 7.ppsx

Ch. 7: Potential Energy and

Energy Conservation

Newton’s Laws represent one set of tools

for solving mechanics problems. We’ll now

(7.1) Gravitational Potential

Energy

• _{as an object falls, its}

speed increases

• _{therefore it gains}

kinetic energy

Two Interpretations

a) Work-Energy Theorem: gravity does work to increase the object’s K

W_grav = K mg s = K

mg (y₁ – y₂) ½ mv₂2 – ½ mv 12

b) Gravitational Potential Energy: there is energy associated with an object’s weight and position above the ground

U = mgy

a.k.a. “total

mechanical energy”

• _{as an object falls, U decreases and K increases:}

–U = K

U₁ – U₂ = K₂ – K₁ K₁ + U₁ = K₂ + U₂

½ mv₁2 + mgy

1 = ½ mv22 + mgy2

• _{the sum of K and U is constant, or conserved, if}

gravity is the only force acting

Example 7.1

You throw a 0.15-kg baseball straight up in the air, giving it an initial velocity of 20 m/s. Use conservation of energy to find how high it goes.

Example 7.4

In Ch. 3, we derived the following expression for the max height of a projectile:

Now, derive it using conservation of energy.

h = v₀2 sin2  0

2g v₀

Riding a “Loop-the-Loop”

Problem 7.46

A car on a roller

coaster rolls without friction around the track. It starts from

rest at point A at a height h above the bottom of the loop.

Conservation of Energy

• _{if we call “total mechanical energy” E = K + U,}

then the law (simply) becomes:

K

+ U

= K

+ U

Effect of other Forces

• work done by forces other than gravity will

change the total mechanical energy of a system

E

+ W

= E

Note:

If W_other > 0, E₂ > E₁ < 0,

= 0,

E₂ < E₁ E₂ = E₁

Where does the energy go in these two cases?

½ mv₁2 + mgy

1 + Wother = ½ mv22 + mgy2

Example 7.7

A worker gives a 12-kg crate a push and lets go. a) Find the magnitude of the force of friction

(7.2) Elastic Potential Energy

• _{as a spring stretches (or compresses), work is}

done on it and it gains elastic potential energy

U

= ½

kx

• U_el must be included in the law of conservation of mechanical energy

½ mv₁2 + mgy

1 + ½ kx12 + Wother =

½ mv₂2 + mgy

Example 7.11

In the “worst-case” design scenario, an elevator with broken cables crashes into a spring,

Bungee Jump Lab

Objective: Use conservation of energy to design a safe bungee jump

* The jumper should come to rest just before hitting the ground!

1. Create a procedure

(7.3) Conservative and

Non-Conservative Forces

•

conservative forces:

1) allow for conversion between

kinetic and potential energies

2) do work that is reversible

•

when only conservative forces do work, total

Examples:

gravity:

spring:

friction:

(7.4) Force and Potential Energy

• _{conservative forces can always be expressed in}

terms of a potential energy function

dx

dU

F





• _{given an expression for the potential energy as a}

To check, let’s examine F

and F

. . .

U

= ½

kx

1. The potential energy function U(x) for a spring is

This can be interpreted graphically as follows:

U vs. x for spring? U vs. y for gravity?

Problem 7.33

A force parallel to the

x

-axis acts on a

particle moving along the

x

-axis. This force

produces a potential energy

U

(

x

) =



x

,

where



= 1.20 J/m

. What is the force

(magnitude and direction) when the particle

(7.5) Energy Diagrams

• _{you can learn a lot about an object’s motion by}

Major points of interest:

1. vertical distance between U and E represents K

2. force on object equals negative slope of U(x)

Position Force Description x = 0 F

x > 0 F x < 0 F

equilibrium

3. The total mechanical energy dictates what kind of motion the object can have

Total Energy Motion

E₁ E₂ > E₃

object trapped in “potential well” with turning points x_a and x_b

object trapped in “potential well” with turning points x_c and x_d