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Dynamics

Colin McDonald, PhD

February 15th, 2013

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Chapter

14:

Kinetics

of

a

Particle:

Work

and

Energyo Chapter Objectives:

Calculate the work of a force Apply the principle of work and energy to a particle or system

of particles

Determine the power generated by a machine, engine, or

motor

Calculate the mechanical efficiency of a machine

Understand the concept of conservative forces and determine

the potential energy of such forces Apply the principle of conservation of energy

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14.5

Conservative

Forces

and

Potential

Energy

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Applications

o The weight of the sacks resting on this platform causes potential

energy to be stored in the supporting springso As each sack is removed, the platform will rise slightly since

some of the potential energy within the springs will be

transformed into an increase in gravitational potential energy of

the remaining sackso If the sacks weigh 100 lb and the

equivalent spring constant is k= 500

lb/ft, what is the energy stored in the

springs?

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Applications

o The roller coaster is released from rest at the top of the hill A. As

the coaster moves down the hill, potential energy is transformedinto kinetic energy

o What is the velocity of the coaster when it is at B and C?

o Also, how can we determine the minimum height of hill A so that

the car travels around both inside loops without leaving track?

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Conservative

Force

o A conservative force is one whose work is independent of the

path followed, depending only on the forces initial and finalpositions on the path

Examples:

Weight of a particle: work done by weight depends only

vertical displacement Force developed by a spring: spring force depends only

on the springs elongation or compression

o A non-conservative force is one whose work depends on the

path followed

Example:

Friction force: the longer the path, the greater the work

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Energy

o Recall: the energy of a particle has two components:

Kinetic energy: measure of particles capacity to do work Associated with the motion of the particle

Potential energy: measure of the amount of work a

conservative force will do when it moves from a given

position to some datum

Associated with the position of the particle measured

from a fixed datum

o Two types of potential energy are important in mechanics:

Potential energy created by gravity (weight)

Potential energy created by an elastic spring

o The conservative potential energy of a particle/system is typically

written using the potential function V

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Gravitational

Potential

Energy

o The potential function for a gravitational force (Vg), e.g., weight

(W = mg),

is the force multiplied by its elevation from a datum:

o If a particle is located a distancey

above an arbitrary datum: The particles weight has positive

gravitational potential energy

i.e.,W has the capacity of doing

positive work when the particleis moved back to the datum

o Similarly, if a particle is located a

distancey below this arbitrary datum

Vg is negative

Eq. (14-13)

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Elastic

Potential

Energy

o When an elastic spring is elongated or compressed a distance s

from its un-stretched position, elastic potential energy (Ve) isstored in the spring:

This energy is:

o Elastic potential energy is always

positive

When deformed, the force of

the spring has the capacity foralways doing positive work on

the particle when returned to its

initial position

Eq. (14-14)

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Potential

Functiono In general, if a particle is subjected to both gravitational and

elastic forces, the particles potential energy can be expressedas a potential function:

Vdepends on the location of the particle with respect to a

selected datum

o The work done by a conservative force in moving the particle

from one point to another point is measured by the difference of

this function, that is:

Eq. (14-15)

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Potential

Functiono For example, if a particle of weightW is suspended from a

spring, the potential function is written as:

o If the particle is moved from position s1(relative to a selected datum located at itsun-stretched length) to a new position s2:

The work of the particles weight and

the spring force is:

1

2

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14.6

Conservation

of

Energy

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Conservation

of

Energyo When a particle is acted upon by a system of conservative forces,

the work done by these forces is conserved i.e., the sum of the kinetic and potential energies remains

constant

o This is referred to as the conservation of energy, and is written

mathematically as:

Where:

T1 and T2: kinetic energies at states 1 and 2 (

V1 and V2: potential energies at states 1 and 2

Eq. (14-21)

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Conservation

of

Energyo For example, if a ball of weightW is dropped from a height h

above the ground (datum):

At height h: potential energy is max, kinetic energy is zero

Just before hitting ground: potential energy is zero and kinetic

energy is related to the mass of the ball and its velocity

In each case, the totalenergy* is:

*Refer to the text for a proof of

this equation

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Problem

Example

Io Given: a girl weighing 125-lbs rides a bicycle from point A to B

Velocity at A is 10 ft/s, girl stops pedaling at Ao Find: the velocity and the normal force at B

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Problem

Example

IIo Given: the 4 kg collar, C, has a velocity of 2 m/s at A

The spring constant is 400 N/m The un-stretched length of the spring is 0.2 m

o Find: the velocity of the collar at B