Chapter 8. Potential Energy and Energy Conservation

Introduction

In Ch 7 – Work- Energy theorem. We learnt that total work done on an object translates to change in it’s Kinetic Energy

In Ch 8 – we will consider a systems

composed of several objects that interact with one another.

What is the difference?

There is an obvious difference between these two situations. The work the weight lifter did has been stored in the new configuration of the Earth-barbell system, and the work done by the woman separating the crates seem to be lost rather than stored away.

How do we determine whether the work done by a particular type of force is “stored” or “used up.”?

We will need to see if the Force that does the work is -  Conservative -  Non-Conservative or Dissipative

Stores Energy

Dissipates or uses up energy

The Path Independence Test for a Gravitational Force

The net work done on the skier as she travels down the ramp is given by

It does not depend on the shape of the ramp but only on the vertical component of the gravitational force and the initial and final positions of her center of mass.

Path Dependence of Work Done by a Friction Force

•  The work done by friction along that path 1→2 is given by

W1→2 = F . D1→2

•  The work done by the friction force along path 1→4→3→2 is given by

W1→2 = ||F . D1→4|| + || F . D2→3|| + || F . D3→4|| + || F . D4→1 ||

Conservative Forces and Path Independence

•  Conservative forces are the forces that do path independent work;

•  Non-conservative /Dissipative forces are the forces that do path dependent work;

The work done by a conservative force along any closed path is zero.

Test of a System's Ability to Store Work Done by Internal Forces:

The work done by a Conservative Force can be

stored in the system as potential energy, and the work done by a Non-Conservative Force will be “used up”

cU WΔ = −Conservative Forces ONLY

Conservative Forces are like forces that transfer money from Bank to Wallet or Wallet to bank but never spends it ( never “uses up” )

EXAMPLE : Cheese on a Track Figure a shows a 2.0 kg block of slippery cheese that

slides along a frictionless track from point 1 to point 2. The cheese travels through a total distance of 2.0 m along the track, and a net vertical distance of 0.80 m. How much work is done on the cheese by the gravitational force during the slide?

15.68J

Determining Potential Energy Values Consider a particle-like object that is part of a system in

which a conservative force acts. When that force does work W on the object,

the change in the potential energy associated with the system is the negative of the work done

Gravitational Potential Energy

GRAVITATIONAL POTENTIAL ENERGY

•  The gravitational potential energy U is the energy that an object of mass m has by virtue of its position relative to the surface of the earth. That position is measured by the height h of the object relative to an arbitrary zero level:

•  SI Unit of Gravitational Potential Energy: joule (J)

GU mgh=

Elastic Potential Energy

we choose the reference configuration to be when the spring is at its relaxed length and the block is at x.

or

Example A 2.0 kg sloth hangs 5.0 m

above the ground (Fig. 8-6). a) What is the gravitational potential energy U of the sloth–Earth system if we take the reference point y=0 to be (1)  at the ground, (2) at a balcony floor that is 3.0 m above the ground, (3) at the limb, and (4) 1.0 m above the limb? Take the gravitational potential energy to be zero at y=0. (b) The sloth drops to the ground. For each choice of reference point, what is the change in the potential energy of the sloth–Earth system due to the fall?

a: 98J; 39 J; 0J; -20J b: -98J

What is mechanical energy of a system?

The mechanical energy is the sum of kinetic energy and potential energies:

mec

sys sys sysE K U= +

For example,

2 21 12 2

mec

sys sys sysE K U mv mgh kx= + = + +

Internal Forces (Conservative)

External Forces (non conservative)

Fgrav Fspring Fnuclear Felectrical

Fapp Ffrict Fair Ftension Fnormal

Conservation of Mechanical Energy In a system where

(1) no work is done on it by external forces and (2) only conservative internal forces act on the system elements, then the internal forces in the system can cause energy to be transferred between kinetic energy and potential energy, but their sum, the mechanical energy Emec of the system, cannot change.

0mecsys sys sysE K UΔ = Δ +Δ =

Example - A Daredevil Motorcyclist A motorcyclist is trying to leap across the

canyon shown in Figure by driving horizontally off the cliff at a speed of 38.0 m/s. Ignoring air resistance, find the speed with which the cycle strikes the ground on the other side.

46.15 m/s

EXAMPLE : Bungee Jumper A 61.0 kg bungee-cord jumper

is on a bridge 45.0 m above a river. The elastic bungee cord has a relaxed length of L = 25.0 m. Assume that the cord obeys Hooke's law, with a spring constant of 160 N/m. If the jumper stops before reaching the water, what is the height h of her feet above the water at her lowest point?

L + d + h = 45m and L = 25 m E = K + Ug + Ue = Const; Δ E = ΔK +ΔUg +Δ Ue = 0 ΔUg= – mgy = – 61 kg · 9.8m/s2 ·*(L+d) ΔUe= kd2/2 = 160 N/m · d2/2 80d2 – 600d – 15000 = 0; d2 – 7.5d – 187.5 = 0 d = 18 m and d = -10.5 m h = 45m – 25m – 18m = 2m

Work done by Ext Forces

•  No Friction involved W = ΔK + ΔU Comes from W-E theorem e.g. Wspring – Wgravity = ΔK •  Friction involved W = ΔK + ΔU + Δ(Ff.d) W = ΔK + ΔU + ΔE thermal

Conservation of Energy in terms of Work Done

W = ΔK + ΔU + ΔE thermal + ΔE internal

W = ΔEmechanical+ ΔE thermal + ΔE internal

Total energy E of a System can change only by amounts of energy that are transferred to and from the system.

Isolated System

An isolated system: is a system where there is no net work is done on the system by external forces.

ΔEmec+ ΔE th + ΔE int = 0 ΔEmec = Emec, Final - Emec, initial

Emec, Final = Emec, initial - ΔE th - ΔE int

EXAMPLE •  In Fig., a 2.0 kg package of tamales slides along a

floor with speed v1=4.0 m/s. It then runs into and compresses a spring, until the package momentarily stops. Its path to the initially relaxed spring is frictionless, but as it compresses the spring, a kinetic frictional force from the floor, of magnitude 15 N, acts on it. The spring constant is 10 000 N/m. By what distance d is the spring compressed when the package stops?

0.5 Kd2 = 0.5 mv2 - fkd 5000 d 2 = 16 -15 d , d = 0.056 m

Emec, Final = Emec, initial - ΔE th - ΔE int

Package+Spring+floor+wall -> isolated system as a whole

Forces 1)normal – does no work 2) grav force - does no work 3) Spring force does work on the package decreasing its KE and increasing the PE 4) Spring force also pushes the rigid wall 5) Friction increases thermal energy