physics 111: mechanics lecture 7 wenda cao njit physics department

Physics 111: Mechanics Lecture 7

Wenda Cao

NJIT Physics Department

November 3, 2008

Potential Energy andEnergy Conservation

Work Kinetic Energy Work-Kinetic Energy Theorem Gravitational Potential Energy Elastic Potential Energy Work-Energy Theorem Conservative and Non-conservative Forces Conservation of Energy

November 3, 2008

Definition of Work W The work, W, done by a constant force on an

object is defined as the product of the component of the force along the direction of displacement and the magnitude of the displacement

F is the magnitude of the force Δ x is the magnitude of the object’s displacement is the angle between

xFW )cos(

and F x

November 3, 2008

Work Done by Multiple Forces

If more than one force acts on an object, then the total work is equal to the algebraic sum of the work done by the individual forces

Remember work is a scalar, so this is the algebraic sum

net by individual forcesW W

rFWWWW FNgnet )cos(

November 3, 2008

Kinetic Energy and Work Kinetic energy associated with the

motion of an object Scalar quantity with the same unit as

work Work is related to kinetic energy

2

2

1mvKE

xFmvmv net )cos(2

1

2

1 20

2

net fiW KE KE KE

November 3, 2008

Work done by a Gravitational Force

Gravitational Force Magnitude: mg Direction: downwards to the

Earth’s center Work done by

Gravitational Force

20

2

2

1

2

1mvmvWnet

cosW F r F r

cosrmgWg

November 3, 2008

Potential Energy Potential energy is associated

with the position of the object Gravitational Potential Energy is

the energy associated with the relative position of an object in space near the Earth’s surface

The gravitational potential energy

m is the mass of an object g is the acceleration of gravity y is the vertical position of the mass

relative the surface of the Earth SI unit: joule (J)

mgyPE

November 3, 2008

Reference Levels A location where the gravitational

potential energy is zero must be chosen for each problem The choice is arbitrary since the change in the

potential energy is the important quantity Choose a convenient location for the zero

reference height often the Earth’s surface may be some other point suggested by the problem

Once the position is chosen, it must remain fixed for the entire problem

November 3, 2008

Work and Gravitational Potential Energy

PE = mgy

Units of Potential Energy are the same as those of Work and Kinetic Energy

figravity PEPEW

)(

0cos)(cos

if

fig

yymg

yymgyFW

November 3, 2008

Extended Work-Energy Theorem

The work-energy theorem can be extended to include potential energy:

If we only have gravitational force, then

The sum of the kinetic energy and the gravitational potential energy remains constant at all time and hence is a conserved quantity

net fiW KE KE KE

figravity PEPEW

gravitynet WW

fiif PEPEKEKE

iiff KEPEPEKE

November 3, 2008


We denote the total mechanical energy by

Since

The total mechanical energy is conserved and remains the same at all times

PEKEE

iiff KEPEPEKE

ffii mgymvmgymv 22

2

1

2

1

November 3, 2008

Problem-Solving Strategy Define the system Select the location of zero gravitational

potential energy Do not change this location while solving the

problem Identify two points the object of interest

moves between One point should be where information is given The other point should be where you want to

find out something

November 3, 2008

Platform Diver A diver of mass m drops

from a board 10.0 m above the water’s surface. Neglect air resistance.

(a) Find is speed 5.0 m above the water surface

(b) Find his speed as he hits the water

November 3, 2008

Platform Diver (a) Find is speed 5.0 m above the

water surface

(b) Find his speed as he hits the water

ffii mgymvmgymv 22

2

1

2

1

ffi mgyvgy 2

2

10

smgyv if /142

02

10 2 fi mvmgy

smmmsm

yygv fif

/9.9)510)(/8.9(2

)(2

2

November 3, 2008

Spring Force Involves the spring

constant, k Hooke’s Law gives the force

F is in the opposite direction of x, always back towards the equilibrium point.

k depends on how the spring was formed, the material it is made from, thickness of the wire, etc. Unit: N/m.

dkF

November 3, 2008

Potential Energy in a Spring Elastic Potential Energy:

SI unit: Joule (J) related to the work required to

compress a spring from its equilibrium position to some final, arbitrary, position x

Work done by the spring

22

2

1

2

1)( fi

x

xs kxkxdxkxWf

i

2

2

1kxPEs

sfsis PEPEW

November 3, 2008


The work-energy theorem can be extended to include potential energy:

If we include gravitational force and spring force, then

net fiW KE KE KE

figravity PEPEW

sgravitynet WWW

0)()()( sisfifif PEPEPEPEKEKE

siiisfff KEKEPEPEPEKE

sfsis PEPEW

November 3, 2008


We denote the total mechanical energy by

Since

The total mechanical energy is conserved and remains the same at all times

sPEPEKEE

isfs PEPEKEPEPEKE )()(

2222

2

1

2

1

2

1

2

1fffiii kxmgymvkxmgymv

November 3, 2008

A block projected up a incline

A 0.5-kg block rests on a horizontal, frictionless surface. The block is pressed back against a spring having a constant of k = 625 N/m, compressing the spring by 10.0 cm to point A. Then the block is released.

(a) Find the maximum distance d the block travels up the frictionless incline if θ = 30°.

(b) How fast is the block going when halfway to its maximum height?

November 3, 2008


Point A (initial state): Point B (final state):

mcmxyv iii 1.010,0,0

m

smkg

mmN

mg

kxd i

28.1

30sin)/8.9)(5.0(

)1.0)(/625(5.0

sin

2

2

221

2222

2

1

2

1

2

1

2


0,sin,0 fff xdhyv

sin2

1 2 mgdmgykx fi

November 3, 2008


Point A (initial state): Point B (final state):

mcmxyv iii 1.010,0,0

sm

ghxm

kv if

/5.2......

2

2222

2

1

2

1

2

1

2


0,2/sin2/?, fff xdhyv

)2

(2

1

2

1 22 hmgmvkx fi ghvx

m

kfi 22

mmdh 64.030sin)28.1(sin

November 3, 2008

Types of Forces Conservative forces

Work and energy associated with the force can be recovered

Examples: Gravity, Spring Force, EM forces

Nonconservative forces The forces are generally

dissipative and work done against it cannot easily be recovered

Examples: Kinetic friction, air drag forces, normal forces, tension forces, applied forces …

November 3, 2008

Conservative Forces A force is conservative if the work it does

on an object moving between two points is independent of the path the objects take between the points The work depends only upon the initial and final

positions of the object Any conservative force can have a potential

energy function associated with it Work done by gravity Work done by spring force

fifig mgymgyPEPEW

22

2

1

2

1fisfsis kxkxPEPEW

November 3, 2008

Nonconservative Forces A force is nonconservative if the work it

does on an object depends on the path taken by the object between its final and starting points. The work depends upon the movement path For a non-conservative force, potential energy

can NOT be defined Work done by a nonconservative force

It is generally dissipative. The dispersal of energy takes the form of heat or sound

sotherforceknc WdfdFW

November 3, 2008


The work-energy theorem can be written as:

Wnc represents the work done by nonconservative forces Wc represents the work done by conservative forces

Any work done by conservative forces can be accounted for by changes in potential energy

Gravity work Spring force work

net fiW KE KE KE

cncnet WWW

22

2

1

2

1fifis kxkxPEPEW

fifig mgymgyPEPEW fic PEPEW

November 3, 2008


Any work done by conservative forces can be accounted for by changes in potential energy

Mechanical energy include kinetic and potential energy

22

2

1

2

1kxmgymvPEPEKEPEKEE sg

)()( iiffnc PEKEPEKEW

)()( ififnc PEPEKEKEPEKEW

PEPEPEPEPEW iffic )(

ifnc EEW

November 3, 2008

Problem-Solving Strategy Define the system to see if it includes non-

conservative forces (especially friction, drag force …) Without non-conservative forces With non-conservative forces

Select the location of zero potential energy Do not change this location while solving the problem

Identify two points the object of interest moves between One point should be where information is given The other point should be where you want to find out

something

2222

2

1

2

1

2

1

2

1iiifff kxmgymvkxmgymv


)2

1

2

1()

2

1

2

1( 2222

iiifffsotherforce kxmgymvkxmgymvWfd

November 3, 2008

A block of mass m = 0.40 kg slides across a horizontal frictionless counter with a speed of v = 0.50 m/s. It runs into and compresses a spring of spring constant k = 750 N/m. When the block is momentarily stopped by the spring, by what distance d is the spring compressed?

Conservation of Mechanical Energy


2222

2

1

2

1

2

1

2


002

1

2

100 22 mvkd

cmvk

md 15.12

002

1

2

100 22 mvkd

November 3, 2008

Changes in Mechanical Energy for conservative forces A 3-kg crate slides down a ramp. The ramp is 1 m in length and inclined at an angle of 30° as shown. The crate stats from rest at the top. The surface friction can be negligible. Use energy methods to determine the speed of the crate at the bottom of the ramp.

N)2

1

2

1()

2

1

2

1( 2222

iiifff kxmgymvkxmgymv

)00()002

1( 2 if mgymv

0,5.030sin,1 ii vmdymd

smgyv if /1.32

?,0 ff vy

)2

1

2

1()

2

1

2

1( 2222


November 3, 2008

Changes in Mechanical Energy for Non-conservative forces A 3-kg crate slides down a ramp. The ramp is 1 m in length and inclined at an angle of 30° as shown. The crate stats from rest at the top. The surface in contact have a coefficient of kinetic friction of 0.15. Use energy methods to determine the speed of the crate at the bottom of the ramp.

N

fk

)2

1

2

1()

2

1

2

1( 2222


)00()002

1(0 2 ifk mgymvNd

?,5.030sin,1,15.0 Nmdymd ik

0cos mgN

ifk mgymvdmg 2

2

1cos

smdygv kif /7.2)cos(2

November 3, 2008

Changes in Mechanical Energy for Non-conservative forces A 3-kg crate slides down a ramp. The ramp is 1 m in length and inclined at an angle of 30° as shown. The crate stats from rest at the top. The surface in contact have a coefficient of kinetic friction of 0.15. How far does the crate slide on the horizontal floor if it continues to experience a friction force.

)2

1

2

1()

2

1

2

1( 2222


)002

1()000(0 2 ik mvNx

?,/7.2,15.0 Nsmvik

0 mgN

2

2

1ik mvmgx

mg

vx

k

i 5.22

2

November 3, 2008

Block-Spring Collision A block having a mass of 0.8 kg is given an initial velocity vA

= 1.2 m/s to the right and collides with a spring whose mass is negligible and whose force constant is k = 50 N/m as shown in figure. Assuming the surface to be frictionless, calculate the maximum compression of the spring after the collision.

msmmN

kgv

k

mx A 15.0)/2.1(

/50

8.0max

002

100

2

1 22max Amvmv

2222

2

1

2

1

2

1

2


November 3, 2008

Block-Spring Collision A block having a mass of 0.8 kg is given an initial velocity vA

= 1.2 m/s to the right and collides with a spring whose mass is negligible and whose force constant is k = 50 N/m as shown in figure. Suppose a constant force of kinetic friction acts between the block and the surface, with µk = 0.5, what is the maximum compression xc in the spring.

)002

1()

2

100(0 22 Ack mvkxNd

)2

1

2

1()

2

1

2

1( 2222


ckAc mgxmvkx 22

2

1

2

1

cxdmgN and

058.09.325 2 cc xx mxc 093.0

November 3, 2008

Energy ReviewKinetic Energy

Associated with movement of members of a system

Potential Energy Determined by the configuration of the

system Gravitational and Elastic

Internal Energy Related to the temperature of the system

November 3, 2008

Conservation of EnergyEnergy is conserved

This means that energy cannot be created nor destroyed

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

November 3, 2008

Practical Case E = K + U = 0

The total amount of energy in the system is constant.

2222

2

1

2

1

2

1

2


November 3, 2008

Practical Case K + U +Eint = W + Q + TMW + TMT + TET + TER

The Work-Kinetic Energy theorem is a special case of Conservation of Energy K + U = W

November 3, 2008

Ways to Transfer Energy Into or Out of A System

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

Mechanical Waves – allow a disturbance to propagate through a medium

Heat – is driven by a temperature difference between two regions in space

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

Electrical Transmission – transfer is by electric current

Electromagnetic Radiation – energy is transferred by electromagnetic waves

November 3, 2008

Connected Blocks in Motion Two blocks are connected by a light string that passes over a

frictionless pulley. The block of mass m1 lies on a horizontal surface and is connected to a spring of force constant k. The system is released from rest when the spring is unstretched. If the hanging block of mass m2 fall a distance h before coming to rest, calculate the coefficient of kinetic friction between the block of mass m1 and the surface.

22 2

10 kxghmNxk

PEKEWfd sotherforce

hxmgN and

)02

1()0( 2

2 kxghmPEPEPE sg

221 2

1khghmghmk gm

khgm

k1

2 21

November 3, 2008

Power Work does not depend on time interval The rate at which energy is transferred is

important in the design and use of practical device

The time rate of energy transfer is called power

The average power is given by

when the method of energy transfer is work

WP

t

November 3, 2008

Instantaneous Power Power is the time rate of energy transfer.

Power is valid for any means of energy transfer

Other expression

A more general definition of instantaneous power

vFt

xF

t

WP

vFdt

rdF

dt

dW

t

WP

t

0lim

cosFvvFP

November 3, 2008

Units of PowerThe SI unit of power is called the

watt 1 watt = 1 joule / second = 1 kg . m2 / s3

A unit of power in the US Customary system is horsepower 1 hp = 550 ft . lb/s = 746 W

Units of power can also be used to express units of work or energy 1 kWh = (1000 W)(3600 s) = 3.6 x106 J

November 3, 2008

A 1000-kg elevator carries a maximum load of 800 kg. A constant frictional force of 4000 N retards its motion upward. What minimum power must the motor deliver to lift the fully loaded elevator at a constant speed of 3 m/s?

Power Delivered by an Elevator Motor

yynet maF ,

0 MgfT

NMgfT 41016.2

W

smNFvP4

4

1048.6

)/3)(1016.2(

hpkWP 9.868.64

physics 111: mechanics lecture 7 wenda cao njit physics department

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