Uniform Circular Motion, Acceleration

 A particle moves with a constant speed in a circular path of radius r with an acceleration:

a c



v

2

r



a

the center of the circle  The centripetal acceleration is always perpendicular to the velocity

Uniform Circular Motion, Force

   associated with the centripetal acceleration The force is also directed toward the center of the circle Applying Newton’s Second Law along the radial direction gives 

F



ma c



m v

2

r

Uniform Circular Motion, cont

   A force causing a centripetal acceleration acts toward the center of the circle It causes a change in the direction of the velocity vector If the force vanishes, the object would move in a straight-line path

tangent

to the circle  See various release points in the active figure

Motion in a Horizontal Circle

 The speed at which the object moves depends on the mass of the object and the tension in the cord   The centripetal force is supplied by the tension T=mv 2 /r hence

v



Tr m

Motion in Accelerated Frames

 A

fictitious force

frame of reference results from an accelerated  A fictitious force appears to act on an object in the same way as a real force, but you cannot identify a second object for the fictitious force  Remember that real forces are always interactions between two objects

“Centrifugal” Force

    From the frame of the passenger (b), a force appears to push her toward the door From the frame of the Earth, the car applies a leftward force on the passenger The outward force is often called a

centrifugal

force  It is a fictitious force due to the centripetal acceleration associated with the car’s change in direction In actuality, friction supplies the force to allow the passenger to move with the car  If the frictional force is not large enough, the passenger continues on her initial path according to Newton’s First Law

“Coriolis Force”

 This is an apparent force caused by changing the radial position of an object in a rotating coordinate system The result of the rotation is the curved path of object Ball in figure to the right, winds, rivers and currents on earth. For winds we get the prevailing wind pattern below.

Fictitious Forces, examples

 Although fictitious forces are not real forces, they can have real effects  Examples:   Objects in the car do slide You feel pushed to the outside of a rotating platform  The Coriolis force is responsible for the rotation of weather systems, including hurricanes, and ocean currents

Introduction to Energy

 The concept of energy is one of the most important topics in science and engineering  Every physical process that occurs in the Universe involves energy and energy transfers or transformations  Energy is not easily defined

Work

 The work,

W

, done on a system by an agent exerting a constant force on the system is the product of the magnitude

F

magnitude D

r

of the force, the of the displacement of the point of application of the force, and cos q, where q is the angle between the force and the displacement vectors

Work, cont.



W

=

F

D

r

cos q 

F .

D

r

 The displacement is that of the point of application of the force   A force does no work on the object if the force does not move through a displacement The work done by a force on a moving object is zero when the force applied is perpendicular to the displacement of its point of application

Work Example

  The normal force and the gravitational force do no work on the object  cos q = cos 90 ° = 0 The force

F

is the only force that does work on the object

Units of Work

  Work is a scalar quantity The unit of work is a joule (J)   1 joule = 1 newton .

J = N · m ( Fr) 1 meter  The sign of the work depends on the direction of the force relative to the displacement   Work is positive when projection of same direction as the displacement onto is in the Work is negative when the projection is in the opposite direction

Work Done by a Varying Force

 Assume that during a very small displacement, D

x

,

F

is constant   For that displacement,

W

~

F

D

x

For all of the intervals,

W



x



f x i F x

D

x

Work Done by a Varying Force, cont



x

lim D  0

x x



f i F x



x x f i F dx x

 Therefore,

W

 

x x f i F dx x

 The work done is equal to the area under the curve between

x i

and

x f

Work Done By A Spring

   A model of a common physical system for which the force varies with position The block is on a horizontal, frictionless surface Observe the motion of the block with various values of the spring constant

Hooke’s Law

  The force exerted by the spring is 

F s

= -

kx x

is the position of the block with respect to the equilibrium position (

x

= 0) 

k

is called the spring constant or force constant and measures the stiffness of the spring This is called Hooke’s Law

Hooke’s Law, cont.

   When

x

is positive (spring is stretched),

F

is negative When

x

is 0 (at the equilibrium position),

F

is 0 When (spring is compressed),

F x

is negative is positive

Hooke’s Law, final

 The force exerted by the spring is always directed opposite to the displacement from equilibrium  The spring force is sometimes called the

restoring force

 If the block is released it will oscillate back and forth between –

x

and

x

Hooke’s Law consider the spring

     When

x

is positive (spring is stretched),

F s

is negative When

x

is 0 (at the equilibrium position),

F s

is 0 When

x

is negative (spring is compressed),

F s

is positive Hence the

restoring force

F s

=

F s = -kx

  

Work Done by a Spring

Identify the block as the system and see figure below The work as the block moves from

x i

= -

x

max to

x f

= 0 is ½ kx 2 Note: The total work done by the spring as the block moves from –

x

max to

x

max is zero see figure also  Ie. From the General definition

W s

 

W

Or  

F

s



W net



d

r

  x f  x i ( 

kx

i



dx

i F

) 

d

r



x 0

 (

max



kx

)

dx W s

 

x x f i F dx x

   0

x

max     1 2

kx

2 max

ie

.



x n dx



x n

 1 /(

n

 1 )

Work Done by a Spring,in general

  Assume the block undergoes an arbitrary displacement from

x = x i

to

x = x f

The work done by the spring on the block is

W s

 

x x f i

   1 2

kx i

2  1 2

kx f

2    If the motion ends where it begins, W = 0 NOTE the work is a change in the expression 1/2kx 2 We say a change in elastic potential energy..in general a energy expression is defined for various forces and the work done changes that energy.

Kinetic Energy and Work Kinetic Energy Theorem

 Kinetic Energy is the energy of a particle due to its motion 

K

= ½

mv

2 

K

is the kinetic energy  

m

is the mass of the particle

v

is the speed of the particle  A change in kinetic energy is one possible result of doing work to transfer energy into a system

Kinetic Energy

 Calculating the work:

W

 

x f x i



F dx

 

x i x f ma dx W

 

W W net



v v i f mv dv

 

K f

1 2

mv f

2 

K i

IE. a=dv/dt adx=dv/dt dx  1

mv i

2 2  D

K

=dv dx/dt=vdv  The Work-Kinetic Energy Theorem states S

W

=

K f

–

K i

= D

K

Hence K=1/2 mv 2 is a a natural for energy expression..

And the last equation is called the Work-Kinetic Energy Theorem Again we note that the work done changes an energy expression … in this case a change in Kinetic energy The speed of the system increases if the work done on it is positive The speed of the system decreases if the net work is negative Also valid for changes in rotational speed

Potential Energy in general

 Potential energy is energy related to the configuration of a system in which the components of the system interact by forces   The forces are internal to the system Can be associated with only specific types of forces acting between members of a system

Gravitational Potential Energy NEAR SURFACE OF EARTH ONLY

 The system is the Earth and the book   Do work on the book by lifting it slowly through a vertical displacement D  D

y

ˆ

j

The work done on the system must appear as an increase in the energy of the system

Gravitational Potential Energy, cont

 There is no change in kinetic energy since the book starts and ends at rest  Gravitational potential energy is the energy associated with an object at a given location above the surface of the Earth

W

    D

r

W

 (

mg

ˆ

j

)  

y f



y i

 ˆ

j

W



mgy f



mgy i

Gravitational Potential Energy, final

 The quantity

mgy

is identified as the gravitational potential energy,

U g

 

U g

=

mgy

THIS IS ONLY NEAR THE EARTH’s surface ……………WHY???????

 Units are joules (J)  Is a scalar  Work may change the gravitational potential energy of the system 

W

net = D

U

g

Conservative Forces and Potential Energy

 Define a potential energy function,

U

, such that the work done by a conservative force equals the decrease in the potential energy of the system  The work done by such a force,

F

, is

W C

 

x i x f F dx x

 D

U

 D

U

is negative when

F

direction and

x

are in the same

Conservative Forces and Potential Energy

  The conservative force is related to the potential energy function through

F x

 

dU dx

The

x

component of a conservative force acting on an object within a system equals the negative of the potential energy of the system with respect to

x

 Can be extended to three dimensions

Conservative Forces and Potential Energy – Check

 Look at the case of a deformed spring

F s

 

dU s dx

 

d dx

  1

kx

2 2    

kx

 This is Hooke’s Law and confirms the equation for U  U is an important function because a conservative force can be derived from it

Energy Diagrams and Equilibrium

   Motion in a system can be observed in terms of a graph of its position and energy In a spring-mass system example, the block oscillates between the turning points,

x

= ±

x

max The block will always accelerate back toward

x

= 0

Energy Diagrams and Stable Equilibrium

   The

x

= 0 position is one of

stable equilibrium

Configurations of stable equilibrium correspond to those for which

U

(

x

) is a minimum

x

=

x

max and

x

= -

x

max are called the turning points

Energy Diagrams and Unstable Equilibrium

   

F x

= 0 at

x

= 0, so the particle is in equilibrium For any other value of

x

, the particle moves away from the equilibrium position This is an example of

unstable equilibrium

Configurations of unstable equilibrium correspond to those for which

U

(

x

) is a maximum

Neutral Equilibrium



Neutral equilibrium

when

U

occurs in a configuration is constant over some region  A small displacement from a position in this region will produce neither restoring nor disrupting forces

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 A word from our sponsors:

CONDUCTION, CONVECTION, RADIATION

More Ways to Transfer Energy Into or Out of A System

  

Matter Transfer

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

Electrical Transmission

electric current – transfer is by

Electromagnetic Radiation

– energy is transferred by electromagnetic waves

Two New important Potential Energies

  In the universe at large Gravitational force as defined by Newton prevails Ie..

F = -Gm 1 m 2 /r 2

m the masses G a universal constant and r distance between the masses (negative is attractive force)  In the atomic world the electric force dominates defined as

F=kq 1 q 2 /r 2

here r is the distance between the electric charges represented by q and k a universal constant  Charges can be + or  The Constant values.G,k depend upon units used

Gravitational and Electric Potential energies (3D)

W C

 

x i x f F dx x

 D

U

With r replacing x we get and using the gravitational and electric forces equations and S for integration from point initial to final

W =

S

F G dr = - Gm 1 m 2 =

S

1/r 2 dr = -Gm 1 m 2 ( 1/r f -1/r i ) W =

S

F e dr = kq 1 q 2 =

S

1/r 2 dr = kq 1 q 2 (1/r f -1/r i ) Or potential energies for these forces go as 1/r Note from above that F = -dU/dr with U G = Gm 1 m 2 /r U e = kq 1 q 2 /r we get back the 1/r 2 forces

Conservation of Energy



Energy 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!

Isolated System

 For an isolated system, D E mech = 0   Remember E mech = K + U This is

conservation of energy

for an isolated system with no nonconservative forces acting   If nonconservative forces are acting, some energy is transformed into internal energy Conservation of Energy becomes D E system  = 0 E system is all kinetic, potential, and internal energies  This is the most general statement of the isolated system model

Isolated System, cont

        ( example book falling) The changes in energy D E system Or D K + D U=0 D K= D U = 0 Ie. K f - K i = -(U f –U i ) can be written out and rearranged K f + U f = K i + U i Remember, this applies only to a system in which conservative forces act Or 1/2mv f 2 +mgh f =1/2mgv i 2 +mgh i

Example – Free Fall example 8-1

 Determine the speed of the ball at y above the ground  Conceptualize  Use energy instead of motion  Categorize   System is isolated Only force is gravitational which is conservative

Example – Free Fall, cont

 Analyze   Apply Conservation of Energy

K f



+ U K i gf = K i + U gi

= 0, the ball is dropped  Solving for v f

v f



v i

2  2  

y

  Finalize  The equation for v f is consistent with the results obtained from kinematics

For the electric force

 Total energy   Is K+U=1/2mv 2 +kq 1 q 2 /r Specifically in a hydrogen atom using charge units e (CALLED ESU we get rid of K) and the proton and electron both have the same charge =e  Or total energy for electron in orbit  =1/2mv 2 +e 2 /r we will use this in chapter 3

Instantaneous Power

 Power is the time rate of energy transfer  The

instantaneous power

is defined as  

dE dt

 Using work as the energy transfer method, this can also be written as 

avg



W

D

t

Power

 The time rate of energy transfer is called power  The average power is given by    

P



W

D

t

when the method of energy transfer is work Units of power: what is a Joule/sec called ?

Answer WATT! 1 watt=1joule/sec

Instantaneous Power and Average Power

 The instantaneous power is the limiting value of the average power as D t approaches zero 

t

lim D  0

W

D

t



dW dt d

r

dt

 The power is valid for any means of energy transfer  NOTE: only part of F adds to power ?

Units of Power

 The SI unit of power is called the watt  1 watt = 1 joule / second = 1 kg .

m 2 / s 2  A unit of power in the US Customary system is horsepower  1 hp = 746 W  Units of power can also be used to express units of work or energy  1 kWh = (1000 W)(3600 s) = 3.6 x10 6 J

W T Example 8.10 m elev =1600kg passengers =200kg A constant retarding force =4000 N How much power to lift at constant rate of 3m/s How much power to lift at speed v with a=1.00 m/s s f USE S F =0 in first part and =ma in second then use Next equation 

t

lim D  0

W

D

t



dW dt d dt

r