Course website: faculty.uml.edu/pchowdhury/95.141/ www ...faculty.uml.edu/pchowdhury/95.141/Lectures/LECTURE11.pdf · Last Lecture Chapter 7 Work done by a Force Scalar (Dot) product

CHOWDHURY 95.141 PHYSICS I SPRING 2013 LECTURE 11

Course website: faculty.uml.edu/pchowdhury/95.141/

www.masteringphysics.com

Course: UML95141SPRING2013

Lecture Captureh"p://echo360.uml.edu/chowdhury2013/physics1Spring.html

95.141 Mar 4 , 2013 PHYSICS I Lecture 11

Last Lecture Chapter 7 Work done by a Force Scalar (Dot) product of two vectors Kinetic Energy Work-Kinetic Energy theorem

Today Chapter 6 Newton’s Law of Universal Gravitation Kepler’s Laws Satellite Orbits Weightlessness Variations of g on earth


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Clicker Quiz: Gravity on Earth

Two objects are dropped simultaneously from 2m above the ground at the equator and at the North pole. Which hits the ground first? A) Object at equator B) Object at North pole C) They hit simultaneously


Variable forces & accelerations •  Equations of Motion…..Force…Free-body diagrams…

Newton’s Laws……Do we need more concepts? •  The moon suddenly loses its tangential velocity around

the earth. How long will it take to hit the earth? What will be its speed when it hits the earth?

•  You start from rest on a roller coaster ride with lots of ups and downs. What will be your speed at various points along the ride?

•  You tie an elastic bungee cord around your ankle and jump off a high bridge. What will be the lowest point in your fall?

•  Variable forces…variable accelerations…cannot use the constant acceleration formulae that we have learned


Work and Energy •  New concepts that make variable force problems easy •  Eventually concepts of energy more powerful in physics •  Work done on an object by a force •  If a constant force F moves object by a displacement d,

work done by force is given by W = F||d, where F|| is the component of the force along d

•  Work has units of N-m, or Joules (J), and is a scalar !!


Example A force of 10 N, applied in the direction shown, moves a box across a horizontal distance of 10 m. How much work does the force do on the box?

NF 10=

! = 60!

!d =10m

= 50 J


Scalar Product of Two Vectors •  Both Force and Displacement are vectors. •  Work, a scalar, comes from the product of

two vectors. •  Three ways to multiply vectors

– Multiplication by a scalar

– Scalar (or dot) product

– Vector (or cross) product

vectorAc →

scalarBA →•

vectorBA →×


Scalar (Dot) Product of 2 Vectors The scalar product of two vectors is written as:

Work done by a force along a displacement:

dF

⊥ dF||

Work: scalar product of force and displacement Special cases: or


Clicker Quiz The work done by the earth on the moon as the moon makes one full revolution around the earth at constant speed v is:

–  A) positive

–  B) negative –  C) zero


Scalar Product Math •  Commutative property ABBA

•=•

( ) CABACBA

•+•=+•

kAjAiAA zyxˆˆˆ ++=

kBjBiBB zyxˆˆˆ ++=

•  Distributive property •  In component form:

!A•!B = (Axi + Ay j + Azk)•(Bxi +By j +Bzk)

= AxBx (i • i )+ AxBy (i • j)+ AxBz (i • k)

+AyBx ( j • i )+ AyBy ( j • j)+ AyBz ( j • k)

+AzBx (k • i )+ AzBy (k • j)+ AzBz (k • k)

!A•!B = AxBx + AyBy + AzBz

(i • i ) = ( j • j) = (k • k) =1 (i • j) = (i • k) = ( j • k) = 0


Example A constant force F acts on an object as it moves from position x1 to x2. What is the work done by this Force?

kjix ˆ3ˆ4ˆ21 −+= kjix ˆ6ˆ3ˆ52 +−=

kjiF ˆ2ˆ5ˆ4 ++−=

!d = !!x = !x2 "

!x1 = 3i " 7 j + 9k!F •!d = (!4)(3)+ (5)(!7)+ (2)(9) = !29J


Energy •  One of the most powerful concepts in science,

can solve complex problems … •  Energy and work have the same units •  Work is done “on” and object “by” a force •  Work is associated with the force •  Energy is a property of the object •  An object “gains” or “loses” energy depending on

whether the net work done on it is positive or negative

•  The change in energy “of” an object is equal to the work done “on” the object


Translational Kinetic Energy •  Kinetic: associated with motion •  Translational: motion in a line or trajectory (as

opposed to circular/rotational motion) •  Point objects have translational KE only •  Extended objects can have rotational KE


Kinetic Energy •  A constant horizontal force F acts on an object on a

frictionless surface. The box accelerates, i.e. speeds up. •  We want to define a property of the object (called kinetic

energy, or energy of motion) which changes as work is done on it by the force.

•  We also want that change in the kinetic energy to be exactly equal to the net work done on it.

•  Work and kinetic energy has to have the same units.

F F

d

vi vf


Work and Kinetic Energy

F F

d

vi vf

F =mavf2 = vi

2 + 2ad

vf2 = vi

2 + 2 Fmd

K f !Ki =W

12mvf

2 !12mvi

2 = Fd

12mvf

2 =12mvi

2 +Fd

mvf2 =mvi

2 + 2Fd

K =12mv2

K f !Ki = "K =Wnet

Define kinetic energy

Work-Kinetic Energy Principle

The net work done on an object is equal to the change in the object’s kinetic energy.


Example Net Work required to accelerate a 1000kg car from (a) rest to 20 m/s?

(b) from 20 m/s to 40 m/s? (c) from 40 m/s to zero?

W = !KE = (12)m[vf

2 " vi2 ]= 1000

2[202 " 0]= 200,000J = 200kJ

W = !KE = (12)m[vf

2 " vi2 ]= 1000

2[402 " 202 ]= 600, 000J = 600kJ

W = !KE = (12)m[vf

2 " vi2 ]= 1000

2[0" 402 ]= "800,000J = "800kJ


The work done by the earth on an apple as the apple falls towards the earth with increasing speed v is positive.

So kinetic energy increases.

Work & Kinetic Energy


The work done by the earth on the moon as the moon makes one full revolution around the earth at constant speed v is zero.

So there is no change in kinetic energy.

What about Friction?

Wfriction = !K = K f "Ki

K f = 0

Wfriction = !9 J

The work-kinetic energy theorem still holds.

And friction does negative work.

An object of mass 2 kg with an initial velocity of 3 m/s is brought to a stop by a constant frictional force of 6 N. How much work does the force of friction do on the object?

Ki = 12 mv

2i = 1

2 !2 kg! (3 m/s)2 = 9 J


Clicker Quiz

A) one force B) two forces C) three forces D) four forces E) no forces are doing work

The red box is being pulled up a rough incline (where friction is present) by a rope connected to a pulley. How many forces are doing work on the red box?


Summary

•  Work done by a Force •  Scalar (Dot) product of two vectors •  Kinetic Energy •  Work-Kinetic Energy theorem


Documents

Course website: faculty.uml.edu/pchowdhury/95.141/ www ...faculty.uml.edu/pchowdhury/95.141/Lectures/LECTURE11.pdf · Last Lecture Chapter 7 Work done by a Force Scalar (Dot) product