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Kinetic Energy, Work, Power, and Potential Energy

- 8.01
- W05D1

Todays Reading Assignment W05D1

- Young and Freedman 6.1-6.4
- Math Review Module Scalar Product

Kinetic Energy

- Scalar quantity (reference frame dependent)
- SI unit is joule
- Change in kinetic energy

Concept Question Work and Kinetic Energy

- Compared to the amount of energy required to

accelerate a car from rest to 10 mph (miles per

hour), the amount of energy required to

accelerate the same car from 10 mph to 20 mph is - (1) the same
- (2) twice as much
- (3) three times as much
- (4) four times as much
- (5) unsure.

Work Done by a Constant Force for One Dimensional

Motion

- Definition
- The work W done by a constant force with an

x-component, Fx, in displacing an object by ?x

is equal to the x-component of the force times

the displacement

Concept Question Pushing against a wall

- The work done by the contact force of the wall

on the person as the person moves away from the

wall is - positive.
- negative.
- zero.
- impossible to determine from information given in

question and the figure.

Concept Question Work and Walking

- When a person walks, the force of friction

between the floor and the person's feet

accelerates the person forward. The work done by

the friction force is - positive.
- negative.
- zero.

Pushing a Stalled Car

Table Problem Work Done by Gravity Near the

Surface of the Earth

- Consider an object of mass m near the surface

of the earth falling directly towards the center

of the earth. The gravitational force between the

object and the earth is nearly constant. Suppose

the object starts from an initial point that is a

distance y0 from the surface of the earth and

moves to a final point a distance yf from the

surface of the earth. How much work does the

gravitational force do on the object as it falls?

Work done by Non-Constant Force One Dimensional

Motion

- (Infinitesimal) work is a scalar
- Add up these scalar quantities to get the total

work as area under graph of Fx vs x

Concept Question Work due to Variable Force

- A particle starts from rest at x 0 and moves

to x L under the action of a variable force

F(x), which is shown in the figure. What is the

particle's kinetic energy at x L/2 and at x

L? - (Fmax)(L/2), (Fmax)(L)
- (2) (Fmax)(L/4), 0
- (3) (Fmax)(L), 0
- (4) (Fmax)(L/4), (Fmax)(L/2)
- (5) (Fmax)(L/2), (Fmax)(L/4)

Table Problem Work Done by the Spring Force

- Connect one end of a spring of length l0 with

spring constant k to an object resting on a

smooth table and fix the other end of the spring

to a wall. Stretch the spring until it has length

l and release the object. - How much work does the spring do on the object

as a function of x l - l0, the distance the

spring has been stretched or compressed?

Worked Example Work Done by Several Forces

- A block of mass m slides along a horizontal

table with speed v0. At x 0 it hits a spring

with spring constant k and begins to experience a

friction force. The coefficient of kinetic

friction is given by m. How far did the spring

compress when the block first momentarily comes

to rest?

Recall integration formula for acceleration with

respect to time

- The x-component of the acceleration of an object
- is the derivative of the x-component of the

velocity - Therefore the integral of x-component of the

acceleration with respect to time, is the

x-component of the velocity

Integration formula for acceleration with respect

to displacement

- The integral of x-component of the acceleration

with respect to the displacement of an object, is

given by - Multiply both sides by the mass of the object

giving integration formula

Work-Kinetic Energy Theorem One Dimensional Motion

- Substitute Newtons Second Law (in one dimension)

- in definition of work integral which then becomes
- Apply integration formula to get work-kinetic

energy theorem

Concept Question Work-Energy

An object is dropped to the earth from a height

of 10m. Which of the following sketches best

represent the kinetic energy of the object as it

approaches the earth (neglect friction)?

- a
- b
- c
- d
- e

Concept Question

- Two objects are pushed on a frictionless

surface from a starting line to a finish line

with equal constant forces. One object is four

times as massive as the other. Both objects are

initially at rest. Which of the following

statements is true when the objects reach the

finish line? - The kinetic energies of the two objects are

equal. - Object of mass 4m has the greater kinetic energy.

- Object of mass m has the greater kinetic energy.
- Not information is given to decide.

Worked Example Work-Energy Theorem for Inverse

Square Gravitational Force

- Consider a magnetic rail gun that shoots an

object of mass m radially away from the surface

of the earth (mass me). When the object leaves

the rail gun it is at a distance ri from the

center of the earth moving with speed vi . What

speed of the object as a function of distance

from the center of the earth?

Power

- The average power of an applied force is the rate

of doing work - SI units of power Watts
- Instantaneous power

Work and the Dot Product

Dot Product

- A scalar quantity
- Magnitude
- The dot product can be positive, zero, or

negative - Two types of projections the dot product is the

parallel component of one vector with respect to

the second vector times the magnitude of the

second vector

Dot Product of Unit Vectors in Cartesian

Coordinates

For unit vectors We have Generally

Dot Product in Cartesian Coordinates

Kinetic Energy and Dot Product

- Velocity
- Kinetic Energy
- Change in kinetic energy

Work Done by a Constant Force

- Definition Work
- The work done by a constant force on

an object is equal to the component of the force

in the direction of the displacement times the

magnitude of the displacement - Note that the component of the force in the

direction of the displacement can be positive,

zero, or negative so the work may be positive,

zero, or negative

Concept Question Work and Gravity 1

- A ball is given an initial horizontal velocity

and allowed to fall under the influence of

gravity near the surface of the earth, as shown

below. The work done by the force of gravity on

the ball is

(1) positive (2) zero (3) negative

Worked Example Work Done by a Constant Force in

Two Dimensions

Force exerted on the object Components Cons

ider an object undergoing displacement Work

done by force on object

Table Problem Work Constant Forces and Dot

Product

An object of mass m, starting from rest, slides

down an inclined plane of length s. The plane is

inclined by an angle of ? to the ground. The

coefficient of kinetic friction is µ.

- Use the dot product definition of work to

calculate the work done by the normal force, the

gravitational force, and the friction force as

the object displaces a distance s down the

inclined plane. - For each force, is the work done by the force

positive or negative? - What is the sum of the work done by the three

forces? Is this positive or negative?

Concept Question Work and inverse square gravity

- A comet is speeding along a hyperbolic orbit

toward the Sun. While the comet is moving away

from the Sun, the work done by the Sun on the

comet is

(1) positive (2) zero (3) negative

Work Done Along an Arbitrary Path

Work done by force for small displacement Work

done by force along path from A to B

Work-Energy Theorem in Three-Dimensions

As you will show in the problem set, the one

dimensional work-kinetic energy theorem

generalizes to three dimensions

Work Path Dependent Line Integral

Work done by force along path from A to B

In order to calculate the line integral, in

principle, requires a knowledge of the path.

However we will consider an important class of

forces in which the work line integral is

independent of the path and only depends on the

starting and end points

Conservative Forces

- Definition Conservative Force If the work done

by a force in moving an object from point A to

point B is independent of the path (1 or 2), - then the force is called a conservative force

which we denote by . Then the work done only

depends on the location of the points A and B.

Example Gravitational Force

- Consider the motion of an object under the

influence of a gravitational force near the

surface of the earth - The work done by gravity depends only on the

change in the vertical position

Potential Energy Difference

- Definition Potential Energy Difference between

the points A and B associated with a conservative

force is the negative of the work done by

the conservative force in moving the body along

any path connecting the points A and B.

Potential Energy Differnece Constant Gravity

- Force
- Work
- Potential Energy
- Choice of Zero Point Choose and choose

. Then - Potential Energy

Worked Example Change in Potential Energy for

Inverse Square Gravitational Force

- Consider an object of mass m1 moving towards

the sun (mass m2). Initially the object is at a

distance r0 from the center of the sun. The

object moves to a final distance rf from the

center of the sun. For the object-sun system,

what is the change in potential during this

motion?

Worked Example Solution Inverse Square Gravity

- Force
- Work done
- Potential Energy
- Change
- Zero Point
- Potential Energy
- Function

Table Problem Change in Potential Energy Spring

Force

- Connect one end of a spring of length l0 with

spring constant k to an object resting on a

smooth table and fix the other end of the spring

to a wall. Stretch the spring until it has length

l and release the object. Consider the

object-spring as the system. When the spring

returns to its equilibrium length what is the

change in potential energy of the system?

Potential Energy Difference Spring Force

- Force
- Work done
- Potential Energy
- Change
- Zero Point
- Potential Energy

Work-Energy Theorem Conservative Forces

- The work done by the force in moving an object

from A to B is equal to the change in kinetic

energy - When the only forces acting on the object are

conservative forces - then the change in potential energy is
- Therefore

Table Problem Asteroid about Sun

- An asteroid of mass m is in a non-circular

closed orbit about the sun. Initially it is a

distance ri from the sun, with speed vi. What is

the change in the kinetic energy of the asteroid

when it is a distance is rf, from the sun?

Next Reading Assignment W05D2

- Young and Freedman 7.1-7.5,12.3
- Experiment 3 Energy Transformations