Lecture 16

General Physics (PHY 2130)

•  Energy

  Conservative and non-conservative forces   Elastic potential energy   Power

http://www.physics.wayne.edu/~apetrov/PHY2130/

Lightning Review

Last lecture: 1.  Work and energy:

  work: connection between forces and energy

  potential energy Review Problem: Tarzan swings on a 30.0-m-long vine

initially inclined at an angle of 37.0° with the vertical. What is his speed at the bottom of the swing (a) if he starts from rest? (b) if he pushes off with a speed of 4.00 m/s?

Review problem

Tarzan swings on a 30.0-m-long vine initially inclined at an angle of 37.0° with the vertical. What is his speed at the bottom of the swing (a) if he starts from rest? (b) if he pushes off with a speed of 4.00 m/s?

yi

Work Done by Varying Forces

• Recall: the work done by a variable force acting on an object that undergoes a displacement is equal to the area under the graph of F versus x

5

x (m)

Fx (N) F3

F2

F1

x3 x2 x1

The work done by F1 is ( )0111 −= xFW

Example: What is the work done by the variable force shown below?

The net work is then W1+W2+W3.

The work done by F2 is ( )1222 xxFW −=The work done by F3 is ( )2333 xxFW −=

Potential Energy Stored in a Spring •  Involves the spring constant (or force constant), k

• Hooke’s Law gives the force

•  F = - k x

•  F is the restoring force •  F is in the opposite direction of x •  k depends on how the spring was formed, the

material it is made from, thickness of the wire, etc.

7

Example: (a) If forces of 5.0 N applied to each end of a spring cause the spring to stretch 3.5 cm from its relaxed length, how far does a force of 7.0 N cause the same spring to stretch? (b) What is the spring constant of this spring?

(a) For springs F∝x. This allows us to write .2

1

2

1

xx

FF

=

Solving for x2: ( ) cm. 9.4cm 5.3N 5.0N 0.7

11

22 =⎟

⎠

⎞⎜⎝

⎛== xFFx

N/cm. 43.1cm 3.5N 0.5

1

1 ===xFk

(b) What is the spring constant of this spring? Use Hooke’s law:

N/cm. 43.1cm 4.9N 0.7

2

2 ===xFkOr

8

Example: An ideal spring has k = 20.0 N/m. What is the amount of work done (by an external agent) to stretch the spring 0.40 m from its relaxed length?

Fx (N)

x (m) x1=0.4 m

kx1

( )( ) ( )( ) J 6.1m 4.0N/m 0.2021

21

21

curveunder Area22

111 ====

=

kxxkx

W

Potential Energy in a Spring • Elastic Potential Energy

•  related to the work required to compress a spring from its equilibrium position to some final, arbitrary, position x

2

21 kxPEs =

( )

.21

20:

220

2,1cos

:,cos

2

0

xkxkxWThus

kxFFFF

butxFW

spr

xx

spr

=−

−=

−=

+=

+==

=

θ

θ

This is called elastic potential energy:

Conservation of Energy including a Spring •  If needed, the PE of the spring is added to both sides of

the conservation of energy equation

• 

fsgisg PEPEKEPEPEKE )()( ++=++

11

Example: A box of mass 0.25 kg slides along a horizontal, frictionless surface with a speed of 3.0 m/s. The box encounters a spring with k = 200 N/m. How far is the spring compressed when the box is brought to rest?

m 11.0

021

210 22

=⎟⎟⎠

⎞⎜⎜⎝

⎛=

+=+

+=+

=

vkmx

kxmv

KUKUEE

ffii

fi

Given: m = 0.25 kg k = 200 N/m vi = 3.0 m/s vf = 0 m/s Find:

x = ?

Idea: we are given velocity, mass and spring constant. Let’s use conservation of energy: kinetic energy pf the box was transformed into elastic potential energy of the spring.

Nonconservative Forces with Energy Considerations

• When nonconservative forces are present, the total mechanical energy of the system is not constant

•  The work done by all nonconservative forces acting on parts of a system equals the change in the mechanical energy of the system

EnergyW ativenonconserv Δ=

Nonconservative Forces and Energy •  In equation form:

• The energy can either cross a boundary or the energy is transformed into a form not yet accounted for

• Friction is an example of a nonconservative force

)()(

)(

iiffnc

fiifnc

PEKEPEKEWorPEPEKEKEW

+−+=

−−−=

Transferring Energy

• By Work • By applying a force • Produces a displacement

of the system

Transferring Energy

• Heat •  The process of

transferring heat by collisions between molecules

Transferring Energy

• Mechanical Waves •  a disturbance propagates

through a medium • Examples include sound,

water, seismic

Transferring Energy

• Electrical transmission •  transfer by means of

electrical current

Transferring Energy

• Electromagnetic radiation •  any form of electromagnetic

waves •  Light, microwaves, radio waves

Problem Solving with Nonconservative Forces

• Define the system • Write expressions for the total initial and final energies • Set the Wnc equal to the difference between the final and

initial total energy •  Follow the general rules for solving Conservation of

Energy problems

Power • Often also interested in the rate at which the energy transfer takes place

• Power is defined as this rate of energy transfer • 

• SI units are Watts (W)

• 

vFtWP ==

2

2

smkg

sJW •==

Power, cont. • US Customary units are generally hp (horsepower)

•  need a conversion factor

• Can define units of work or energy in terms of units of power: •  kilowatt hours (kWh) are often used in electric bills

W746slbft550hp1 ==

22

Example: A race car with a mass of 500.0 kg completes a quarter-mile (402 m) race in a time of 4.2 s starting from rest. The car’s final speed is 125 m/s. What is the engine’s average power output? Neglect friction and air resistance.

watts103.921

5

2

av

×=Δ

=Δ

Δ=

Δ

Δ+Δ=

Δ

Δ=

t

mv

tK

tKU

tEP

f

Given: m=500.0 kg s = 402 m t = 4.2 s vf = 125 m/s vi = 0 m/s Find:

P = ?

Idea: to compute power we need energy and time. Time is given, while energy (kinetic) can be computed

Notice that the distance information was not needed.

Center of Mass

•  The point in the body at which all the mass may be considered to be concentrated •  When using mechanical energy, the change in potential energy is

related to the change in height of the center of mass

Momentum and Collisions

Momentum •  From Newton’s laws: force must be present to change an

object’s velocity (speed and/or direction)   Wish to consider effects of collisions and corresponding

change in velocity

 Method to describe is to use concept of linear momentum

×

scalar vector

Linear momentum = product of mass velocity

Golf ball initially at rest, so some of the KE of club transferred to provide motion of golf ball and its change in velocity

Momentum

• Vector quantity, the direction of the momentum is the same as the velocity’s

• Applies to two-dimensional motion as well

yyxx mvpandmvp ==

vmp =

Size of momentum: depends upon mass depends upon velocity

Impulse

•  In order to change the momentum of an object (say, golf ball), a force must be applied

• The time rate of change of momentum of an object is equal to the net force acting on it

• 

• Gives an alternative statement of Newton’s second law •  (F Δt) is defined as the impulse •  Impulse is a vector quantity, the direction is the same as

the direction of the force

tFporamtvvm

tpF net

ifnet Δ=Δ=

Δ

−=

Δ

Δ= :)(

Graphical Interpretation of Impulse

•  Usually force is not constant, but time-dependent

•  If the force is not constant,

use the average force applied •  The average force can be

thought of as the constant force that would give the same impulse to the object in the time interval as the actual time-varying force gives in the interval

( )i

i it

impulse F t area under F t curveΔ

= Δ =∑

If force is constant: impulse = F Δt

Example: Impulse Applied to Auto Collisions • The most important factor is the collision time or the time it takes the person to come to a rest •  This will reduce the chance of dying in a car crash

• Ways to increase the time • Seat belts • Air bags

  The air bag increases the time of the collision and absorbs some of the energy from the body

ConcepTest

Suppose a ping-pong ball and a bowling ball are rolling toward you. Both have the same momentum, and you exert the same force to stop each. How do the time intervals to stop them compare?

1. It takes less time to stop the ping-pong ball. 2. Both take the same time. 3. It takes more time to stop the ping-pong ball.

ConcepTest

Suppose a ping-pong ball and a bowling ball are rolling toward you. Both have the same momentum, and you exert the same force to stop each. How do the time intervals to stop them compare?

1. It takes less time to stop the ping-pong ball. 2. Both take the same time. 3. It takes more time to stop the ping-pong ball.

 

Note: Because force equals the time rate of change of momentum, the two balls loose momentum at the same rate. If both balls initially had the same momenta, it takes the same amount of time to stop them.

Problem: Teeing Off

A 50-g golf ball at rest is hit by “Big Bertha” club with 500-g mass. After the collision, golf leaves with velocity of 50 m/s.

a)  Find impulse imparted to ball b)  Assuming club in contact with

ball for 0.5 ms, find average force acting on golf ball

Problem: teeing off

Given: mass: m=50 g = 0.050 kg velocity: v=50 m/s Find: impulse=? Faverage=?

1. Use impulse-momentum relation:

2. Having found impulse, find the average force from the definition of impulse:

( )( )smkg

smkgmvmvpimpulse if

⋅=

−=

−=Δ=

50.2050050.0

Nssmkg

tpFthustFp

3

3

1000.5105.0

50.2,

×=

×

⋅=

Δ

Δ=Δ⋅=Δ

−

 

  Note: according to Newton’s 3rd law, that is also a reaction force to club hitting the ball:

( )iiff

ifif

R

VMvmVMvm

orVMVMvmvm

ortFtF

+=+

−−=−

Δ⋅−=Δ⋅

,

, of club

CONSERVATION OF MOMENTUM

Conservation of Momentum • Definition: an isolated system is the one that has no external forces acting on it

• A collision may be the result of physical contact between two objects

•  “Contact” may also arise from the electrostatic interactions of the electrons in the surface atoms of the bodies

Momentum in an isolated system in which a collision occurs is conserved (regardless of the nature of the forces between the objects)

Conservation of Momentum

The principle of conservation of momentum states when no external forces act on a system consisting of two objects that collide with each other, the total momentum of the system before the collision is equal to the total momentum of the system after the collision

Conservation of Momentum • Mathematically:

• Momentum is conserved for the system of objects •  The system includes all the objects interacting with each other • Assumes only internal forces are acting during the collision • Can be generalized to any number of objects

ffii vmvmvmvm 22112211 +=+

Problem: Teeing Off (cont.)

Let’s go back to our golf ball and club problem:

( )( ) sm

kgsmkgvv

smkgvvm

smvmsmkgp

if

if

55.0

50.2so,50.2:Club

50gramm50,50.2:Ball

−=⋅−

=−

⋅−=−

=Δ

=⋅=Δ

factor of 10 times smaller

ConcepTest

Suppose a person jumps on the surface of Earth. The Earth

1. will not move at all 2. will recoil in the opposite direction with tiny velocity 3. might recoil, but there is not enough information provided to see if that could happened

ConcepTest

Suppose a person jumps on the surface of Earth. The Earth

1. will not move at all 2. will recoil in the opposite direction with tiny velocity 3. might recoil, but there is not enough information provided to see if that could happened

 

Note: momentum is conserved. Let’s estimate Earth’s velocity after a jump by a 80-kg person. Suppose that initial speed of the jump is 4 m/s, then:

smkgsmkgV

smkgVMpsmkgp

Earth

EarthEarth

2324 103.5

106320

so,320:Earth320:Person

−×−=×

⋅−=

⋅−==Δ

⋅=Δ

tiny negligible velocity, in opposite direction