Department of Physics and Applied PhysicsPHYS.1410 Lecture 14 Danylov

Lecture 14

Chapter 10

Energy DissipationSpring potential energy

Course website:http://faculty.uml.edu/Andriy_Danylov/Teaching/PhysicsI

Physics I

Energy dissipation

Department of Physics and Applied PhysicsPHYS.1410 Lecture 14 Danylov

Today we are going to discuss:

Chapter 10: Energy Principle with Losses: Section 10.7-8

Chapter 11: Newton’s 2nd law (more general form): Section 11.1 Linear Momentum: Section 11.1 Impulse: Section 11.1

IN IN THIS CHAPTER, you will learn to use the concepts of impulse and momentum.

Department of Physics and Applied PhysicsPHYS.1410 Lecture 14 Danylov

Gravitational Potential EnergyConsider a block sliding down on a frictionless surface under the influence of gravity

x

y

sd

1y gm

1K

2y

2K

dymgWy

yG

2

1

The work done by gravity depends only on coordinates of the final and initial positions,

so gravitational force is conservative

)( 12 yymg

mgyU Gravitational potential energy

)( 12 UU

UWG

mgyUU 0Actually, in general it isReference point

0UReference level

KW U

K2 U2 K1 U1

E K UTotal Mechanical Energy

Conservation of Mechanical Energy

Department of Physics and Applied PhysicsPHYS.1410 Lecture 14 Danylov

The roller-coaster car starts from rest at the top of the hill. The height of the hill is 40 m. Calculate a) the speed of the car at the bottom of

the hill;b) at what height it will have half this

speed.

Roller coasterExample

ConcepTest Water Slide IA) Paul

B) Kathleen

C) both the same

Paul and Kathleen start from rest at the

same time on frictionless water slides

with different shapes. At the bottom,

whose velocity is greater?

Conservation of Energy (for any of them):

fi EE

221 mvmgh ghv 2

Ref. level U=0

ffii UKUK i

f

therefore:

because they both start from the same height (h), they have the same velocityat the bottom.

Department of Physics and Applied PhysicsPHYS.1410 Lecture 14 Danylov

Spring Potential Energy

What is the potential energy of a spring compressed from equilibrium by a distance x?

kxFsp

Uspring 12

kx2

Use a relation between potential energy and work:

Potential energy of a spring

Work done by a spring (from the previous class,

Lecture 13) 22

2 ifsp xxkW Let’s combine them

22

2)( ifif xxkUU

From here you can see that the PE of a spring is

Where x is a displacement from an equilibrium of a spring

Department of Physics and Applied PhysicsPHYS.1410 Lecture 14 Danylov

A 2 kg mass, with an initial velocity of 5m/s, slides down the frictionless trackshown below and into a spring withspring constant k=250 N/m.How far is the spring compressed?

Brick/spring on a trackExample

Department of Physics and Applied PhysicsPHYS.1410 Lecture 14 Danylov

How to solve problems using the energy approach if there are losses of mechanical energy

in a system due to presence of a friction force?

Department of Physics and Applied PhysicsPHYS.1410 Lecture 14 Danylov

Energy conservation with nonconservative forcesConsider an object experiences conservative and nonconservative forces:

NCCnet FFF

Since the work-KE principle works for ANY forces, the total work done on the object

x

y

sd

i

fCF

NCFRecall, from the previous class, if a

force is conservative, then

KWnet

CWU

KWU NC

if KK

NCiiff WUKUK NCif WUU )(

(Work done by friction is usually negative)

KWW NCC

We are going to have a friction force as a nonconservative force

Total Mech Energyat the final point

Total Mech Energyat the initial point

sfW kfr

Mech. Energy converted to Thermal Energy (losses)

NCif WEE

Department of Physics and Applied PhysicsPHYS.1410 Lecture 14 Danylov

A block of mass m slides down a plane of length l and angle α. Find the speed of the block at the bottom of the incline plane assuming that it starts from rest and the coefficient of friction μ is constant.

(Thermal energy is generated)

mgα

α

kf

Block sliding down Inclined PlaneExample

Department of Physics and Applied PhysicsPHYS.1410 Lecture 14 Danylov

Review

mgyU Gravitational potential energy

iiff UKUK

E K UTotal Mechanical Energy

Conservation of Mechanical Energy

221 mvK Kinetic energy

Uspring 12

kx2Potential energy of a spring

NCiiff WUKUK

Without losses (no friction)

With losses (with friction)

ConcepTest Water Slide II

Paul and Kathleen start from rest at

the same time on frictionless water

slides with different shapes. Who

makes it to the bottom first?

Even though they both have the same final velocity, Kathleen is at a lower height than Paul for most of her ride. Thus, she always has a larger velocity during her ride and therefore arrives earlier!

A) Paul

B) Kathleen

C) both the same

http://phys23p.sl.psu.edu/phys_anim/mech/ramped.avi Ref. level U=0

Department of Physics and Applied PhysicsPHYS.1410 Lecture 14 Danylov

Thank you