Potential Energy andConservation of Energy

PHY11 Lesson 2

Potential Energy U

potential energy (U) – energy associated with the position or arrangement of a system of one or more objects.

• gravitational potential energy (U) – energy associated with the state of separation between two objects.

• elastic potential energy (U) – energy associated with the state of compression or extension of an elastic object.

Work and Potential Energy Consider an object of mass m thrown upward with an initial velocity vo and initial kinetic energy Ko =½ mvo

2.

As the object rises to a height d the work done by the gravitational force Fg is

mgdWg As the object rises, its kinetic energy decreases.

The decrease in kinetic energy of the object is transferred to its gravitational potential energy.

vo

Fg

Ko

K< Ko

v<vo

Fgd

WU

Work and Potential Energy (Elastic) Consider a block of mass m attached to an ideal spring. The block is sent moving to the right.

x =0

m

mFs

x =0

mFs

x =0

As the block moves to the right, the work done on it by the spring force is negative. The kinetic energy of the block decreases.

The decrease in kinetic energy of the block becomes the increase in the elastic potential energy of the spring.

WU

As the block moves to the left, the work done on it by the spring force is positive and U is negative..

WU

Conservative Forces

conservative force – a force that, when it acts on an object/system, does not change the total energy (K+U) of the object/system.

KWg

Consider an object that is thrown vertically upward.

)( oo UUKK

Fg

2

21

oo mvK

0 oo mgyU

Fgy

0221 mvK mgyU

UWg

UKUK oo

Conservation of Mechanical Energy

Work Done by a Conservative Force in a Closed Path

Consider an object thrown vertically upward.

On its way upward, the work done by gravity is

Fg

Fg

2

21

oo mvK

0 oo mgyU

y

0221 mvK mgyU UWg KWg

Work Done by a Conservative Force in a Closed Path

Consider an object thrown vertically upward.

On its way upward, the work done by gravity is

Fg

221 mvK

0mgyU

yFg

02

21 oo mvK

oo mgyU

On its way back down, the work done by gravity is

The net work done by gravity is

0 gg WW

The net work done by a conservative force on a particle moving around any closed path is zero.

UWg KWg

UWg KWg

Potential Energy

Recall the work done by a general force is x

x xo

dxFW

UW

The change in potential energy of an object is

Gravitational Potential Energy

)()( o

y

y

y

yyymgdymgdymgU

o

f

i

Consider an object moving vertically (along y-axis). The change in it potential energy is

)( oo yymgUUU

x

x xo

dxFU

mgyyU )(

Elastic Potential Energy

x

x

x

x oo

xdxkdxkxU )(

Consider a block-spring system that is displaced from the equilibrium position.

2

2

1)( kxxU

UW The work done by the spring force is

22

2

1

2

1okxkxU

Work Done on a System by an External Force

Consider an external force F applied to a body of mass m which is on a horizontal and frictionless plane.

m F

vo

m F

d

v

Work is the energy transferred to or from a system by means of an external force acting on that system.

Work Done (No Friction)

The work done by the external force is

UKW

Work Done (With Friction)

Consider an external force F applied to a body of mass m which is on a horizontal and frictionless plane.

UKWW f

m F

vo

f m F

d

v

f

The work done on the system is

UKfdFd )(

fWUKW

ANS: (a)0, 0.98 J (b) 2.8 m/s (c) constant: weight not constant: N and f (d) 5.1 N

Sample Problem 1

ANS: 1.72 m

Sample Problem 2

ANS: (a) 3.11 m/s (b) 0.821 m

Sample Problem 3

ANS: 42 m/s

Sample Problem 4

ANS: 42 m/s

Sample Problem 4