Example: A 20 kg block is fired horizontally across a frictionless surface. The block strikes a platform that is attached to a spring at its equilibrium position. If the spring has a spring constant of 1200 N/m and is compressed by 10 cm before the block stops, what speed did the block strike the platform with?

v

x

x

ss xdFW0

20

2

2

1

2

1kxkx

20

2

2

1

2

1mvmvKEW

2

022

02

2

1

2

1

2

1

2

1kxkxmvmv

Work done by the block on the spring

Work done on the block by the spring to change the speed of block

0022

0 2

1

2

1kxmv

m

kxv

2

0

sWW Work done on a system is considered positive work - system gains energyWork done by a system is considered negative work - system loses energy

s

m77.0

Power

It is sometime more useful to discuss the rate at which the energy is transferred. This is called power. Whenever you hear rate it means changing with respect to time.

t

E

t

WP

dt

dEP

dt

dUP

dt

rdF

dt

rdF

vF

P – Power [W]W – Work [J]DE – Change in energy [J]Dt – Time interval [s]

[P] = W = Watts = J/s = Nm/s = kgm2/s3

1 horsepower (hp) = 746 W

CH 6: Potential Energy and energy

Conservation

Potential EnergyWhat is potential energy? Energy stored in a system

The energy is stored when conservative forces do work on a system. The energy can be released and then stored again without the system permanently losing energy to the environment.

Gravitational Potential Energy

We will discuss two common conservative forces, gravitational forces and elastic forces.

Work is done against gravity to store energy in a system. The force, which is equal to the gravitational force, must be applied to the system to move it a vertical distance against gravity.

y

y

g ydFW0

y

y

gdyF0

cos1

y

y

g dyF0

y

ymgy

0 0yymg

0If we define the initial vertical position to be zero.

00 hhmgyymgUW g

mghmgyU g

Work is done to change the potential energy of the system.

Ug – Gravitational potential energy [J]m – mass [kg]g – gravitational acceleration [m/s2]h – height above what is defined to be zero height [m]

This expression is only valid near the surface of the Earth!

Example: Determine the potential energy of the 10 kg cart for each case.a) The cart is at point A and the reference is point E.b) The cart is at point B and the reference is point E.c) The cart is at point A and the reference is point C.d) The cart is at point E and the reference is point D.

A

B

C

D

E

A is 20 m above EB is 10 m above EC is 15 m above ED is 5 m above E

a) EAg hhmgU mmmg 020 mmg 20 J2000

b) EBg hhmgU mmmg 010 mmg 10 J1000

c) CAg hhmgU mmmg 1520 mmg 5 J500

d) DEg hhmgU mmmg 50 mmg 5 J500

Elastic Potential Energy

Energy is stored in a spring when a force is applied against the restoring force of the spring to stretch or compress the spring. The force required to stretch the spring would be equal in magnitude to the restoring force.

x

x

xdFW0

cos

0

x

x

sdxF x

x

dxkx0

x

x

dxkx0

x

x

xk0

2

2

1

202

2xx

k

20

2

2

1

2

1kxkxUW s

0

If we define x0 to be the equilibrium position which is zero.

Work is done to change the potential energy of the system.

2

2

1kxU s

Us – Elastic potential energy [J]k – spring constant [N/m]x – distance from equilibrium [m]

The result obtained for the elastic potential energy describes a similar situation as can be described for gravitational potential energy. The work done by an applied force to move an object a distance from their equilibrium position against a pre-existing force will store energy within the specified system.

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