Graphical Integration

ENGR 103 Project #3

Graphical Integration

First Law of Thermodynamics

• Energy can be converted from one form to another, but total energy remains

constant (energy cannot be created or destroyed).

http://www.googlelearn.com/first-law-of-thermodynamics/

Forms of Energy

• Potential Energy

• Spring PE = ½ kx2

• Gravitational PE = mgz

• Kinetic Energy

• KE = ½ mv2

• How does a toy dart gun work?

Dart Gun

• Stored Energy in Spring by applying a force through a displacement (work)

• Converted potential energy in spring to kinetic energy in dart and moving

components of gun

• Converted KE dart into PE dart

• Where did the rest of the energy go?

• Where did we get the magic formula “Spring PE = ½ kx2” ?

Spring PE = ½ kx2

Work

• For a constant force applied over a displacement

• Work = Force • displacement

• Notice it is the area under the curve

Work on a Spring (Stored PE)

• Work = force x displacement, but

• Force in spring is a function of displacement

• Constant slope = k (linear spring)

• Work compressing spring = area under the curve

• Force = x • k

• Therefore Work (Area of Triangle)

PE = ½ kx2

Work on a non-linear spring?

• Area under the curve

• Don’t have a formula?

• No problem Graphical Integration!

A non-linear spring in action

Technology is Wonderful

• Cave Man Rod (tree limb)

• Bait = 0.06 lbf

• Velocity just before release = 32 ft/sec

• k = Very high (can neglect)

• Energy in lure at point of release?

• High Tech Graphite Rod

• Bait = 0.06 lbf with velocity just before the rod “snaps” of 44 ft/s (Graphite is lighter than a stick!)

• Increased Velocity of bait at release from rod PE (“unloading” or “snapping” of the rod)

• k = Not constant (non-linear spring)

• Spring energy in rod just before point of release?

• What effect will this have on bait speed (casting distance)?

Spring Constant of Rod

Deflection[ft] Force [lbf]

0 0

0.1 0.015

0.2 0.12

0.3 0.405

0.4 0.96

0.5 1.875

0.6 3.24

0.7 5.145

0.8 7.68

0.9 10.935

1 15

1.1 19.965

0

5

10

15

20

25

0 0.2 0.4 0.6 0.8 1 1.2

F o rc

e

Rod Deflection

Fishing Rod Stiffness

Energy 1. Calculate the spring energy stored in the rod just prior to release of the bait as a function of the

bend in the rod (an expert caster will create maximum deflection in the rod).

2. Calculate the speed of the bait assuming all the spring energy in the rod is transferred to the lure as Kinetic Energy. Assume the velocity imparted by the cast motion before the rod “snaps” is 44 ft/s, so the total energy of the lure is KE @ 44 ft/s + spring energy stored in rod.

3. Using a projectile calculator http://www.mrmont.com/teachers/physicsteachershelper-proj.html Calculate the cast distance of a caveman rod versus a modern rod.

Grading

• Documentation including neatness 25%

• Calculations 25%

• Sig Figs & units 10%

• Q1 Rod Energy 15%

• Q2 Bait speed 15%

• Q3 Cast distance 10%