WorkWork--Energy Theorem. Energy Theorem.

• Work done by Force

• Energy

1

• Work-Energy Theorem

• “Lazy” forces

Work. Work. 1D.1D.

Units:Units: [W]=Nm=kg m2/s2=J(oule)

Constant ForceConstant Force

2

Units:Units: [W]=Nm=kg m2/s2=J(oule)

Variable ForceVariable Force

Elementary work

Total work

=AREA

Work. Work. Sample Problems.Sample Problems.

1. An apple falls on Sir Isaac Newton sitting under an apple tree.

Find the work done by gravity on the apple, whose weight is 1N.

How much work would Newton do, if he would slowly lift the apple

from the ground to his mouth?

3

2. How much work would you have to do to slowly stretch a (relaxed) spring

with spring constant k by distance x?

1m

1m

0.1kg

Energy. Energy. Definition and Examples.Definition and Examples.

EnergyEnergy = ability to do work

1. Kinetic energy - due to motion

2. Potential energy - due to interaction with other objects

Today!

Next time

Units:Units: [E]=J

4

i) Heavier objects have more K

ii) Faster objects have more K

Exercise: (GO LIONS!)Find kinetic energy of a football (m = 0.4kg) flying with a speed of 100 km/h.

Compare to kinetic energy of a football player (m = 80kg) running with a speed of 10 km/h.

Work Work –– Energy Theorem. Energy Theorem.

x

y From Newton’s II Law:

From kinematics (on x-axis):

5

From kinematics (on x-axis):

• Work done by a force changes kinetic energy of the block

• Block can do work spending its kinetic energy

WorkWork--Energy Theorem. Energy Theorem. Sample Problems.Sample Problems.

1. From what height does a football need to fall

to have a speed of 100 km/h?

6

2. A block sliding across a frictionless table

encounters a region of rough surface with the coefficient

of friction µ. What was the initial speed of the block if it

traveled distance d before a complete stop?

WorkWork--Energy Theorem. Energy Theorem. Sample Problems.Sample Problems.

3. A bullet (m=20g) shot with a speed of 800 m/s hits an oak tree

and sticks 4cm inside it.

What is the average force acting on the bullet from the tree?

7

WorkWork--Energy Theorem. Energy Theorem. Summary.Summary.

Forces doing Forces doing positivepositive work speed objects up (work speed objects up (increaseincrease kinetic energy)kinetic energy)

Forces doing Forces doing negativenegative work slow objects down (work slow objects down (decreasedecrease kinetic energy)kinetic energy)

8

Can friction ever do a positive work? Can friction ever do a positive work?

Generalization to 3Generalization to 3dd. . Dot Product.Dot Product.

Dot productDot product

9

Exercise: You are pushing a heavy box across the floor. Draw a free body diagram.

Identify forces doing positive, negative and zero work.

Generalization to 3Generalization to 3dd. . Sample problems.Sample problems.

Sisyphus was cursed to roll a huge boulder up a hill.Sisyphus was cursed to roll a huge boulder up a hill.

Assuming the hill to be ideally Assuming the hill to be ideally

flat and the boulder toflat and the boulder to

10

How much work does the force of gravity do? (assume angle How much work does the force of gravity do? (assume angle θθ))

What about the normal force?What about the normal force?

flat and the boulder toflat and the boulder to

be ideally round, neglect friction be ideally round, neglect friction

and compute work doneand compute work done

by Sisyphus when he pushes the by Sisyphus when he pushes the

boulder (mass boulder (mass mm) by ) by

distance distance dd..

Gyroscopic (Lazy) forces. Gyroscopic (Lazy) forces.

Forces which are always perpendicular to velocity(e.g. magnetic force on a moving charged particle, Coriolis’ force)

For a very short (infinitesimal) time ∆t,

instantaneous displacement is .

Elementary work

11

Elementary work

Gyroscopic forces do Gyroscopic forces do NOTNOT change kinetic energy (speed).change kinetic energy (speed).

They only rotate the velocity of the object.They only rotate the velocity of the object.

Exercise: Does this conclusion agree with kinematics?

Next time. Next time.

1. Conservative and non-conservative forces.

2. Potential Energy

12

3. Energy Conservation Law .