WORK & ENERGYReview and then some…

Work & EnergyConservative, Non-conservative, and non-constant Forces

Energy Defined again

Total Mechanical Energy Etotal= PE + KE

PE = mgh

KE = ½ mv2

It is always transferred and so the total energy in a closed system is conserved.

The conservation of Mechanical Energymgh1 + ½ mv1

2 = mgh2 + ½ mv22

Work Energy TheoremWork to get an object to change velocity is the work-energy theoremWork = DKE, F • Dd = ½ mvf

2 - ½ mvi2

Since Work, Potential, and Kinetic energies, are measured in Joules, they can transfer.

Conservative ForcesTotal mechanical energy can usually be solved with

these two main equations.

FDd = ½ mvf2 - ½ mvi

2

mghi + ½ mvi

2 = mghf + ½ mvf

2

Constant Force – distance GraphConsider the equation for work…

In your notes, predict what a Force vs. Displacement (Work) graph looks like if the force is constant.

Can you tie the graph to the equation?

Springs

What would happen to force as you pull on a spring?

Spring – Work GraphF

x

Work

d

F = kx

The force changes as the displacement changes.

It takes more force to stretch the spring a big distance rather than a small distance… so we get a triangle.

Graph to EquationWhat do you think the equation for work done by a

spring is? Think about the graph.What does the area under the curve represent?What shape does this make?Spring force (Fs) is equal to what? (use k for slope)

Ws= ½Fsx

Fs= kx, so Ws= ½kx2

Work and SpringsSprings do work but they don’t have a constant

force.

For a spring:F = -kx (Hooke’s law)

k is the spring constant for that spring (N/m)x is how much the spring is stretched/compressed

(aka Δx)F is the restoring force

Spring FunA 1.50kg object hangs motionless from a

spring with a force constant of k = 250N/m. How far is the spring stretched from its equilibrium length?

-0.0589m

Spring Fun x2A backpack full of books weighing 52.0N rests on a

horizontal table. A spring with a force constant of 350N/m is attached to the backpack and pulled horizontally. If the spring is pulled until it stretches 2.00cm and the

pack remains at rest, what is the magnitude of the force of friction keeping the backpack in place?

7.00NThe backpack begins to slide just as the spring is

stretched beyond 2.00cm. What is the coefficient of static friction?

0.135

Spring WorkA 1.20kg block is held against a spring with

spring constant (k) = 1.00x104N/m, compressing it a distance of 0.150m. How fast is the block moving after it is released and the spring pushes it away (the instant it is no longer touching the spring)?

13.7m/s

Spring Potential EnergyIf you compress or stretch a spring and hold

it there, it is considered PE (stored energy).

Soooooooooooooo:

Ws= ½ Fsx = ½ kx2 = PEs

Spring Potential PracticeWhen a force of 120.0N is applied to a

certain spring, it causes a stretch of 2.25cm. What is the potential energy of this spring when it is compressed by 3.50cm?

Clue?Solve for k

k = 5330N/mPEs = 3.26J

Conservation of EnergyYour total initial energy MUST equal your

total final energy. There is no way around this fact!

E = E’Without Friction:

PE + KE = PE’ + KE’With Friction:

PE + KE = PE’ + KE’ + Loss to Friction (in J)

GraduationFor graduation, you install a spring with k=100.0 N/m

under your 0.120kg cap. The spring is compressed 15.5cm.When released your cap is launched straight upward with

what initial velocity?How high above the release point will your cap travel,

assuming no friction?If it reaches a height of only 0.85m, what energy was “lost” to

air friction?

v = 4.47m/sh = 1.02m

Estart=1.20J, Etop = 1.00J, Efriction = 0.20J

More?

A 1.9-kg block slides down a frictionless ramp, as shown. The top of the ramp is 1.5 m above the ground; the bottom of the ramp is 0.25 m above the ground.

Suppose the ramp is not frictionless. Find the distance d for the case in which friction on the ramp does -9.7J of work on the block before it becomes airborne.

Solution

nc B B A AW K U K U Determine the horizontal velocity at point B (end of ramp) using energy.

v = 3.78

d = 0.85m

Now use 2-d motion to determine d.