Announcements

• CAPA Set #9 due Friday at 10 pm

• This week in Section: Lab #3 – Momentum

• Reading – Chapter 7 on Momentum

Two paths lead to the top of a big hill. Path #1 is steep and direct and Path #2 is twice as long but less steep.

Both are rough paths and you push a box up each.

How much more potential energy is gained if you take the longer path?

A) none B) twice as much

C) four times as much D) half as much

h hd 2d

d 2 h2 4d 2 h2

#1 #2θ φ

PE mgh in both cases.

Clicker Question Room Frequency BA

Consider the same two paths as in the previous problem. Is the amount of work done the same along each path?A) YesB) No, more work is done on the steeper path.C) No, more work is done on the longer path.D) Probably not, but the details are needed to know which is larger.

|W f | Ffd Nd (mgcos )d

mgd 2 h2

dd mg d 2 h2

h hd 2d

d 2 h2 4d 2 h2

#1 #2θ φ

|W f | Ffd N (2d) (mgcos )(2d)

mg(2d)2 h2

2d2d mg 4d 2 h2

Clicker Question Room Frequency BA

Force = -kxHooke’s “Law”

k = “spring constant” units = N/mx = displacement of spring

SpringsSpring in “relaxed” position

exerts no force

Compressed spring pushes back

Force

Stretched spring pulls back

Force

PE = ½ kx2

Stored Energy in Springs

Compressed spring has elastic potential energy

Stretched spring also has elastic potential energy

If no change in any other energy, then DPE = Wext =Fspring x d

However Fspring = -kx changes as we stretch the spring

Use average Fspring = -½ kx

Elastic potential energy:

F = -kx, PE = ½kx2

A force of magnitude F stretches a spring through a displacement x. The force is then increased so that displacement is doubled.

What is the magnitude of force needed to double the displacement and what will be the effect on the potential

energy of doubling displacement?

A) Force doubles, PE doubles.B) Force doubles, PE quadruples.C) Force quadruples, PE doubles.D) Force doubles, PE halves.E) Force quadruples, PE quarters.

Clicker Question Room Frequency BA

Which of the following types of potential energy can be negative?

A) Gravitational Potential Energy

B) Elastic Potential Energy

C) Both of the above

D) Neither of the above

PEelastic 1

2kx2 0

Clicker Question Room Frequency BA

PEgravity = mgy

A block of mass m is released from rest at height H on a frictionless ramp. It strikes a spring with spring

constant k at the end of the ramp.

How far will the spring compress (i.e. x)?

KEi + PEi + Wfrict + Wexternal = KEf + PEf

0 0

0 + mgH + 0 + 0 = 0 + (0 + ½ kx2)

k

mgHx

2

gravitationalelastic

Elastic Potential Energy – Does Not Have to Look Like a Spring

What is Power?

P average power = work

time

The average power can be written in terms of the

force and the average velocity:

Power is a rate of energy per time

Units are Joules / second = Watts

Raise a box of mass 2 kg at constant velocity by a height of 6 meters over a time interval of 2 seconds.

6m

2 kg

How much energy is required to do the work?

Wext = DPE = mgh since DKE=0

Wext = 2 kg x 9.8 m/s x 6 m = 117 Joules

How much power is required to do the work in that amount of time?

Power = Wext / time = 117 J / 2 s = 59 Watts

Also Wext = Fext x d Fext = 117J/6m = 19.6 N

Also Power = Fext v = 19.6 N x (6m/2s) = 59 Watts

The average household in the USA uses 2.7 billion Joules of electrical energy each month.

Approximately how many 100 Watt lightbulbs would need to be left on all the time to amount to that energy

consumption? xxA) 1 lightbulbB) 10 lightbulbsC) 100 lightbulbsD) 1000 lightbulbsE) 10,000 lightbulbs

Clicker Question Room Frequency BA

A kiloWatt-hour is a unit of energy (Power x time).1 kW-hour = 3.6 x 106 Joules

Energy = Power x time = (10 x 100 W) x (30x24x60x60 s)

≈ 2.6 x 109 Joules

m1 m2

v1iv2i

Before the collision:

After the collision:

m1

v1f m2

v2f

v1f = final velocity of mass 1

v2f = final velocity of mass 2

Collisions

These collisions should obey Newton’s Laws

Total kinetic energy remains constant during the collision

KE is conserved or ΔKE = 0

No energy converted to thermal or potential energy

Elastic Collision

ffii KEKEKEKE 2121 2

222

112

222

11 2

1

2

1

2

1

2

1ffii vmvmvmvm

Macroscopic objects never have perfectly elastic collisions but it’s often a good approximation.

Inelastic Collision: KE is not constant; some is converted to another kind of energy;

typically heat or deformation.

Totally Inelastic Collision: KE is not conserved, and the objects stick together after collision.

Demonstration

Kind of like kinetic energy (associated with motion).

However, not in units of Joules (instead kg m/s) and so not an energy.

In addition to total energy being conserved in every collision, the total momentum vector is also conserved!

momentum p

mv (a vector)

p

1i p

2i p

1 f p

2 fConservation of Momentum

mv1i mv2i mv1 f mv2 f(in one-dimension)

New Conserved Quantity Momentum