Physics I 95.141 LECTURE 13 10/20/10

Department of Physics and Applied Physics95.141, F2010, Lecture 13

Physics I95.141

LECTURE 1310/20/10


Circular Motion Problem

• (A) What is the centripetal acceleration/Force of/on the bullet? (5pts)

• (B) Where does this Force come from? (95 pts)


Circular Motion Problem

• R~1300m• mbullet=.03kg

• vbullet=300m/s0.15m

40m


Lecture 12 Review• Translational Kinetic Energy

• Work Energy Theorem– The net work done on an object corresponds to the change in

translational kinetic energy of that object (as long as this energy does not go into internal energy…compressed spring, for instance)

• Conservative vs. Non-Conservative Forces– The work done by a conservative force to move an object from

point A to B depends only on the position of A and B, not path or velocity.

2

21mvKE

21

22 21

21 mvmvKEWWnet


Potential Energy

• In the last class we defined Energy as the ability to do work.

• In particular, we discussed Translational Kinetic Energy, the energy associated with motion.

• However, there are numerous other types of Energy– We know batteries can do work– We know a coiled spring can do work– We know a mass at some height, attached to a pulley can do

work

• All of these are examples of systems that have the potential to do work, and we can associate with them a potential energy.


Gravitational Potential Energy

• Say we start with a mass, and raise it, at constant velocity, to a height h.

• How much work do we do?

• How much work does gravity do?



• The block now has the potential to do work….• Say we drop the brick, at y=0, we can find the

Kinetic Energy of the brick by the work-energy theorem:



• At y=0, the block can do an amount of work equal to it’s kinetic energy

• Imagine the brick being used to drive a stake into the ground:



• Summary– Raising the brick gives it the potential to do work, that

potential energy is given by:

– As the brick falls, its potential energy is converted into kinetic energy



• We assign the letter U to the gravitational potential energy

• The change in gravitational potential energy is then:



• The gravitational potential energy is associated with the Force between the Earth and the object.

• How do we determine what y is?

• It is the change in Potential Energy that we are usually concerned with


Example Problem• A 1000kg roller coaster moves from point 1 to points 2 and 3. • What is the potential energy of the roller coaster at points 2 and 3

relative to point 1?• What is the change in potential energy from points 2 to 3?


Example Problem• A 1000kg roller coaster moves from point 1 to points 2 and 3. • What is the potential energy of the roller coaster at points 2 and 3

relative to point 3?• What is the change in potential energy from points 2 to 3?


General Potential Energy

• Gravitational potential energy is defined as:– The negative of the work done by gravity when the

object moves from height y1 to y2.

• In general, we can define the change in potential energy associated with a particular Force F as the negative of the work done by that Force.


Elastic Potential Energy

• What is the potential energy of a spring compressed from equilibrium by a distance x?


Relating Force and Potential Energy

• Say we are given a Force as a function of position

• We can then write the change in potential energy associated with this (conservative) Force as:


Example

• Suppose we are given the potential energy as:

• What is the Force F as a function of x for this potential energy?

bxeAxxU 2)(


3D Example

• In 3D

• So if

kzUj

yUi

xUzyxF ˆˆˆ),,(

xzxyzyxU 43),,(


Conservation of Energy

• For a conservative system (only conservative forces do work) where energy is transformed between kinetic and potential

• Work Energy Principle

• Relation between potential energy and work:



• Combining the work-energy principle and our definition of potential energy, we see that:

• We can define the total mechanical energy of the system as:

• We can then see that the total energy of the system

• Is constant!



• As long as no non-conservative forces do work, the total mechanical energy of the system is a conserved quantity!

• Principle of conservation of mechanical energy:– If only conservative forces are doing work, the total

mechanical energy of a system neither increases or decreases in any process. It is conserved.


Power of Energy Conservation

• Imagine dropping a mass m from a height h above the ground.

• Solve for speed of the mass at the ground using our equations of kinetic motion


Power of Energy Conservation• Imagine dropping a mass m from a height h

above the ground.• Now solve for speed of the mass at the ground

using energy conservation



• But now, imagine sliding a mass m released from rest on the frictionless track shown below.

• Solve for speed of the mass at the ground using our equations of kinetic motion

h



• But now, imagine sliding a mass m released from rest on the frictionless track shown below.

• Solve for speed of the mass at the ground using conservation of energy

h



• We can even solve this if the mass is given an initial velocity vo.

• Solve for speed of the mass at the ground using conservation of energy

h


Example Problem• A 2 kg mass, starting at rest, slides down the frictionless

track shown below and into a spring with spring constant k=250N/m. How far is the spring compressed by the mass?

2m


Example Problem (Easier)• A 2 kg mass, starting at rest, slides down the frictionless

track shown below and into a spring with spring constant k=250N/m. How far is the spring compressed by the mass?

2m


Example Problem• A 2 kg mass, with an initial velocity of 5m/s, slides down

the frictionless track shown below and into a spring with spring constant k=250N/m. How far is the spring compressed by the mass?

2m


Spring Energy

• What is spring constant of catapult?• What is energy stored in spring?• what is Kinetic Energy of Watermelon?• What is velocity of watermelon?

• Assume– MassWoman=65kg– MassMelon=2kg– Δx=1.5m


Spring Energy

• What is spring constant of catapult?

1.5mθ=30°


Spring Energy• Need to know Force!• Free body diagram…

• With Force, we can now find k!


Spring Energy

• What is energy stored in spring?• What is Kinetic Energy of Watermelon?


Spring Energy

• What is velocity of watermelon?• Assume

– MassMelon=2kg


What does Energy vs Time look like?

US

K

E


What Did We Learn Today?

• Potential Energy

• Conservation of Mechanical Energy

• Concept of Energy Conservation is a powerful way to approach what might seem like complex problems!

Physics I 95.141 LECTURE 13 10/20/10

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