Physics for Scientists and · PDF filePhysics for Scientists and Engineers ... mgh mv mg R K U K U v gR R v mg m i gi f gf 2 2.5 2 1 ... When the trigger is pulled,

Spring, 2008 Ho Jung Paik

Physics for Scientists and Engineers

Chapter 10Energy

31-Mar-08 Paik p. 2

Introduction to EnergyEnergy is one of the most important concepts in science although it is not easily defined

Every physical process that occurs in the Universe involves energy and energy transfers

The energy approach to describing motion is particularly useful when the force is not constant

An approach will involve Conservation of EnergyThis could be extended to biological organisms, technological systems and engineering situations

31-Mar-08 Paik p. 3

Energy of Falling Object

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ifif

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−−=

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)(21

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Kinematic eq. Gravity is the only force acting

Multiplying & putting ,

Define : kinetic energy: gravitational potential energy

Then, : Conservation of Energy

31-Mar-08 Paik p. 4

Kinetic EnergyKinetic energy is the energy associated with the motion of an object

Kinetic energy is always given by K = ½mv 2, where v is the speed of the object

Kinetic energy is a scalar, independent of direction of vKinetic energy cannot be negative

Units of energy: 1 kg m2/s2 ≡ 1 JouleEnergy has dimensions of [M L2 T−2]

31-Mar-08 Paik p. 5

Potential EnergyPotential energy is the energy associated with the relative position of objects that exert forces on each other

A potential energy can only be associated with specific types of forces, i.e. conservative forces

There are many forms of potential energy: gravitational, electromagnetic, chemical, nuclear

Gravitational potential energy is associated with the distance above Earth’s surface: Ug = mgy

31-Mar-08 Paik p. 6

Total Mechanical EnergyThe sum of kinetic and potential energies is the mechanical energy of the system: E = K + ∑Ui

When conservative forces act in an isolated system, kinetic energy gained (or lost) by the system as its members change their relative positions is balanced by an equal loss (or gain) in potential energy

This is the Conservation of Mechanical EnergyIf there is friction or air drag, the total mechanical energy is not conserved, i.e. constant

31-Mar-08 Paik p. 7

Gravitational Potential EnergyThe system consists of Earth and a book

Initially, the book is at rest (K = 0) at y = yb (Ug = mgyb)

When the book falls and reaches ya,Ka + mgya = 0 + mgyb

⇒ Ka = mg (yb - ya) >0

The kinetic energy that was converted from potential energy depends only on the relative displacement yb - ya

It doesn’t matter where we take y = 0

31-Mar-08 Paik p. 8

Example 1: Launching a PebbleBob uses a slingshot to shoot a 20.0 g pebble straight up with a speed of 25.0 m/s.

How high does a pebble go?

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31-Mar-08 Paik p. 9

Energy Conservation in 2-D

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31-Mar-08 Paik p. 10

Example 2: Downhill SleddingChristine runs forward with her sled at 2.0 m/s. She hops onto the sled at the top of a 5.0 m high, very slippery slope.What is her speed at the bottom?

Notice that (1) the mass cancelled out and (2) only Δy is needed. We could have taken the bottom of the hill as y = 10.0 m and the top as y = 15.0 m, and gotten the same answer.

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Energy Conservation: PendulumAs the pendulum swings, there is a continuous exchange between potential and kinetic energies

At A, all the energy is potential

At B, all of the potential energy at A is transformed into kinetic energy

Let zero potential energy be at B

At C, the kinetic energy has been transformed back into potential energy


Example 3: Ballistic PendulumA 10 g bullet is fired into a 1200 g wood block hanging from a 150 cm long string. The bullet embeds itself into the block, and the block then swings out to an angle of 40°.What was the speed of the bullet?


Example 3, contThis problem has two parts: (a) the bullet and the block make a perfectly inelastic collision, in which momentum is conserved, and (b) the total mass as a system undergoes a pendulum motion, in which energy is conserved.

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Example 4: Roller Coaster

A roller-coaster car is released from rest at the top of the first rise and then moves freely with negligible friction. The roller coaster has a circular loop of radius Rin a vertical plane. (a) Suppose first that the car barely makes it around the loop:

at the top of the loop the riders are upside down and feel weightless, i.e. n = 0. Find the height of the release point above the bottom of the loop, in terms of R so that n = 0.


Example 4, cont

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Example 4, cont(b) Show that the normal force on the car at the bottom of the

loop exceeds the normal force at the top of the loop by six times the weight of the car.

The normal force on each rider follows the same rule. Such a large normal force is very uncomfortable for the riders. Roller coasters are therefore built helical so that some of the gravitational energy goes into axial motion.

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Hooke’s LawForce exerted by a spring is always directed opposite to the displacement from equilibrium

Hooke’s Law: Fs = −kΔsΔs is the displacement from the equilibrium positionk is the spring constant and measures the stiffness of the springF is called the restoring force

If it is released, it will oscillate back and forth


Measuring the Force Constant

( )

. determine to and Measure

N/m

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Energy of Object on a Spring

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Elastic Potential EnergyPotential energy of an object attached to a spring is

Us =½ k(Δs)2

The elastic potential energy depends on the square of the displacement from the spring’s equilibrium position

The same amount of potential energy can be converted to kinetic energy if the spring is extended or contracted by the same displacement


Example 5: Spring Gun ProblemA spring-loaded toy gun launches a 10 g plastic ball. The spring, with spring constant 10 N/m, is compressed by 10 cm as the ball is pushed into the barrel. When the trigger is pulled, the spring is released and shoots the ball out.What is the ball’s speed as it leaves the barrel? Assume friction is negligible.

( ) ( ) ( )

m/s 16.3kg 010.0

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Example 6: Pushing ApartA spring with spring constant 2000 N/m is sandwiched between a 1.0 kg block and a 2.0 kg block on a frictionless table. The blocks are pushed together to compress the spring by 10 cm, then released. What are the velocities of the blocks as they fly apart?


Example 6, cont

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Example 7: Spring and Gravity

A 10 kg box slides 4.0 m down the frictionless ramp shown in the figure, then collides with a spring whose spring constant is 250 N/m. (a) What is the maximum compression of the spring? (b) At what compression of the spring does the box have its

maximum velocity?


Example 7, cont(a) Choose the origin of the

coordinate system at the point of maximum compression. Take the coordinate along the ramp to be s.

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Example 7, cont(b) For this part of the problem, it

is easiest to take the origin at the point where the spring is at its equilibrium position.

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Elastic CollisionsBoth momentum and kinetic energy are conserved (U = 0)

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+=+

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Typically, there are two unknowns (v2f and v1f), so you need both equations: pi = pf and Ki = Kf

The kinetic energy equation can be difficult to useWith some algebraic manipulation, the two equations can be used to solve for the two unknowns


1-D Elastic Collisions

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1-D Elastic Collisions, cont

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=


2-D Elastic CollisionsEnergy conservation: ½ m1v1i

2 = ½ m1v1f2 + ½ m2v2f

2

Momentum conservation:x: m1v1i = m1v1f cosθ +m2v2f cosφy: 0 = m1v1f sinθ +m2v2f sinφ

Note that we have 3 equations and 4 unknowns

We cannot solve for the unknowns unless we have one final angle or final velocity given


Example 8: Billiard Ball A player wishes to sink target ball in the corner pocket. The angle to the corner pocket is 35°. At what angle is the cue ball deflected?

°+−=

°+=

+=

35sin sin 0 35 coscos

:onconservati Momentum21

21

21

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211

211

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θ

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Example 8, cont

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22

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Internal EnergySo far we have considered situations where there is no friction

In the absence of friction, the mechanical energy is conserved: E = K + U = constant

Where friction is present, some energy is converted to internal energy, usually heat

Here K + E ≠ constant


Example 9: Bullet Fired into BlockA 5.00-g bullet moving with an initial speed of 400 m/s is fired into and passes through a 1.00-kg block. The block, initially at rest on a frictionless, horizontal surface, is connected to a spring of force constant 900 N/m. The block moves 5.00 cm to the right after impact. Find (a) the speed at which the bullet emerges from the block and (b) the mechanical energy converted into internal energy in the collision.


Example 9, cont(a) Energy conservation for the spring:

Momentum conservation in the collision :

(b)

The lost energy is due to the friction between the bullet and block. The block heats up a little.

22

21

21 kxMVi =

mv MV mv ii +=

m/s 100kg105.00

m/s) kg)(1.5 (1.00m/s) kg)(40010(5.003

3

=×

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−=

−

−

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21

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−×=Δ+Δ=Δ−

−UKE

m/s 1.50kg 1.00

m) N/m)(0.05 (900 22

===MkxVi


Energy DiagramsParticle under a free fall


Energy Diagrams, contParticle attached to a spring


General Energy Diagram

Equilibria are at points where dU/dx = 0. At equilibria, F = 0. Away from equilibria, dU/dx ≠ 0 and F ≠ 0.


Molecular BondingPotential energy associated with the force between two neutral atoms in a molecule can be modeled by the Lennard-Jones function:


Force Acting in a Molecule

The force is repulsive (positive) at small separationsThe force is zero at the point of stable equilibrium

This is the most likely separation between the atoms in the molecule (2.9 x 10-10 m at minimum energy)

The force is attractive (negative) at large separationsAt great distances, the force approaches zero


Example 10: Hemispherical Hill

A sled starts from rest at the top of the frictionless, hemispherical, snow-covered hill shown in the figure. (a) Find an expression for the sled's speed when it is at

angle φ. (b) Use Newton's laws to find the maximum speed the sled

can have at angle φ without leaving the surface. (c) At what angle φmax does the sled "fly off" the hill?


Example 10, cont

( )

o2.4832cos

32cos ,cos)cos1(2

two, theEquating . and anglearbitrary an for have We(c)cos 0

hill. theleaves sled the0, When increases. as decreases

cos cos , (b)

cos12

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(a)

1maxmaxmaxmax

max

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22

1

202

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0202

11

212

1

0011

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gRgR

vvgRvn

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Documents

Physics for Scientists and · PDF filePhysics for Scientists and Engineers ... mgh mv mg R K U K U v gR R v mg m i gi f gf 2 2.5 2 1 ... When the trigger is pulled,