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Potential Energy, Conservation of Energy, and Energy

Diagrams

8.01 W06D2

Today’s Reading Assignment:

Chapter 14 Potential Energy and Conservation of Energy, Sections 14.1-14.7

Announcements

Problem Set 5 due Oct 16 at 9 pm in box outside 26-152

Review: Conservative Forces

Definition: Conservative Force If the work done by a force in moving an object from point A to point B is independent of the path (1 or 2), then the force is called a conservative force which we denote by . Then the work done only depends on the location of the points A and B.

c (path independent)B

cA

W d≡ ⋅∫F r

cF

2

Potential Energy Difference

c

B

cA

U d WΔ ≡ − ⋅ = −∫F r

Definition: Potential Energy Difference between the points A and B associated with a conservative force is the negative of the work done by the conservative force in moving the body along any path connecting the points A and B.

cF

Potential Energy Difference: Constant Gravity

Force: Work: Potential Energy: Choice of Zero Point: Choose . Potential Energy Function: Let then

F = mg = Fy j = −mg j

W = FyΔy = −mgΔy

ΔU = −W = mgΔy = mg y f − yi( )

U (0) ≡ 0

U ( y) = mgy; U (0) = 0

y f = y

Worked Example: Change in Potential Energy for Inverse Square

Gravitational Force

Consider an object of mass m1 moving towards the sun (mass m2). Initially the object is at a distance r0 from the center of the sun. The object moves to a final distance rf from the center of the sun. For the object-sun system, what is the change in potential energy during this motion?

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Worked Example Solution: Inverse Square Gravity

Force:

Work done:

Potential Energy Change:

Zero Point:

Potential Energy Function

1 2

1 2, 2

ˆm mGm mr

= −F r

W =

F ⋅dr =

ri

rf

∫ −Gm1m2

r 2

⎛⎝⎜

⎞⎠⎟

drri

rf

∫ =Gm1m2

rri

rf

= Gm1m2

1rf

− 1ri

⎛

⎝⎜

⎞

⎠⎟

ΔU = −W = −Gm1m2

1rf

− 1ri

⎛

⎝⎜

⎞

⎠⎟

U (∞) = 0

U (r) = −

Gm1m2

r; U (∞) = 0

Concept Question: Potential Energy and Inverse Square Gravity

A comet is speeding along a hyperbolic orbit toward the Sun. While the comet is moving away from the Sun, the change in potential energy of the Sun-comet system is:

(1) positive (2) zero (3) negative

Table Problem: Change in Potential Energy Spring Force

Connect one end of a spring of length l0 with spring constant k to an object resting on a smooth table and fix the other end of the spring to a wall. Stretch the spring until it has length l and release the object. Consider the object-spring as the system. When the spring returns to its equilibrium length what is the change in potential energy of the system?

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Potential Energy Difference: Spring Force

Force:

Work done:

Potential Energy Change:

Zero Point:

Potential Energy

F = Fx i = −kx i

W = −kx( )dx = − 1

2x=xi

x=x f

∫ k x f2 − xi

2( )

ΔU = −W = 1

2k x f

2 − xi2( )

U (0) = 0

U (x) = 1

2kx2; U (0) = 0

x = displacement from equilibrium

Work-Energy Theorem: Conservative Forces

The work done by the force in moving an object from A to B is equal to the change in kinetic energy When the only forces acting on the object are conservative forces then the change in potential energy is Therefore

W ≡

F ⋅ dr

A

B

∫ =12

mvB2 −

12

mvA2 ≡ ΔK

ΔU = −W

−ΔU = ΔK

F =Fc

Forms of Energy

•  kinetic energy •  gravitational potential energy •  elastic potential energy •  thermal energy •  electrical energy •  chemical energy •  electromagnetic energy •  nuclear energy •  mass energy

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Energy Transformations

Falling water releases stored ‘gravitational potential energy’ turning into a ‘kinetic energy’ of motion.

Human beings transform the stored chemical energy of food into catabolic energy Burning gasoline in car engines converts ‘chemical energy’ stored in the atomic bonds of the constituent atoms of gasoline into heat

Stretching or compressing a spring stores ‘elastic potential energy’ that can be released as kinetic energy

Mechanical Energy

When a sum of conservative forces are acting on an object, the potential energy function is the sum of the individual potential energy functions with an appropriate choice of zero point potential energy for each function

Definition: Mechanical Energy The mechanical energy function is the sum of the kinetic and potential energy function

Emech ≡ K +U

U =U1 +U2 + ⋅ ⋅ ⋅

Conservation of Mechanical Energy

When the only forces acting on an object are conservative

Equivalently, the mechanical energy remains constant in time

ΔK + ΔU = 0

E f

mech = K f +U f = Ki +Ui = Eimech

(K f − Ki ) − (U f −Ui ) = 0

ΔEmech = E f

mech − Eimech = 0

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Demonstration: Experiment 2 Cart-Spring on an Inclined Plane

A cart of mass m slides down a plane that is inclined at an angle θ from the horizontal. The cart starts out at rest. The center of mass of the cart is a distance d from an unstretched spring with spring constant k that lies at the bottom of the plane. A plot of distance vs. time shows the change in the mechanical energy of the system.

Energy Diagrams

Identify system Identify state of system before interaction and state of final states. Draw energy diagrams for these states including: 1) Choice of zero point P for potential energy for each interaction in which potential energy difference is well-defined. 2) Draw speed of all objects in system 3) Identify initial and final mechanical energy Emech = K+U

Demo slide: Loop-the-Loop B95

A ball rolls down an inclined track and around a vertical circle. This demonstration offers opportunity for the discussion of dynamic equilibrium and the minimum speed for safe passage of the top point of the circle.

http://tsgphysics.mit.edu/front/?page=demo.php&letnum=B 95&show=0

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Table Problem: Loop-the-Loop

An object of mass m is released from rest at a height h above the surface of a table. The object slides along the inside of the loop-the-loop track consisting of a ramp and a circular loop of radius R shown in the figure. Assume that the track is frictionless. When the object is at the top of the track (point a) it just loses contact with the track. What height was the object dropped from?

Table Problem: Sliding on Slipping Block

A small cube of mass m1 slides down a circular track of radius R cut into a large block of mass m2 as shown in the figure below. The large block rests on a , and both blocks move without friction. The blocks are initially at rest, and the cube starts from the top of the path. Find the velocity of the cube as it leaves the block.

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Review: Non-Conservative Forces

Definition: Non-conservative force Whenever the work done by a force in moving an object from an initial point to a final point depends on the path, then the force is called a non-conservative force and the work done is called non-conservative work

Fnc

Wnc ≡

Fnc

A

B

∫ ⋅dr

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Non-Conservative Forces

Work done on the object by the force depends on the path taken by the object

Example: friction on an object moving on a level surface

friction

friction friction 0k

k

F NW F x N x

µµ

== − Δ = − Δ <

Change in Energy for Conservative and Non-conservative Forces

Force decomposition:

Work done is change in kinetic energy:

Mechanical energy change: W =

F ⋅dr

A

B

∫ = (Fc +Fnc )

A

B

∫ ⋅dr = −ΔU +Wnc = ΔK

ΔK + ΔU = ΔEmech =Wnc

F =Fc +Fnc

Modeling the Motion Energy Concepts

Change in Mechanical Energy: Identify non-conservative forces. Calculate non-conservative work: Apply Energy Law:

final

nc ncinitial

W d .= ⋅∫ F r

Wnc = ΔK + ΔU = ΔEmech

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Modeling the Motion: Newton’s Second Law

Define system, choose coordinate system. Draw free body force diagrams. Newton’s Second Law for each direction. Example: x-direction Example: Circular motion

totalˆ : .2

x 2

d xF m

dt=i

2totalˆ : .r

vF mR

= −r

Table Problem: Sliding Friction

Block A of mass mA is moving horizontally with speed VA along a frictionless surface. It collides elastically with block B of mass mB that is initially at rest. After the collision block B enters a rough surface at x = 0 with a coefficient of kinetic friction that increases linearly with distance,

where b is a positive constant. At x = d, block B collides with an unstretched spring with spring constant k. For x>d the surface is frictionless. The downward gravitational acceleration has magnitude g. What is the distance the spring is compressed when block B first comes to rest? Express your answer in terms of VA, mA, mB, b, d, g, and k.

µk (x) = bx, 0 ≤ x ≤ d