physics 151: lecture 15, pg 1 today’s topics l potential energy, ch. 8-1 l conservative forces,...

Physics 151: Lecture 15, Pg 1

Today’s TopicsToday’s Topics

Potential Energy, Ch. 8-1

Conservative Forces, Ch. 8-2

Conservation of mechanical energy Ch.8-4


New Topic - Potential EnergyNew Topic - Potential Energy

Consider a ball at some height above the ground.

Some Velocity

No Velocity




What work is done in this process ? (Work done by the earth on the ball)

W = F. x W = mgh cos(0)W = mgh

h




Before the ball falls it has the potential to do an amount of work mgh.

We say the ball has a potential energy of U = mgh.

By falling the ball loses its potential energy, work is done on the ball, and it gains some kinetic energy,

W = K = 1/2 mv2 = -U = mgh

h


Lecture 15,Lecture 15, ACT 1ACT 1Work Done by GravityWork Done by Gravity

The air track is at an angle of 30 degrees with respect to horizontal. The cart (with mass 1 kg) is released. It bounces back and forth on the track. It falls 1 meter down the track, then bounces back up to its original position. How much total work is done by gravity on the cart when it reaches its original position.

1 meter

30 degrees

A) 5 J B) 10 J C) 20 J D) 0 J


Some DefinitionsSome Definitions

Conservative Forces - those forces for which the work done does not depend on the path taken, but only the initial and final position.

Potential Energy - describes the amount of work that can potentially be done by one object on another under the influence of a conservative force

W = -U

only differences in potential energy matter.


Potential EnergyPotential Energy

For any conservative force F we can define a potential energy function U in the following way:

W = FF.dr r = - U

U = U2 - U1 = - W = - FF.drrr1

r2

r1

r2 U2

U1

See text: 8.1

The work done by a conservative force is equal and opposite to the change in the potential energy function.

This can be written as:


Question - 1Question - 1

For a force to be a conservative force, when applied to a single test body:

a. it must have the same value at all points in space.

b. it must have the same direction at all points in space.

c. it must be parallel to a displacement in any direction.

d. equal work must be done in equal displacements.

e. no work must be done for motion in closed paths.


A Conservative Force : SpringA Conservative Force : Spring

For a spring we know that Fx = -kx.

F(x) x2

x

x1

-kxrelaxed position

F = - k x1

F = - k x2

See text: 8-1

See Figure 7-7


Spring...Spring...

F(x) x2

Ws

x

x1

-kx

The work done by the spring Ws during a displacement from x1 to x2 is the area under the F(x) vs x plot between x1 and x2.

See text: 8-1

See Figure 7-7

2

1

2

2

2

1

2

2s

2

21

21

W

21

)(

)(

2

1

2

1

2

1

xxkWU

xxk

kx

dxkx

dxxFW

s

x

x

x

x

x

xs



The force a spring exerts on a body is a conservative force because :

1. a spring always exerts a force opposite to the displacement of the body.

2. the work a spring does on a body is equal for compressions and extensions of equal magnitude.

3. the work a spring does on a body is equal and opposite for compressions and extensions of equal magnitude.

4. the net work a spring does on a body is zero when the body returns to its initial position.


Lecture 15,Lecture 15, ACT 2ACT 2Work/Energy for Conservative ForcesWork/Energy for Conservative Forces

The air track is is again at an angle of 30 degrees with respect to horizontal. The cart (with mass 1 kg) is released 1 meter from the bottom and hits the bumper with some speed, v1. You want the cart to go faster, so you release the cart higher. How high do you have to release the cart so it hits the bumper with speed v2 = 2v1?

1 meter

30 degrees

A) 1 m B) 2 m C) 4 m D) 8 m


Conservation of EnergyConservation of Energy If only conservative forces are present, the total energy If only conservative forces are present, the total energy

(sum of potential and kinetic energies) of a system(sum of potential and kinetic energies) of a system is is conserved.conserved.

See text: 8-4

E = K + U = W + U using K = W = W + (-W) = 0 using U = -W

E = K + U

Both K and U can change, but E = K + U remains constant.

E = K + U is constantconstant !!!!!!

Animation_3

Animation_2


ACT - 3ACT - 3

A 0.04-kg ball is thrown from the top of a 30-m tall building (point A) at an unknown angle above the horizontal. As shown in the figure, the ball attains a maximum height of 10 m above the top of the building before striking the ground at point B. If air resistance is negligible, what is the value of the kinetic energy of the ball at B minus the kinetic energy of the ball at A (KB – KA)?

a. 12 Jb. –12 Jc. 20 Jd. –20 Je. 32 J

Animation_1


What speed will skateboarder reach at bottom of the hill ?

R=3 m

. .

m = 25 kgConservation of Total Energy:

Lecture 16, Lecture 16, ExampleExampleSkateboardSkateboard

v ~ 8 m/s (~16mph) !

Does NOT depend on the mass !

. .


What would be the speed if instead the skateboarder jumps to the ground on the other side ?

. .

. .

R=3 m

KINEMATICS:

the same magnitude as before !and independent of mass



A Non-Conservative ForceA Non-Conservative ForceFrictionFriction

Looking down on an air-hockey table with no air,

Path 2

Path 1

For which path does friction do more work ?


A Non-Conservative ForceA Non-Conservative Force

Path 2

Path 1

W1 = -mg d1

W2 = -mg d2

since d2 > d1, -W2 > -W1

Since |W2|>|W1| the puck will be traveling slower at the end of path 2. Work done by a non-conservative force takes energy out of the system.


Lecture 15,Lecture 15, ACT 4ACT 4Work/Energy for Non-Conservative ForcesWork/Energy for Non-Conservative Forces

The air track is is again at an angle of 30 degrees with respect to horizontal. The cart (with mass 1 kg) is released 1 meter from the bottom and hits the bumper with some speed, v1. This time the vacuum/ air generator breaks half-way through and the air stops. The cart only bounces up half as high as where it started. How much work did friction do on the cart ?

1 meter

30 degrees

A) 2.5 J B) 5 J C) 10 J D) –2.5 J E) –5 J F) –10 J


Generalized Work Energy Theorem:Generalized Work Energy Theorem:

Suppose FNET = FC + FNC (sum of conservative and non-conservative forces).

The total work done is: WTOT = WC + WNC

The Work Kinetic-Energy theorem says that: WTOT = K.WTOT = WC + WNC = K WNC = K - WC

But WC = -U

So WNC = K + U = E


Example - 2Example - 2

A 12-kg block on a horizontal frictionless surface is attached to a light spring (force constant = 0.80 kN/m). The block is initially at rest at its equilibrium position when a force (magnitude P = 80 N) acting parallel to the surface is applied to the block, as shown. What is the speed of the block when it is 13 cm from its equilibrium position?

x

Fm

kv1= 0

v2=? 0.78 m/s


QuestionQuestion As an object moves from point A to point B only

two forces act on it: one force is nonconservative and does –30 J of work, the other force is conservative and does +50 J of work. Between A and B,

1. the kinetic energy of object increases, mechanical energy decreases.

2. the kinetic energy of object decreases, mechanical energy decreases.

3. the kinetic energy of object decreases, mechanical energy increases.

4. the kinetic energy of object increases, mechanical energy increases.

5. None of the above.



As an object moves from point A to point B only two forces act on it: one force is conservative and does –70 J of work, the other force is nonconservative and does +50 J of work. Between A and B,

1. the kinetic energy of object increases, mechanical energy increases.

2. the kinetic energy of object decreases, mechanical energy increases.

3. the kinetic energy of object decreases, mechanical energy decreases.

4. the kinetic energy of object increases, mechanical energy decreases.

5. None of the above.


ACT- 2ACT- 2

Objects A and B, of mass M and 2M respectively, are each pushed a distance d straight up an inclined plane by a force F parallel to the plane. The coefficient of kinetic friction between each mass and the plane has the same value . At the highest point,

1. K A > K B .

2. K A = K B .

3. K A < K B .

4. The work done by F on A is greater than the work done on B.

5. The work done by F on A is less than the work done on B.


Recap of today’s lectureRecap of today’s lecture

Conservative Forces and Potential Energy W = -U Conservation of mechanical energy


Let’s now suppose that the surface is not frictionless and the same skateboarder reach the speed of 7.0 m/s at bottom of the hill. What was the work done by friction on the skateboarder ?

R=3 m

. .

m = 25 kg

Conservation ofTotal Energy :


K1 + U1 = K2 + U2

Wf + 0 + mgR = 1/2mv2 + 0

Wf = 1/2mv 2 - mgR

Wf = (1/2 x25 kg x (7.0 m/s2)2 - - 25 kg x 10m/s2 3 m)

Wf = 613 - 735 J = - 122 J

Total mechanical energy decreased by 122 J !

. .

Wf +

physics 151: lecture 15, pg 1 today’s topics l potential energy, ch. 8-1 l conservative forces,...

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