PHYS234 Lecture 6: PHYS 235 - Chapter 6

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harlequinmoose65

5 May 2016

School

University of Waterloo

Department

Physics

Course

PHYS234

Professor

unknown

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Document Summary

In this chapter we focus on an important method of solving certain problems in classical. In many problems we need to determine how a system evolves between an initial state and a final state. For example, consider two locations on a two-dimensional plane. Consider an object moving from one location to the other, and assume the object experiences a friction force when it moves over the surface. One can ask questions such as what is the path between the initial and the final condition that minimizes the work done by the friction force? that can be most easily answered using the calculus of variations. The evolution that can be studied using the calculus of variations is not limited to evolutions in real space. For example, one can consider the evolution of a gas in a pv diagram and ask what is the path between the initial and final state that maximizes the work by the gas? .

Related Questions

Consider three identical spherical conductors. One conductor,labeled A, is given a
charge of 7mC of type Nell. The charge state of the second andthird conductors, B and C, are unknown. The conductors undergo asequence of 3 contacts that occur ONLY between pairs of conductors.After the contacts, the net charge on each conductor is A: (0.25mCof type Nell), B: (1.875mC of type Nell) and C: (1.875mC of typeNell). Letâs make the assumption that the type Paul charge statecorresponds to an excess of electrons.

Because these three spheres are ______________________ the_______________ can be
freely exchanged between any pair of conductors in contact. Sincethese conductive
spheres all have the same size, when the spheres are placed incontact the ______
_____________ will evenly distribute between the two spheres.Because charge is a
property of particles and the particles can not be destroyed ANDthere is no outside
source of charge, the _______________ charge of the ______________must remain
constant at 4mC of type Nell. Based on this information, we candetermine that the
charge of at least one of B or C must be of type_______________.

To determine the order of contacts we must start from the lastcontact and work our way to the first contact. It is important tonote that because we have no initial charge state information forspheres B and C that we have the ability to make an arbitrarydecision about the initial charge state of these two spheres. Inother words, the charge states of sphere B and C areinterchangeable. Third and last Contact: Because spheres ________and _______ have the same charge we can determine that they were______ in contact. If we look at spheres B and C as the system, wefind that the total charge of the sub system consisting of B and Cwas ______________ of type _________.

Second contact: We can safely assume that A touched either B or Cprior to B and C
being brought into contact. Here we can make the arbitrary choicethat B touched A in the second contact. If this is the case, weknow that the charge of B ___________ it touched C must have been______________ of type __________ because it must have the sameamount and type of charge as the final state of _______. Therefore,at this instant we know that the charge on sphere C was________________ of type ____________________ it touched B.

First contact: Now we have an interesting situation. We find thatthe charge state of
sphere C would be consistent with sphere C being initially____________ and being
placed in contact with _______________ since the charge state of Cis exactly
__________ that of ____________ initially. This would mean that thecharge state of
sphere A after contact with ______ would be _____________ of type__________.
However, we already determined that the __________ charge of thesystem containing sphere _______ and sphere _______ in the secondcontact was ____________ of type __________. The only way this ispossible is if sphere B has a charge of __________ of type____________.

Final Results
The initial charge state of sphere Ais:_________________________________________
The initial charge state of sphere Bis:_________________________________________
The initial charge state of sphere Cis:_________________________________________
In the first contact of sphere ___________ to sphere_________________ how many
electrons were transferred from one sphere to the other?
In the second contact of sphere ___________ to sphere_________________ how many
electrons were transferred from one sphere to the other?
In the third contact of sphere ___________ to sphere_________________ how many
electrons were transferred from one sphere to the other?

I am not even sure will to begin with this problem anysuggestions?

I am so lost onwere to even begin. I just need help trying to find the Ñ² (angle)for part part 1 of the experiment if anyone can just explain how todo that and maybe give me a step-by-step example so I can finishthe rest. Thank you! Experiment 6 ~ Friction

Purpose:

In this lab, you will make some basic measurements of friction.You will measure the coefficients of staticfriction between several combinations of surfaces using aninclined plane, and you will measure the coefficient ofkinetic friction between two of the combinationsof surfaces you used in the static friction part of thisexperiment.

Part 1: StaticFriction

Theory:

The coefficient of static frictionm_s can be measured experimentally foran object placed on an inclined plane (a.k.a. ramp, a.k.a. hill).The coefficient of static friction is related to the criticalangle q_c for the ramp, at whichthe object just begins to slide. Using what we havecovered in class, you can derive this relationshipyourself! At this critical angle, static friction preventing theobject from sliding down the hill is just exactly equal to thecomponent of the objectâs weight along the hill. If the componentof the weight along the hill were just a little greater, it wouldovercome friction, and the object would start to slide down. Thefree body diagram for this situation will look like the figuresketched below. Use this diagram to find an equation that relatesm_s toq_c.

F_N = NormalForce = mg cos(Î¸)

F_F =Î¼_s x F_N = Î¼_s mg cos(Î¸)

Î£F_x = mA_x = F_Gx - F_F = ?

F_Gx = mgsin(Î¸)

Hint #1: the angle q of the incline is the same as theangle between the normal force N and the weightmg. Why?

Hint #2: The weight mg can be broken into twocomponents. One component is along the incline(W_H). The other component is âpressinginto the hillâ, and is equal in magnitude to the normal forceN. (This is an example of a situation where the normalforce is not equal to the weight!) Remember that you need to usethe normal force to calculate friction, sincef=mN. Now calculate what the angle ofthe incline should be, so that the component of the weight down thehill is just exactly balanced by friction.

Equipment:

For this lab you will need to share the following ramps andblocks with other lab groups:

Ramps Blocks

Aluminum Aluminum

ParticleBoard Brass

StainlessSteel Plexiglass

Plastic Copper

Rubber

Steel

Wood

Experiment:

Test at least 5 different combinations of materials. Record thematerial of the block and the ramp for each combination.

1. Measure and record the length of theinclined plane you will be using. (This is thehypotenuse of the right triangle!)

2. Place the block on the plane and begin tiltingthe plane just until the point when the block begins to slide.

3. Measure the height of the raisedend of the inclined plane at this point. (This is theopposite side of the right triangle!) You cancalculate the angle from the measurements of the two sides.

4. Repeat the height measurement at least 3 timesfor each combination of materials. Record all the measurements inthe table below. Then,

(a) calculate the angle from the measurements of the hypotenuseand opposite side you just obtained

(b) use the equation you derived above in order to calculate thecoefficient of kinetic friction from the angle.

Block: woodenblock Block: cooper

Ramp:particleboard Ramp: particle board

Length of ramp:0.61(m) Length of ramp: 0.61(m)

Height(m)	Ñ²	Ms	Height(m)	Ñ²	Ms
0.24			0.26
0.25			0.31
0.245			0.345

avg m_s: ______ s : _______ avg m_s: ______ s : _______

Block:aluminum Block:particle board block

Ramp: solidewood Ramp: solide wood

Length of ramp:0.605(m) Length of ramp: 0.605(m)

Height(m)	Ñ²	s	Height(m)	Ñ²	s
0.15			0.23
0.16			0.165
0.16			0.195

avg m_s: ______ s : _______ avg m_s: ______ s : _______

5. Remember that in class we said that friction does not dependon surface area. You can easily test whether or not this iscorrect. Choose a block/ramp combination not already tested, andmake sure the block is not a cube or cylinder. Place the block sothat the side with the larger surface area is in contact with theboard. Now measure the angle at which the block just begins toslide (by measuring the hypotenuse and opposite sides of thecorresponding right triangle). As before, calculate the coefficientof friction from the angle you measured, using your equation.Repeat this process with the side of the block with the smallersurface area in contact with the board. Enter this data in thesmall tables below.

Block (larger end) :_________ Block (smaller end):__________

Ramp:_______________ Ramp: ________________

Length of ramp:______(m) Length of ramp: ______(m)

Height(m)	s	Height(m)	s

avg m_s: ______ s : _______ avg m_s: ______ s : _______

Part 2: KineticFriction

Theory:

You can calculate the coefficient of kinetic friction,m_k, using a variation of the methodyou used for the coefficient of static friction. For thecoefficient of kinetic friction, you can use the same free bodydiagram as the one drawn on the first page. But now, thecombination of W_H and the force offriction will need to add up such that the block will slide at aconstant speed. Think of Newtonâs first and second laws when youset up this equation. Derive a relationship between the criticalangle and the coefficient of kinetic friction.

Experiment:

You can measure m_k using aprocedure similar to the one you used to measurem_s. This time, just pick twocombinations of materials. The combinations you pick for this partof the experiment have to be combinations you alreadyused for the static friction part of the experiment, because thepoint is to compare m_sandm_k!

1. Measure and record the length ofthe inclined plane you will be using.

2. Raise the plane high enough for the blockto begin moving. Begin lowering the plane just until the pointwhen the block begins to slow down. (Alternatively, raise theboard while tapping the block until block moves at a constantrate.)

3. Measure the height of the raisedend of the inclined plane at this point.

4. Repeat the height measurement at least 3times for each combination of materials. Record all themeasurements in the table below. Then,

(a) calculate the angle from the measurements of the hypotenuseand opposite side you just obtained

(b) use the equation you derived above in order to calculate thecoefficient of kinetic friction from the angle.

Block (from part 1) :_________ Block (from part 1):__________

Ramp:_______________ Ramp: ________________

Length of ramp:______(m) Length of ramp: ______(m)

Height(m)	k	Height(m)	k

avg m_k: ______ s : _______ avg m_k: ______ s : _______

Analysis:

Calculate the mean value and standard deviation of thecoefficient of static friction that you measured, for each set ofmaterials.

ramp/board pair		Standard deviation




(larger end)
(smaller end)

Calculate the mean value and standard deviation of thecoefficient of kinetic friction that you measured, for each set ofmaterials.

ramp/board

pair

Standard

deviation

Questions:

1. How do the values of m_s compareto the values of m_k? (Of course, youcan only compare them for the same pairs of materials.)

2. Is the relationship betweenm_sand m_kwhat you expected?

3. Of the two parts of the experiment, measurement ofm_sand measurement ofm_k, which had more sources of error?What were some of the sources of error?

4. Could m_k orm_s ever be greater than 1? Explain whyor why not.

5. Is the coefficient of friction the same as when the block wasstanding on its larger (or smaller) end?

6. Think about your results. Do they make sense when youconsider your everyday experiences?

PHYS234 Lecture 6: PHYS 235 - Chapter 6

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