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Because Earth's rotation is gradually slowing, the length of each day increases: The day at the end of the 1.0 century is 1.0 ms longer than the day at the start of the century. In 44 centuries, what is the total (in hours) of the daily increases in time (that is, the sum of the gain on the first day, the gain on the second day, etc.)?

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Vikas
VikasLv10
17 Mar 2019

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Physics: Principles with Applications
7th Edition, 2013
Giancoli
ISBN: 9780321625922

Related questions


A spring has a relaxed length of 30 cm (0.30 m) and its springstiffness is 12 N/m. You glue a 66 gram block (0.066 kg) to the topof the spring, and push the block down, compressing the spring soits total length is 13 cm. You make sure the block is at rest, thenat time t = 0 you quickly move your hand away. The block begins tomove upward, because the upward force on the block by the spring isgreater than the downward force on the block by the Earth.Calculate y vs. time for the block during a 0.27-second intervalafter you release the block, by applying the Momentum Principle inthree steps each of 0.09-second duration.

We will only consider the y components in the followingcalculations, because there is no change in x or z.

STEP 1

Force: Just after releasing the block, calculate the force exertedon the block by the spring, the force exerted on the block by theEarth, and the net force:
Fspring,y = (...) N
FEarth,y = (...) N
Fnet,y = (...) N

Momentum update: Just after releasing the block, the momentum ofthe block is zero. Calculate the average net force during the nexttime interval by the force you just calculated. At t = 0.09seconds, what will the new momentum and velocity of the blockbe?
py = (...) kg
I am so lost onwere to even begin. I just need help trying to find the Ѳ (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 frictionms 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 qc 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 object’s 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 relatesms toqc.

FN = NormalForce = mg cos(θ)

FF =μs x FN = μs mg cos(θ)

ΣFx = mAx = FGx - FF = ?

FGx = mgsin(θ)

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(WH). The other component is “pressinginto the hill”, 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)

Ѳ

Ms

Height(m)

Ѳ

Ms

0.24

0.26

0.25

0.31

0.245

0.345

avg ms: ______ s : _______ avg ms: ______ 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)

Ѳ

s

Height(m)

Ѳ

s

0.15

0.23

0.16

0.165

0.16

0.195

avg ms: ______ s : _______ avg ms: ______ 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 ms: ______ s : _______ avg ms: ______ s : _______

Part 2: KineticFriction

Theory:

You can calculate the coefficient of kinetic friction,mk, 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 WH and the force offriction will need to add up such that the block will slide at aconstant speed. Think of Newton’s 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 mk using aprocedure similar to the one you used to measurems. 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 msandmk!

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 mk: ______ s : _______ avg mk: ______ 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 ms compareto the values of mk? (Of course, youcan only compare them for the same pairs of materials.)

2. Is the relationship betweenmsand mkwhat you expected?

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

4. Could mk orms 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?

Situations arise in engineering when we want to control the flow rate of a solution. This solution could be pumped into a holding tank, distillation column, heat exchanger, reactor, etc. This means that the flow rate becomes a function of time. When we turn on a pump, the flow rate is (for the most part) constant. The problem is that we may want the flow rate to be very fast at first, and then very slow towards the end of the process. This is especially true in certain chemical reactors where the concentration of reactants is always important. Control systems that could handle this type of situation are very expensive. Pumps (relatively speaking) are cheap. So, having a bank of pumps will usually solve the problem. The pumps are simply turned on or off to regulate the flow. Mathematically, this amounts to taking the volumetric flow rate function and finding a piecewise-linear approximation. Suppose that, because of the application we are running, we need a flow rate given by the function ,where t is in hours and v(t) is in 100 cubic feet per hour (e.g., v(0)=1 which means that at time t=0, we need a flow rate of 1*100=100 cubic feet per hour). Find a linear approximation and a piecewise-linear approximation using first two lines and then three lines. Think about what using more lines would do to the approximation. You will need to find suitable values of t to be able to find your approximation. As a further exercise, try to find values of t for your approximation that keeps the error (i.e., the difference between the approximation and the real function v(t) ) less than 0.1.Hint: to help with this, ask yourself this question: where would more lines be useful - where the function changes a lot or where the function changes a little?]

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