Query 5

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course Phy 232

7/8 1130am

005. `query 5

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Question: query introset plug of water from cylinder given gauge pressure, c-s hole area, length of plug.

Explain how we can determine the velocity of exiting water, given the pressure difference between inside and outside, by considering a plug of known cross-sectional area and length.

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Your Solution:

First of all, we can use the pressure difference to find the net force on the plug by multiplying F = Pressure x Area.

Once we have the net force, we can find the work done by multiplying the force by the distance of the plug, because Work = Force x Distance

The work is equal to the amount of kinetic energy, so we can set the work equal to 1/2*m*v^2 and solve for v.

Before we do that, we must find the mass. We can do this by finding the volume (area x length of plug). The density of water is 1000 kg/m^3, so we can multiply density by volume to get mass.

Once we have mass we can plug it into the Work = KE equation and solve for the velocity.

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Given Solution:

** The net force on the plug is P * A, where A is its cross-sectional area and P the pressure difference.

· If L is the length of the plug then the net force F_net = P * A acts thru distance L doing work `dW = F_net * L = P * A * L.

If the initial velocity of the plug is 0 and there are no dissipative forces, this is the kinetic energy attained by the plug.

The volume of the plug is A * L so its mass is rho * A * L.

· Thus we have mass rho * A * L with KE equal to P * A * L.

Setting .5 m v^2 = KE we have

.5 rho A L v^2 = P A L so that

v = sqrt( 2 P / rho).

STUDENT SOLUTION AND QUESTION

From looking at my notes from the Intro Problem Set, it was a lot of steps to determine the velocity in

that situation.

The Force was determined first by using F = (P * cross-sectional area).

With that obtained Force, use the work formula ‘dW = F * ‘ds using the given length as the ‘ds. Then we had to obtain the Volume of the ‘plug’ by cross-sectional area * length.

That Volume will then be using to determine the mass by using m = V * d, and the density is a given as 1000kg/m^3. With that mass, use the KE equation 1/2(m * v^2) to solve for v using the previously obtained ‘dW for KE since ‘dW = ‘dKE.

Some of my symbols are different than yours. I’m in your PHY 201 class right now and am in

the work and energy sections. So is it wrong to use 1/2(m * v^2) instead of 1/2(‘rho * A * L * v^2)?? Aren’t they basically

the same thing??

INSTRUCTOR RESPONSE

You explained the process very well, though you did miss a step.

m = rho * V (much better to use rho than d, which isn't really a good letter to use for density or distance, given its use to represent the prefix 'change in'). You covered this in your explanation.

V isn't a given quantity; it's equal to the length of the plug multiplied by its cross-sectional area and should be expressed as such. You didn't cover this in your explanation.

However, other than this one missing step, your explanation did cover the entire process. With that one additional step, your solution would be a good one.

In any case, V would be A * L, and m would be rho * V = rho * A * L. So 1/2 m v^2 is the same as 1/2 rho A L v^2.

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Question: prin phy problem 10.3. Mass of air in room 4.8 m x 3.8 m x 2.8 m.

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Your Solution:

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Given Solution:

The density of air at common atmospheric temperature and pressure it about 1.3 kg / m^3.

The volume of the room is 4.8 m * 3.8 m * 2.8 m = 51 m^3, approximately. The mass of the air in this room is therefore

· mass = density * volume = 1.3 kg / m^3 * 51 m^3 = 66 kg, approximately.

This is a medium-sized room, and the mass of the air in that room is close to the mass of an average-sized person.

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Question: prin phy problem 10.8. Difference in blood pressure between head and feet of 1.60 m tell person.

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Your Solution:

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Given Solution:

The density of blood is close to that of water, 1000 kg / m^3. So the pressure difference corresponding to a height difference of 1.6 m is

· pressure difference = rho g h = 1000 kg/m^3 * 9.80 m/s^2 * 1.60 m = 15600 kg / ( m s^2) = 15600 (kg m / s^2) / m^2 = 15600 N / m^2, or 15600 Pascals.

760 mm of mercury is 1 atmosphere, equal to 101.3 kPa, so 1 mm of mercury is 133 Pascals (101.3 kPa / (760 mm) = 133 Pa / mm), so

· 15,600 Pa = 15,600 Pa * ( 1 mm of Hg / 133 Pa) = 117 mm of mercury.

(alternatively 15 600 Pa * (760 mm of mercury / (101.3 kPa ) ) = 117 mm of mercury)

Blood pressures are measured in mm of mercury; e.g. a blood pressure of 120 / 70 stands for a systolic pressure of 120 mm of mercury and a diastolic pressure of 70 mm of mercury.

The density of blood is actually a bit higher than that of water; if you use a more accurate value for the density of blood you will therefore get a slightly great result.

STUDENT QUESTION

I had to find that conversion online because I know that 1atm is 360mmHg but I could find a conversion for atms to

Pascals to make it work from what I knew, so I had to find the 133 online. ???

INSTRUCTOR RESPONSE

You should know that 1 atmosphere is about 100 kPa (more accurately 101.3 kPa but you don't need to know it that accurately), and that this is equivalent to 760 mm of mercury.

Using these two measures it's easy to convert from one to the other and there's no reason to look for or try to remember a conversion directly beteween mm of mercury and Pa.

Specifically the conversion factors are

101.3 kPa / (760 mm of mercury) = 133 Pa / (mm of mercury) and

(760 mm of mercury) / (101.3 kPa) = .0075 mm of mercury / Pa

If you use 100 kPa for the purposes of the problems and tests in this course, as I said before it's OK. If you're ever in a situation outside this class where you need the more accurate figure, it's easy to find.

STUDENT QUESTION

Do you know if our text tells us this conversion??

INSTRUCTOR RESPONSE

The list of equivalent quantities is in Table 10-2 in the 6th edition. This specific conversion isn't given, but the number of Pa and the number of mm of mercury in an atmosphere both are.

STUDENT COMMENT

The conversion of the units here is very confusing to me. The question didn’t ask for a specific unit set, so I just assumed use the one determined by the answer (kg/ms^2=N). My answer is slightly different, because I actually looked up the density of blood, which is slightly higher than water.

INSTRUCTOR RESPONSE

kg/m^3 * (m/s^2) * m = kg / (m * s^2), not kg * m/s^2.

kg m/s^2 is N. To get kg / (m s^2) you would have to divide kg m/s^2 by m^2.

That is, you divide N by m^2, obtaining N / m^2.

N / m^2 is the unit of pressure, also called the Pascal.

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Question: prin phy and gen phy 10.25 spherical balloon rad 7.35 m total mass 930 kg, Helium => what buoyant force

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Your Solution:

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Given Solution:

** The volume of the balloon is about 4/3 pi r^3 = 1660 cubic meters and mass of air displaced is about 1.3 kg / m^3 * 1660 m^3 = 2160 kg.

The buoyant force is equal in magnitude to the force of gravity on the displaced air, or about 2160 kg * 9.8 m/s^2 = 20500 Newtons, approx.. If the total mass of the balloon, including helium, is 930 kg then the weight is 930 kg * 9.8 m/s^2 = 9100 N approx., and the net force is

· Net force = buoyant force - weight = 20,500 N - 9100 N = 11,400 N, approximately.

If the 930 kg doesn't include the helium we have to account also for the force of gravity on its mass. At about .18 kg/m^3 the 1660 m^3 of helium will have mass about 300 kg on which gravity exerts an approximate force of 2900 N, so the net force on the balloon would be around 11,400 N - 2900 N = 8500 N approx.

The mass that can be supported by this force is m = F / g = 8500 N / (9.8 m/s^2) = 870 kg, approx.. **

STUDENT QUESTION

I got part of the problem right. I don’t understand the volume of air displaced…..

INSTRUCTOR RESPONSE

The 1660 m^3 volume of the balloon takes up 1660 m^3 that would otherwise be occupied by the surrounding air.

The surrounding air would be supporting the weight of the displaced air, if the balloon wasn't there displacing it. That is, the surrounding air would act to support 20500 Newtons of air.

The surrounding air is no different for the fact something else is there, instead of the air displaced air. So it supports 20500 Newtons of whatever is there displacing the air it would otherwise be supporting.

This supporting force of 20500 Newtons is therefore exerted on the balloon. We call this the buoyant force of the air on the balloon.

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Question: `q001. Water pressure exerts a force of .8 Newtons, in excess of the pressure exerted by the atmosphere on the other end, on one end of a water 'plug' with cross-sectional area 3 cm^2 and length 5 cm. The 'plug' is forced out of the side of the container by the net force as it moves through its 5 cm length, starting from rest.

How much work will the net force do on the 'plug'?

What will be the KE of the 'plug' as it exits the container?

How fast will the 'plug' be moving as it leaves the container?

Answer the analogous series of questions for a 'plug with cross-sectional area 1 cm^2 and length 2 cm.

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Your Solution:

If the force is 0.8 Newtons, and Work = Force x Distance, then the work will be equal to (0.8 Newtons)(0.05 m) = 0.04 Joules

The kinetic energy of the plug as it exits the container will be equal to the work done, so (1/2)*m*v^2 = 0.04 Joules.

To find the exiting velocity, we must first find the mass. The volume of the plug would be the area x length, so V = (0.03 m^2)(0.05 m) = 0.0015 m^3

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1 cm^2 is not .03 m^2. You can't cover a 1-meter square with 100 1-cm squares.

cm^2 = (.01 m)^2 = .0001 m^s.

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Using the density of water = 1000 kg/m^3, we can determine the mass to = (density)(volume) = (1000 kg/m^3)(0.0015 m^3) = 1.5 kg

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You can visualize 3 cm^2 and 5 cm, and it's clear that the mass of water that would fit into that cylinder is much less that 1.5 kg.

I've pointed out the source of this error above.

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To find how fast the plug will be moving, we can set the work done equal to kinetic energy and solve for v.

(1/2)*(1.5 kg)*v^2 = 0.04 Joules

v = 0.232 m/s^2

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Your units won't be m/s^2.

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Your units won't be m/s^2 here either.

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For a plug with cross sectional area 1 cm^2 and length 2 cm:

Work = force x distance = (0.8 Newtons)(0.02 m) = 0.016 Joules

The kinetic energy will be equal to the work done, so 1/2*m*v^2 = 0.016 Joules

Volume = (0.01 m^2)(0.02 m) = 0.0002 m^3

Mass = density x volume = 1000 kg/m^3 x 0.0002 m^3 = 0.2 kg

(1/2)(0.2 kg)v^2 = 0.016 Joules

v = 0.4 m/s^2

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Question: `q002. A closed glass jar with a half-liter capacity has a mass of 200 grams. If it is submerged in water what will be the buoyant force acting on it, and at the instant it is released from rest what will be the net force on it and its acceleration?

The drag force of water on the jar is 1 N s^2 / m^2 * v^2, where v is the velocity of the jar, then at what speed will it be rising when the net force acting on it becomes zero?

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Your Solution:

The buoyant force can be found by the equation Fb = (density)(acceleration due to gravity)(volume of water displaced).

Volume = 1/2 L = (1/2)(0.001 m^3) = 0.0005 m^3

Buoyant force = (1000 kg/m^3)(9.81 m/s^2)(0.0005 m^3) = 4.9 N

The net force would be the buoyancy force subtracted by the weight of the object.

So, net force = 4.9 N - (density object)(accleration due to gravity)(volume of object)

The density of the object would be mass/volume = 0.2 kg / 0.0005 m^3 = 400 kg/m^3

Net force = 4.9 N - (400 kg/m^3)(9.81 m/s^2)(0.0005 m^3) = 4.9 N - 1.96 N = 2.9 N

Its acceleration would be equal to Force/Mass = 2.9 N / 0.2 kg = 14.5 m/s^2

We can find the velocity of the jar when the net force becomes zero through the equation W = Fb + D, with W = weight of object, Fb = force of buoancy, D = drag force.

W = 1.96 N, Fb = 4.9 N, D = 1 Ns^2/m^2 x v^2

1.96 N = 4.9 N + 1 Ns^2/m^2 x v^2

2.94 m^2/s^2 = v^2

v = 1.7 m/s

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Very good.

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Question: univ 12/58 / 14.57 was 14.51 (14.55 10th edition) U tube with Hg, 15 cm water added, pressure at interface, vert separation of top of water and top of Hg where exposed to atmosphere.

Give your solution to this problem.

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Your Solution:

For the gauge pressure, it would be the atomosphere pressure in addition to the pressure of the water.

The pressure of the water would be (density)*(gravitational acceleration)*(height) = 1000 kg/m^3*9.8m/s^2*0.15m = 1470 Pascals

So the final pressure would be 1 atm + 1470 Pascals.

To find the height of the mercury column, we can take into account the differing densities of water and mercury.

The density of mercury is 13600 kg/m^2, which is about 13.6 times that of water.

If you take the distance of water, 15 cm, and divide it by the amount mercury is more dense than water, you will get 15 cm/13.6 = 1.1 cm. This will end up being the difference in the height.

To get the height of the mercury, subtract 1.1 cm from 15 cm to get 13.9 cm.

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Given Solution:

** At the interface the pressure is that of the atmosphere plus 15 cm of water, or 1 atm + 1000 kg/m^3 * 9.8 m/s^2 * .15 m = 1 atm + 1470 Pa.

The 15 cm of water above the water-mercury interface must be balanced by the mercury above this level on the other side. Since mercury is 13.6 times denser than water there must therefore be 15 cm / (13.6) = 1.1 cm of mercury above this level on the other side. This leaves the top of the water column at 15 cm - 1.1 cm = 13.9 cm above the mercury-air interface.

Here's an alternative solution using Bernoulli's equation, somewhat more rigorous and giving a broader context to the solution:

Comparing the interface between mercury and atmosphere with the interface between mercury and water we see that the pressure difference is 1470 Pa. Since velocity is zero we have P1 + rho g y1 = P2 + rho g y2, or

rho g (y1 - y2) = P2 - P1 = 1470 Pa.

Thus altitude difference between these two points is

y1 - y2 = 1470 Pa / (rho g) = 1470 Pa / (13600 kg/m^2 * 9.8 m/s^2) = .011 m approx. or about 1.1 cm.

The top of the mercury column exposed to air is thus 1.1 cm higher than the water-mercury interface. Since there are 15 cm of water above this interface the top of the water column is

15 cm - 1.1 cm = 13.9 cm

higher than the top of the mercury column.

NOTE BRIEF SOLN BY STUDENT:

Using Bernoullis Equation we come to:

'rho*g*y1='rho*g*y2

1*10^3*9.8*.15 =13.6*10^3*9.8*y2

y2=.011 m

h=y1-y2

h=.15-.011=.139m

h=13.9cm. **

GENERAL STUDENT QUESTION

I have completely confused myself on the equations for liquids and gas and cant figure out if they are interchangeable or I have just been leaning the wrong equations for different variables. I mentioned a few throughout the excercise. For example

P= F/A = mg/A = rhoAgh/A = rhogh

but this is the equation for PE as well?

However in some notes PE = rho A g L then other times it = rho g h

Is the first equation only used for fluids and the second for gas? ""

INSTRUCTOR RESPONSE

P = F / A is the definition of pressure (force per unit of area)

In a fluid, the fluid pressure at depth h is rho g h.

This quantity can also be interpreted as the PE per unit volume of fluid at height h, and is what I call the 'PE term' of Bernoulli's Equation

Bernoulli's Equation that equation applies to a continuous moving fluid, and one version reads

1/2 rho v^2 + rho g h + P = constant..

The 1/2 rho v^2 term is called the 'KE term' of the equation, and represents the KE per unit of volume. This equation represents conservation of energy, in a way that is at least partially illustrated by the exercises below.

The equation you quote, PE = rho A g L, could apply to a specific situation with a fluid in a tube with cross-sectional area A and length L, but that would be a specific application of the definition of PE. This is not a generally applicable equation.

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Question: `q003. The cap is screwed onto jar half-full of water, and the jar is place on a level surface, on its side. The cap of the jar has diameter 8 cm. How much force will water pressure exert on the cap? Note that it is necessary to set up an integral to solve this problem.

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Your Solution:

Pressure = (density)(acceleration due to gravity)(water depth)

Since the water height is going to change based on where you measure, you can set up an integral such as:

P = (density)(accleration due to gravity)(h) dh

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Right approach, but the length of the strip at height h changes with h, due to the round shape of the lid.

Your analysis would be good for a square lid, but it comes up a little short for the round lid.

Good attempt, nevertheless.

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P = (1000 kg/m^3)(9.81 m/s^2)(h) dh

P = (9810 kg/m^2S^2)(h) dh

I would then solve the integral from 0 to 4 (since it is half full).

P = [(9810 kg/m^2S^2)/2](h)^2 dh = (4905 kg/m^2S^2)(h)^2

P(4)-P(0) = ((4905 kg/m^2S^2)(4)^2) - ((4905 kg/m^2S^2)(0)^2) = 78480 Pa

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Self-Critique Rating:(4905 kg/m^2S^2)(h)^2

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Self-critique (if necessary):

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Self-critique rating:

@&

Good overall.

You should give that last problem another try. Your approach is good but you missed one detail.

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