A few sample physics questions for Reiku

Discussion in 'Free Thoughts' started by James R, Jan 23, 2012.

  1. James R Just this guy, you know? Staff Member

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    This thread is for Reiku, who has asked the physicists on sciforums to test his basic understanding of physics and mathematics at 1st year university level, or thereabouts.

    I invite the physicists here to post appropriate questions at that level, and for Reiku to post his answers.

    After Reiku has posted a solution to any particular question, then I invite anybody else who is interested to also post their solution.

    Posted questions should be numbered consecutively so people can refer to them later in thread.

    I ask that nobody posts any solution to a question until Reiku has first posted his best solution.
     
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  3. parmalee peripatetic artisan Valued Senior Member

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    A suggestion for someone more ambitious and less lazy than myself: since Reiku deigns himself sufficiently expert to write a book on it, how 'bout someone create a corrolary thread for questions on philosophy and philosophy of mind?
     
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  5. James R Just this guy, you know? Staff Member

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    Question 1

    Find the eigenvalues and corresponding eigenvectors of the following matrix:

    \(\left(\matrix{1 & 3 \\ 4 & 2}\right)\)

    Question 2

    The gravitational field inside a hollow spherical shell of mass \(m\) is zero, and outside the shell the field is as if all the mass were at the centre of the sphere.

    (a) Show that the gravitational field inside the Earth at radius \(r\) is given by \(g=-Kr\), where \(K=\frac{4\pi G \rho}{3}\), \(G\) is the universal gravitational constant and \(\rho\) is the density of the Earth (taken to be uniform).
    ---

    Imagine that a straight tunnel is drilled through the centre of the Earth from surface to opposite surface. A small mass \(m\) is positioned inside the hole at distance \(r\) from the centre of the Earth and released from rest.

    (b) Write an expression in terms of the parameters defined above for the force on the mass when it is in the given position.

    (c) Explain why the mass will oscillate about the centre of the Earth and specify the type of motion that it will undergo.

    (d) Show that if the mass is released into the hole at the Earth's surface, it will reach the opposite side of the Earth after a time \(t=\pi\sqrt{1/K}\).
    ---

    Question 3

    A sinusoidal wave is travelling along a string with a wave speed of 40 cm/s. The displacement of an element of the string located at \(x=10~cm\) is found to vary with time according to the following equation:

    \(y(t) = (5.0~cm)\sin [1.0 - (4.0~s^{-1})t]\)

    If the linear density of the string is \(4.0~g/cm\) find:

    (a) The amplitude of the wave.
    (b) The frequency of the wave.
    (c) The wavelength of the wave.
    (d) The general equation giving the transverse displacement of the string as a function of position and time.
    (e) An expression for the transverse velocity of the string element at \(x=10~cm\) as a function of time.
    (f) The tension in the string.
     
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  7. parmalee peripatetic artisan Valued Senior Member

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    @ James:

    Personally, I think question 3 is too easy--I mean, I can do that one and I haven't taken a physics course beyond high school or a math course beyond a year-and-a-half of calculus.

    And, you know, if I were gonna write a book on it...

    Please Register or Log in to view the hidden image!



    Just my humble opinion.
     
  8. James R Just this guy, you know? Staff Member

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    Question 4

    A travelling car does a right turn on a horizontal road by following a circular path of radius 30 m.

    (a) If the coefficient of static friction is \(\mu_s = 0.6\), what is the fastest speed the car can travel without slipping? Give your answer in km/hr.

    (b) Suppose the (curved) road is instead banked at angle \(\theta\) from the horizontal. Determine the bank angle such that the car will be able to take the turn at 40 km/hr without slipping, even if the road is wet or icy.

    ---

    Question 5

    1.00 mol of an ideal gas (\(\gamma = 1.4\)) is initially at a pressure of 1.00 atm and a temperature of zero degrees Celcius (state 1). The gas is heated at constant volume to a temperature of 150 degrees Celcius (state 2). Then the gas is allowed to expand adiabatically until its pressure returns to 1.00 atm (state 3). Finally, it is compressed at constant pressure back to the original state (state 1).

    (a) Find the temperature of the gas after the adiabatic expansion (i.e. in state 3).

    (b) Calculate the total amount of energy that enters or leaves the system as heat during the entire process.

    (c) Is this cycle an example of a heat engine or a refrigerator? Explain.

    ---

    That's probably enough to start with.

    Remember to show all relevant working in answers.
     
  9. James R Just this guy, you know? Staff Member

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    parmalee:

    The question was taken from an actual 1st-year university physics exam for a standard-level course.

    I'm not trying to post difficult questions to stump Reiku. I am posting fair questions that he would surely find easy if he is equipped to cope with the Dirac equation, Clifford algebras and the like.

    Question 3 is obviously a straightforward wave question and Reiku should have absolutely no difficulty answering it as long as his level of knowledge is what he claims it to be.

    And you're right - a lot of high-school students would be able to cope with that question with no problems.
     
  10. Reiku Banned Banned

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    11,238
    Right, ok. Read through it, shouldn't be too hard; there is one notation I haven't seen before, but already I think I have a vague idea what it is. I will be back later once I have typed it up.
     
  11. Reiku Banned Banned

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    Mind you, I think I will fail on the mole question and ideal gas. I have done absolutely no gas dynamics in my life.
     
  12. Reiku Banned Banned

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    I have but one question,

    \(y(t) = (5.0~cm)\sin [1.0 - (4.0~s^{-1})t]\)

    What is (5.0cm), (1.0) and (4.0s^-1) standing for, I assume that the frequency is (4.0s^-1) right?
     
  13. Reiku Banned Banned

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    11,238
    Question 1

    Well, I know that the eigenvalues of a 2x2 matrix can be obatined from the quadratic formula. But there are simpler ways.

    The eigenvalue is often denoted as \(\lambda\).

    So we have,

    \(\left(\matrix{1 & 3 \\ 4 & 2}\right)\)

    The eigenvalues, could be given by values of r which we know make the det of a matrx equal to zero.

    \(\left(\matrix{1-r & 3 \\ 4 & 2-r}\right)\)

    \((1-r)(2-r)-3-4 = 0\)

    So the Eigenvalues equal \(\lambda_1 = -2\) and \(\lambda_2 = 5\).

    Question 2 (a)

    Right, well straight away I can see Gauss' Law equation divided by 3, probably 3 comes from the dimensions working on a three dimensional sphere (or atleast an approximation).

    I haven't admittedly ever seen it expressed like this, equalled to K. What is K?

    The gravitational field of the earth is \(\frac{\mu}{r} = \frac{GM}{r}\) where \(\mu\) is the gravitational parameter. So in your equation, it describes the volume times the density.

    So is, \(K\) simply another way to write

    \(\frac{\mu}{r^2}= \frac{4\pi G \rho}{3}\)

    ? This is just the gravitational field of an homogeneous sphere. Now, \(g\) is traditionally given as

    \(g= \frac{GM}{r^2}\)

    So I assume that

    \(g= \frac{4 \pi G \rho}{3}\)

    So if we have

    \(K= \frac{4 \pi G \rho}{3}\)

    and

    \(g = -Kr\)

    So, up till now, if you haven't guessed, I've been guessing

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    .... there is an equation which comes to mind which is similar to the appearance of the negative \(K\) in the context I gave it, with the radius \(r\) and that is

    \(-\frac{GM(r)}{r^2} = -\frac{4 \pi G \rho}{3}\)

    Hows that?

    Question 2 (b)

    I am to write an expression in terms of the parameters for the force on a mass when at any single position?

    Well, force is generally given by the equation \(F=\frac{GM^2}{r^2}\).

    So the expression would be

    \(\frac{GM^2}{r^2}\)

    ? Not sure if this is what you wanted?

    Question 2 (c)

    ''Explain why the mass will oscillate about the centre of the Earth and specify the type of motion that it will undergo.''

    As an object (take it to be ourselves) falls past the surface of the earth, the portion of the Earth's mass which is above you is further from the center than you are. The force of gravity which attracts you is contributed from the center of the Earth. Since the surface is no longer contributing to pull you back, you will continue to fall towards the center of the Earth.

    As you fall however, mass tends to zero, as the radius tends to zero. As you continue to fly towards the center, there will be kinetic energy left over from your decent, so you do not settle in the center of the spherical mass. Instead, you will oscillate back and forth around the center, for each time you turn back and move past the center, the same rules apply: the surface is no longer pulling at you, but rather the center of mass, the origin of the Earth's radius is attracting your body.

    Question 2 (d)

    The distance traversed by an object falling for some time is if I am correct, usually given by the equation

    \(d = \frac{1}{2} gt^2\)

    Solving for the time, we have an equation which looks (only) similar to your equation, but are essentially the same

    \(t = \sqrt{\frac{2d}{g}}\)

    Where I have surmised before \(K = \frac{GM}{r^2}\) and that \(g = \frac{GM}{r^2}\) So I've replaced this with the acceleration due to gravity and I am going to replace your equation numerator with the radius.

    \(t=\pi \sqrt{\frac{r}{g}}\)

    If the acceleration of the gravity is \(9.8 m/s\) and (I've had to look this up) the radius \(6,353\) km.

    But I feel I have unecesserily made this complicated. This is essentially a problem concerning Hooks Law.

    Taken that the spring constant would be \(\frac{k}{M}\) would be \(\frac{9.8 m/s}{6300 km}\) works out then that to fall to the other side of the earth would take about 42 mins, considering the period of oscillation is

    \(t = 2\pi \sqrt{\frac{Mr}{Mg}} = 2\pi \sqrt{\frac{r}{g}}\)
     
    Last edited: Jan 23, 2012
  14. Reiku Banned Banned

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    I'll leave it at that for now, I must go.
     
  15. James R Just this guy, you know? Staff Member

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    Reiku:

    Please post your full answer. This is basic, first-year stuff, remember. You've said you could easily skip all this stuff and jump straight into a graduate course in physics. You already know all the basics, don't you?

    (edit: this is with reference to post #9. I hadn't seen your later post. More on those later.)
     
  16. Reiku Banned Banned

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    That last equation came out wrong, fixed it. Now I really do need to go for now.
     
  17. Reiku Banned Banned

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    And actually what I said James, is that arguably with what I know, the first year would be a breeze. I never meant to intend I ''knew'' everything. That is your own intepretation of what I said.
     
  18. James R Just this guy, you know? Staff Member

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    Let's start with Question 1:

    I'm not sure why you switched notation from \(\lambda\) to r, but it doesn't matter.

    There's a typo in your equation for r. Obviously you mean 3 times 4 rather than 3 minus 4. But...

    These are the correct eigenvalues.

    So far so good, but that's only half the question.

    What are the corresponding eigenvectors?

    ---

    There's no time limit on this. Take your time.
     
  19. Reiku Banned Banned

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    Sorry the Eigenvectors are

    V_1 =

    -0.707
    0.707

    V_2 =

    0.6
    -0.8


    And I will be buisy for the rest of the day James.
     
  20. Hercules Rockefeller Beatings will continue until morale improves. Moderator

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    That post was made ~11.00am (Reiku local time).

    Now, from ~7.30pm to 11.00pm (Reiku local time) that evening you made 11 posts in various other threads.

    So, it begs the question, if you had the time to participate in those threads after your busy day, why didn’t you knock off the remaining exam questions?
     
  21. Gustav Banned Banned

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    priorities obviously
    little green men are so much more exciting than boring math shit
     
  22. CptBork Valued Senior Member

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    Man those 2x2 matrices can be real brainbusters sometimes, can't they? :roflmao:
     
  23. AlphaNumeric Fully ionized Registered Senior Member

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    Precisely. The questions are quite easy, some could be done by a good student in high school, never mind undergrad. I wish I'd had those questions for my first year exams! If Reiku thinks he's competent enough to be doing stuff I didn't get to until my 4th year he shouldn't have any problem doing stuff which is assumed knowledge for new undergrads, like questions 2 and 4. All 5 questions he should have done in under an hour, just knocked them out. Questions 1 and 3 could even be done without putting pen to paper if you're typing the answers as you work through them.

    But let's wait for his excuses, attempts at the rest and then we can grade him. How are you going to do it James? Do we just say "Right/Wrong!" or are you going to give him marks out of 5 or 10 or 20 for the various questions? Will Reiku be provided with model answers afterwards so he can check the marks for himself? How long do we wait before we call time on the first 5 questions?
     

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