Tuesday, May 15, 2018

Midterm2 notes and practice problems. Please prepare and bring your own table for cos.

Please prepare a table of values of the cosine function (with the argument in radians). That is, \(cos(\alpha)\), where \(\alpha\) is in units of radians and ranges from about -pi to pi (or 2 pi if you prefer). You can bring that with you to the midterm. I would suggest including all 1/4 and 1/6 fractions of pi in that range. Here is a rough example of the sort of thing I have in mind, but the one you make should have a better set of values of alpha (all 1/6 and 1/4 fractions of pi) and could be neater and more systematic. You can include a graph of \(cos(\alpha)\) and \(sin(\alpha)\) as well.

You can also bring a similar table and graph for the sin function.




Also here is a draft of the equations you can expect to see at the top
of your exam paper. Please let me know if I have left out anything essential.

The problems will involve, oscillations, collisions, and energy.
Practice problems:
(Note added: I think I may have said in class that problem 3 was particularly important, but actually 4 is just as important. All 4 of the oscillator problems are worthwhile.)
The best way to prepare for the midterm using these practice problems is to try to work the problems using only the equation sheet, your table for cos and sin, a calculator and blank paper. Try to do them without looking at any other notes!) That is really the best preparation.
    For example:
1) Consider a block attached to a spring in SHM with \(\phi_o =-\pi/3\) and T=6 with A =10m.
a) What is x(0), x(1), x(2 seconds)...? What is the equation for x(t)?
b) plot x(t)
c) what is the velocity at t=0? 1 sec?...

2) Consider a block attached to a spring in SHM with \(\phi_o = +\pi/4\) and T=8 with A =10m.
a) What is x(0), x(1), x(2 seconds)...? What is the equation for x(t)?
b) plot x(t)
c) what is the velocity at t=0? 1 sec?...

3) A block of mass 2 kg is attached to a spring with k=5 N/m. It is oscillating (SHM) with an amplitude of 20 meters.  At t=0, you observe that the block is at 14.2 meters and that it is moving to the left.
a) what is \(\omega\)?
b) what is the period of the motion?
c) What is \(\phi_o\)?
d) what is x(t)?  Graph x(t)  What is x(0.5)?  What is x(1)?
e) What is v(0)? What is v(0.5)?

4) A block attached to a spring of spring constant 2 N/m is oscillating with an amplitude of 10 meters and a period of 2 seconds.  At t=0, you observe that the block is moving to the right and that its position at t=0 is 8.7 m.
a) What is \(\phi_o\)? What is the equation for x(t)?
d) Graph x(t)  What is x(1)?  What is x(2)?
e) What is v(0)? What is v(1)?
f) (Can you infer the mass of the block?) What is the mass of the block?

Spoiler alert: For the problem just above, you have p=12 before, so after: 2*v1+2*v2=12 by COM.
Also, before, K.E.= 36 J. So after you get the 2nd equation: v1^2 + v2^2=20 .
Solving those you may get: v1=2 m/s; v2=4 m/s.

For the problem just above (2) the KE before is 320 J, and after the total KE is 140 J.
For this 3rd problem, the KE before and after is 64 J I think.
at

14 comments:

  1. Zack SchlesingerMay 19, 2018 at 4:16 PM

    Thanks for asking. The equation sheet had a problem and I fixed it. Thanks for pointing that out! Does it make more sense now? (We always use the cos function for x(t), with a phase shift.)

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  2. UCSCguyMay 19, 2018 at 8:15 PM

    Will there be a key?

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    Replies
    1. Zack SchlesingerMay 20, 2018 at 7:41 AM

      Let's try creating a key in the comments here. For example, for oscillator problem1 (OP1), I got x(0) = 5 and also x(2) =5. Is that what you got?

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  3. UCSCslugzMay 20, 2018 at 4:31 PM

    For the 1st collision problem, I am getting a pair of 2 different answers. One solution I got was v1= 4 m/s and v2= 2 m/s. The other solution I got was v1= 2 m/s and v2 = 4m/s. Why am I getting a two different answers and which one is correct and why?

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    Replies
    1. Zack SchlesingerMay 20, 2018 at 8:51 PM

      great question! The 2nd solution, v1=2... is correct because otherwise the first block would have to pass thru the 2nd block. Does that make sense?

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    2. UCSCslugzMay 21, 2018 at 1:16 PM

      This comment has been removed by the author.

      Delete
    3. UCSCslugzMay 21, 2018 at 1:20 PM

      I still dont understand how v1=2 m/s....If the first block just passes through the 2nd block, wouldn't the velocity be the same as the initial velocity(6 m/s) because it didn't collide with the 2nd block. The other solution( v1= 4m/s and v2= 2m/s) doesn't illustrate that the block passes through the second block.

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    4. physicsMay 22, 2018 at 7:58 PM

      When he said "would have to pass through the 2nd block" he means the first block can't be V1=4m/s (faster that the block in front of it, block 2) and V2=2 (slower than the block behind it, block 1) bc the first block would literally have to go through the second block (which is impossible). So, the only solution that makes sense is v1= 2m/s and v2= 4m/s.

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  4. UnknownMay 21, 2018 at 9:49 AM

    This comment has been removed by the author.

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  5. Zack SchlesingerMay 21, 2018 at 11:47 AM

    the key is already there for the collision problems, isn't it?
    For the oscillation problems, the key is in the notes posted from the 2nd week we covered oscillations.

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  6. Zack SchlesingerMay 21, 2018 at 12:27 PM

    Look at the cosine function. When is it increasing? When is it decreasing? That is the key thing!

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  7. Zack SchlesingerMay 21, 2018 at 12:28 PM

    Alternatively, you can look at the velocity function, but that is more difficult I think.

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  8. UCSCslugzMay 22, 2018 at 12:53 AM

    You use substitution to solve for one variable. Some for either V1 or V2 in the momentum equation, then plug that into the variable into the kinetic energy equation to solve for the velocity of one block. Then to find the velocity of the other block, plug in the velocity you found into the initial momentum equation you used to substitute into the kinetic energy equation to find the other velocity.

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  9. UCSCslugzMay 22, 2018 at 12:53 AM

    *Solve for either

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