Thank you for taking the time to answer and rate these assessment questions. The Colorado Classical Mechanics Instrument (CCMI) is an instrument designed to test whether sophomores successfully gain some of the key skills in Classical Mechanics I. This survey asks you to both answer these questions (so that we may identify problems with the questions) and indicate whether you think that they adequately address the learning goals of the course.

First, you will be asked to answer the question. Where you are asked to sketch, describe the sketch you would draw as best as you can. Where you asked for an equation, you can type it out in plain text or LaTeX. Below each question we have listed a course learning goal, so you can get a better idea of where we feel each question fits into a classical mechanics course curriculum. After you answer each assessment question, please rate its quality by answering 3 follow up questions.

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Please Note: Once your information comes back to us, we will assign you a random number and will only reveal your answers and opinions in an anonymous manner.

1. QUESTION 1
   

   Without any calculations, write the general solution to the following differential equations. Express your answer with arbitrary constants, if necessary. For this question, A and B are real constants.


(a) \(\ddot{x}=-A^2x\)

(b) \(\frac{dy}{dt}=By\)

(c) Describe or draw a specific physical situation where the differential equation \(\frac{d^2z}{dt^2} = B\) might be applicable. Please give an explanation to justify your answer.


Topic Learning Goal: Students should have the solutions to a handful of ordinary differential equations at their fingertips. These include the most commonly used ordinary differential equations in physics.
1. How well does this question test student achievement of the learning goals?
   not answered
   a) Well
   b) Coud be improved
   c) Not well
   
   
2. Is the information given in this question scientifically accurate?
   not answered
   a) Yes
   b) Not completely
   c) No
   
   
3. Is this question written clearly and precisely?
   not answered
   a) Yes
   b) Could be improved
   c) No
   
   
4. Please make any additional comments or suggestions about this question here:
   



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2. QUESTION 2
   

   An astronaut is in orbit around the Earth at a distance of \(R\) from the center of the Earth. Another astronaut is in a closer orbit at a distance \(R - d\). The difference in the strength of the gravitational field between the astronauts is given by, \(\Delta g = \frac{G\;M_E}{(R-d)^2} - \frac{G\;M_E}{R^2}\) You are interested in predicting the difference in the strength of the gravitational field, when the astronauts are close to each other. Describe how you would simplify \(\Delta g\). For what choice of variable would be justifed in neglecting higher order terms? Explain your choice briefly but clearly.

Response:


Topic Learning Goal: Students should be able to compute the Taylor expansion of simple functions. In a physics problem, they should also be able to figure out which variable to expand in, how many terms to keep for a physically significant answer, and for what values of the expansion variable this expansion is a good approximation of the original function.
1. How well does this question test student achievement of the learning goals?
   not answered
   a) Well
   b) Could be improved
   c) Not well
   
   
2. Is the information given in this question scientifically accurate?
   not answered
   a) Yes
   b) Not completely
   c) No
   
   
3. Is this question written clearly and precisely?
   not answered
   a) Yes
   b) Could be improved
   c) No
   
   
4. Please make any additional comments or suggestions about this question here:
   



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3. QUESTION 3
   
   Below is a plot of the potential energy, in Joules, of a particle free to move on a 2-d plane.
   

(a) For which of these points (A-F) is the particle in stable equilibrium?

(b) Please explain how you decided on the above answer:

(c) Rank the magnitude of the gradient at the above points, from largest to smallest. (If some points have gradients with equal magnitude, please make that clear in your answer.)

(d) Draw vectors that represent the force, \(\vec{F}\), at points A-F on the diagram above. Please make sure that the relative magnitude of your vectors is consistent among the points. Please indicate clearly if \(\vec{F} = 0\) at any of the points.


Topic Learning Goal: (1) Students should be able to explain the physical meaning of the gradient, predict relative direction and magnitude for several points given equipotential lines, and relate the gradient to the 1 dimensional idea of slope. (2) Students should be able to determine the relative magnitude and direction of a force at several points on a set of drawn equipotential lines. (3) Students should be able to recognize equilibrium points on a plot of potential energy, \(U\), and determine if these points are stable given the function \(U(x)\).
1. How well does this question test student achievement of the learning goals?
   not answered
   a) Well
   b) Could be improved
   c) Not well
   
   
2. Is the information given in this question scientifically accurate?
   not answered
   a) Yes
   b) Not completely
   c) No
   
   
3. Is this question written clearly and precisely?
   not answered
   a) Yes
   b) Could be improved
   c) No
   
   
4. Please make any additional comments or suggestions about this question here:
   



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4. QUESTION 4
   

   Given this differential equation: \(a_1\ddot{x}+a_2\dot{x}+a_3 x = 0\), with \(x(0) = 1\) and \(\dot{x}(0) = 0\) The solution (for a particular set of positive coeffcients \(a_1\), \(a_2\), and \(a_3\)) is shown below:

   (a) If x(t) represents the motion of a mass on a spring, and each term (for example, \(a_1\ddot{x}\)) has units of force, what are the units of "\(a_1\)"?
   
   (b) What aspect of the physical situation does "\(a_3\)" describe?
   
   (c) Sketch the solution, on the axes below, if \(a_3\) is slightly smaller (but still positive) in \(a_1\ddot{x} + a_2\dot{x} + a_3 x = 0\) and everything else is the same. (Note the original solution above is drawn below for reference.)
   
   Explain in words briefly what is different between your sketch and the reference case.
   
   (d) If we replaced the "0" in the differential equation \(a_1\ddot{x} + a_2\dot{x} + a_3 x = 0\) with some function \(g(t)\), what would \(g(t)\) represent physically?
   
   
   Topic Learning Goal: (1) Students should be able to explain if a given motion is simple harmonic and why, including explaining the meaning of the terms in the differential equation of harmonic motion with damping and driving forces. (2) Students should be able to predict how changing the various parameters in this equation would change the resulting motion.
   1. How well does this question test student achievement of the learning goals?
      not answered
      a) Well
      b) Could be improved
      c) Not well
      
      
   2. Is the information given in this question scientifically accurate?
      not answered
      a) Yes
      b) Not completely
      c) No
      
      
   3. Is this question written clearly and precisely?
      not answered
      a) Yes
      b) Could be improved
      c) No
      
      
   4. Please make any additional comments or suggestions about this question here:
      
   
   

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5. QUESTION 5
   

   Please fill in the blanks so that the following correctly describe simple harmonic motion:

   (a) A restoring force that is proportional to
   
   (b) \(x(t)\) (position as a function of time) =
   
   (c) Draw \(U(x)\) - potential energy as a function of position. Describe any signifcant features of your graph.
   
   
   
   Topic Learning Goal: Students should be able to determine if a given periodic motion is simple harmonic motion or not.
   1. How well does this question test student achievement of the learning goals?
      not answered
      a) Well
      b) Could be improved
      c) Not well
      
      
   2. Is the information given in this question scientifically accurate?
      not answered
      a) Yes
      b) Not completely
      c) No
      
      
   3. Is this question written clearly and precisely?
      not answered
      a) Yes
      b) Could be improved
      c) No
      
      
   4. Please make any additional comments or suggestions about this question here:
      
   
   

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6. QUESTION 6
   

   A ball slides without friction in the bottom of a sawed off sphere of radius \(a\) (as pictured to the right). The ball is moving downhill with a known speed of magnitude \(v_0\) and is at a known angle from the horizontal of \(\theta\). (a) Draw the vectors \(\hat{r}\) and \(\hat{\theta}\) on the picture to the a right. (b) Express the velocity vector \(\vec{v}\) in the in the \(x-y\) and \(r-\theta\) coordinate systems in terms of \(\mathbf{a}\), \(\mathbf{\theta}\), and \(\mathbf{v_0}\). (i.e., \(\vec{v} =v_x\hat{x} + v_y\hat{y}\) and \(\vec{v} = v_r \hat{r} + v_{\theta} \hat{\theta}\), what are \(v_x, v_y, v_r, v_{\theta}\)?) Please show your work.

   
   (c) For the \(x-y\) coordinate system, check that your answer makes sense by considering some particular value of \(\theta\).
   
   
   
   Topic Learning Goal: Students should be able to project a given vector into components in multiple coordinate systems, and determine which coordinate system is most appropriate for a given problem.
   1. How well does this question test student achievement of the learning goals?
      not answered
      a) Well
      b) Could be improved
      c) Not well
      
      
   2. Is the information given in this question scientifically accurate?
      not answered
      a) Yes
      b) Not completely
      c) No
      
      
   3. Is this question written clearly and precisely?
      not answered
      a) Yes
      b) Could be improved
      c) No
      
      
   4. Please make any additional comments or suggestions about this question here:
      
   
   

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7. QUESTION 7
   

   I have a mass on a frictionless spring. If I pull the mass out and let it go, it oscillates with a natural frequency \(\omega_f\) . I attach this mass to a driving force \(F(t) = B \sin (\omega_d t)\), where \(\omega_d\) can be adjusted. I also add a small amount of friction to the system. Draw a rough graph of the amplitude of the oscillation of the mass as a function of the driving frequency, \(\omega_d\) (assume you measure the amplitude after all transients have died out). Identify any major features. Don't worry about being exact.

   
   Response:
   
   
   Topic Learning Goal: Students should be able to explain the concept of resonance both conceptually and mathematically.
   1. How well does this question test student achievement of the learning goals?
      not answered
      a) Well
      b) Could be improved
      c) Not well
      
      
   2. Is the information given in this question scientifically accurate?
      not answered
      a) Yes
      b) Not completely
      c) No
      
      
   3. Is this question written clearly and precisely?
      not answered
      a) Yes
      b) Could be improved
      c) No
      
      
   4. Please make any additional comments or suggestions about this question here:
      
   
   

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8. QUESTION 8
   

   Which Fourier series is the correct expansion for the function \(f(x)\)?

   
   (a) Choose your answer below.
   A) \(f(x) = 1+ \frac{4}{\pi}\sin \pi x + \frac{4}{\pi}\cos \pi x + \frac{4}{3 \pi}\sin 3 \pi x + \dots\)
   B) \(f(x) = \frac{4}{\pi}\sin \pi x + \frac{4}{3 \pi}\sin 3 \pi x + \frac{4}{5 \pi}\sin 5 \pi x + \frac{4}{7 \pi}\sin 7 \pi x + \dots\)
   C) \(f(x) = 1+ \frac{4}{\pi}\sin \pi x + \frac{4}{3 \pi}\sin 3 \pi x + \frac{4}{5 \pi}\sin 5 \pi x + \dots\)
   D) \(f(x) = \frac{4}{\pi}\sin \pi x + \frac{4}{\pi}\cos \pi x + \frac{4}{3 \pi}\sin 3 \pi x + \frac{4}{3 \pi}\cos 3 \pi x + \dots\)
   E) \(f(x) = 1+ \frac{4}{\pi}\cos \pi x + \frac{4}{3 \pi}\cos \pi x + \frac{4}{5 \pi}\cos 5 \pi x + \dots\)
   
   (b) Please explain your answer above briefly but clearly.
   
   
   Topic Learning Goal: Students should be able to determine from the even or odd symmetry of a function which terms in the Fourier series expansion are zero.
   1. How well does this question test student achievement of the learning goals?
      not answered
      a) Well
      b) Could be improved
      c) Not well
      
      
   2. Is the information given in this question scientifically accurate?
      not answered
      a) Yes
      b) Not completely
      c) No
      
      
   3. Is this question written clearly and precisely?
      not answered
      a) Yes
      b) Could be improved
      c) No
      
      
   4. Please make any additional comments or suggestions about this question here:
      
   
   

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9. QUESTION 9
   

   You plan to solve Laplace's equation in Cartesian coordinates, \(\frac{\partial^2 U}{\partial x^2} + \frac{\partial^2 U}{\partial y^2} = 0\), with given boundary conditions using the technique of separation of variables. How would you separate \(U\)?

   
   A) \(U(x,y)= f(x)+g(y)\)
   B) \(U(x,y)= f(x)g(y)\)
   C) \(U(x,y)= f(xy)\)
   D) \(U(x,y) = X(x,y)Y(x,y)\)
   E) \(U(x,y) = X(x,y)+Y(x,y)\)
   F) None - Separation of variables only applies to ordinary differential equations.
   G) Other - write it here:
   
   Topic Learning Goal: Students should be able to derive the relevant separated ordinary differential equations in Cartesian coordinates from Laplace's equation.
   1. How well does this question test student achievement of the learning goals?
      not answered
      a) Well
      b) Could be improved
      c) Not well
      
      
   2. Is the information given in this question scientifically accurate?
      not answered
      a) Yes
      b) Not completely
      c) No
      
      
   3. Is this question written clearly and precisely?
      not answered
      a) Yes
      b) Could be improved
      c) No
      
      
   4. Please make any additional comments or suggestions about this question here:
      
   
   

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10. QUESTION 10
    

    A particle (mass, \(m\)) is confined to move on the \(x\)-axis. There are two objects on the \(x\)-axis that attract this particle. One object is located at \(x\) = 0 and the attractive force between the object and the particle is proportional to the square of the distance between them (proportionality constant \(c\)). The second object is located at \(x\) = 10, and the attractive force between the object and the particle is inversely proportional to the distance between them (proportionality constant \(k\)). Consider a particle that is confined to the region between the two repulsive objects. Write down a differential equation that describes the position of the particle as a function of time, \(x(t)\).

    
    
    
    Topic Learning Goal: Students should be able to use Newton's laws to translate a given physical situation into a differential equation.
    1. How well does this question test student achievement of the learning goals?
       not answered
       a) Well
       b) Could be improved
       c) Not well
       
       
    2. Is the information given in this question scientifically accurate?
       not answered
       a) Yes
       b) Not completely
       c) No
       
       
    3. Is this question written clearly and precisely?
       not answered
       a) Yes
       b) Could be improved
       c) No
       
       
    4. Please make any additional comments or suggestions about this question here:
       
    
    

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11. QUESTION 11
    

    Consider an infinitely thin cylindrical shell with non-uniform mass per unit area of \(\sigma(\phi,z)\). The shell has height \(h\) and radius \(a\), and is not enclosed at the top or bottom. (a) What is the area, \(dA\), of the small dark gray patch of the shell which has height \(dz\) and subtends an angle \(d\phi\) as shown to the right?

    
    (b) Write down (BUT DO NOT EVALUATE) an integral that would give you the MASS of the entire shell. Include the limits of integration.
    
    
    Topic Learning Goal: (1) Students should be able to translate the physical situation in to an appropriate integral to calculate the gravitational force at a particular point away from some simple mass distributions. (2) Students should be able to choose appropriate area and volume elements to integrate over a given shape.
    1. How well does this question test student achievement of the learning goals?
       not answered
       a) Well
       b) Could be improved
       c) Not well
       
       
    2. Is the information given in this question scientifically accurate?
       not answered
       a) Yes
       b) Not completely
       c) No
       
       
    3. Is this question written clearly and precisely?
       not answered
       a) Yes
       b) Could be improved
       c) No
       
       
    4. Please make any additional comments or suggestions about this question here:
       
    
    

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    Thank you for taking the time to answer these assessment questions.
    
    If you have any additional comments about this survey as a whole, please enter them here:

    
    
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