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### Content from Introductory Physics that Upper-Division Physics Students Still Find Challenging

• Phase shifts. Many students are still troubled by the geometric meaning of $\delta$ in an expression like $A\sin(\omega t + \delta)$. Useful homework problems that address the issue of left/right shifts in a function are:
1. (TrigParameters)

Make sketches of the following functions, by hand, all on the same axes. Briefly describe, using good scientific writing that includes both words and equations, the role that the number two plays in the shape of each graph: \begin{eqnarray} y &=& \sin x\\ y &=& 2+\sin x\\ y &=& \sin(2+x)\\ y &=& 2\sin x\\ y&=& \sin 2x \end{eqnarray}

2. (ThetaParameters)

The function $\theta(x)$ (the Heaviside or unit step function) is a defined as: \begin{eqnarray*} \theta(x) = \left\{ \begin{array}{l l} 1 & \quad \mbox{for $x>0$}\\ 0 & \quad \mbox{for $x<0$}\\ \end{array} \right. \end{eqnarray*} This function is discontinuous at $x=0$ and is generally taken to have a value of $\theta(0)=1/2$.

Make sketches of the following functions, by hand, on axes with the same scale and domain. Briefly describe, using good scientific writing that includes both words and equations, the role that the number two plays in the shape of each graph: \begin{eqnarray} y &=& \theta (x)\\ y &=& 2+\theta (x)\\ y &=& \theta(2+x)\\ y &=& 2\theta (x)\\ y &=& \theta (2x) \end{eqnarray}

3. (TriangleParameters)

Consider the function: \begin{eqnarray*} f(x) = 3x\,\theta(x)\,\theta(1-x)+(6-3x)\,\theta(x-1)\,\theta(2-x) \end{eqnarray*} Make sketches of the following functions, by hand, on the axes with the same scale and domain. Briefly describe, using good scientific writing that includes both words and equations, the role that the number two plays in the shape of each graph: \begin{eqnarray} y &=& f(x)\\ y &=& 2+f(x)\\ y &=& f(2+x)\\ y &=& 2f(x)\\ y &=& f(2x) \end{eqnarray}

• Electrostatic potential. Students often try to apply vector reasoning appropriate to electric fields instead of scalar fields. For example, students may claim that midway between two positive point charges there is no electric potential because the vectors cancel.
• Magnetic Fields. For example, when asked how a series of concentric spinning charged rings could be used to produce a non-zero non-changing magnetic field, some of our juniors answered that different rings would have to be spinning different directions to cancel out the field of other rings. We also found other examples in which students confused a non-zero magnetostatic field with a zero magnetic field.
• Velocity versus angular velocity for rotating objects. For example, when trying to find the current for a spinning ring of radius R and charge Q, some students tried to use angular velocity, $\omega$, instead of $v$ for $I = \lambda v$, while other students tried to relate $\omega$ and $v$ with a factor of $2\pi$ instead of a factor of $R$.
• The combination of Newton's 2nd and 3rd Law for bodies not in contact. When discussing elliptical orbits of two objects of different mass, students will often offer explanations that involve a violation of Newton's 2nd or 3rd law.
• Conservation of linear momentum in combination with center of mass. Especially when rotation of multiple bodies is involved. (This was evident in our “Survivor: Outer Space” activity)
• Conservation of angular momentum when the kinetic energy of an object is changing. e.g. eliptical orbits

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