Course:PHYS350/Mock Midterm

1. A particle of mass $m$ moves under a conservative force with a potential energy function given by

$V(x)=\{\begin{array}{cc}\frac{1}{2}k(a^2-x^2)& \mathrm{f}\mathrm{o}\mathrm{r}|x|<a,\\ 0& \mathrm{f}\mathrm{o}\mathrm{r}|x|\ge a,\end{array}$

where $a$ and $k$ are constants, and $a>0$. What is the force on the particle? Sketch the function $V$, for both cases $k>0$ and $k<0$, and describe the possible types of motion.

Let's take $k=m{\alpha }^2>0$ and we have a particle initial in the region $x<-a$, moving to the right with velocity $v$. When it emerges into the region $x>a$, will it do so earlier or later than if it were moving freely under no force? Find an expression for the time difference. (To do the required integral, try the substitution $x=A\sinh u$ where $A$ is some constant.

2. A particle of mass $m$ is attached to a string of length $l$. The particle moves in a horizontal plane without friction, and the string passes through a hole in the plane and is attached to another object of mass $M$ below the plane. The second only moves vertically under the Earth's gravitational field. Write out the Lagrangian for this system and derive the equations of motion. What quantities are conserved?