CMI CM-I Problem Sheet 7 (2012)

31. Given the force law $\vec{F}(x_1,x_2,x_3)\,=\,-(x_3\,+\,\alpha x_2){\hat\epsilon}_1\ +\ (x_3\,+\,\beta x_1) \hat{\epsilon}_2\,$
$+\,g(x_1,x_2,x_3) \hat{\epsilon}_3$, determine the potential by (a) determining the condition on the constants $\alpha\,,\, \beta$. (b) Is the potential completely determined? Give reasons.

32. Evaluate the line integral
$$\int_{(1,1)}^{(2,3)}\vec{F}.d\vec{r}$$ along a contour which is a straight line from $(1,1)\  \rm{to}\  (2,3)$ in the $(x_1\, ,\, x_2)$ plane. $\vec{F}\ =\ x_1^2\, \hat{\epsilon}_1\ +\ x_1\,x_2\ \hat{\epsilon}_2.$

33. A sphere of radius R with it's centre at origin has uniform mass density. Find $V(r)$ where $r\, \leq\, R$. The mass of the sphere is $M$.

34. Suppose one wants to fire a rocket so that it falls into the sun. One way is to make it's speed zero in the sun's rest frame. Another (the second way)is to fire it so that it reaches a maximum distance $R_2\ =\ \alpha R_1$, ($R_1$ is the radius of the earth's orbit) at which place rockets fire so that it's speed reduces to zero and which makes it fall eventually into the sun.

1. Find the change in velocity needed to be supplied by the engine of the rocket in the first case.
2. Find the sum of the changes in the speed needed at earth and at $R_2$ as a function of $\alpha $. Show that this is a decreasing function of $\alpha$. Comment on your answer keeping the fuel efficiency in mind.

Assume the earth's gravity and air resistance  can be neglected and earth's orbit is circular.

35. A beam of particles pass through a target  made of thin foil of  a very small thickness $\Delta x$ having N particles per unit volume. Let the collision cross section be $\sigma$. If the intensity of the beam  ( number of particles per unit are per unit time) is I to start with what would be the decrease in the intensity $\Delta$ in terms of $N,\,\Delta x,\rm{and}\, \sigma$?

Suppose the thickness of the target is $X$ show that the intensity of the beam after passing through the target is given by $$ \rm{I}(X) = \rm{I}\, e^{-\rm{N}\sigma \rm{X}} $$