Worksheet #5 - PHY102 (Spr. 2005)

Solving Equations and Differential Equations II
Due Thursday 17th February

Last week you learned about the following functions which, for physicists, are among the most useful in Mathematica:

Solve - solves polynomial equations analytically
NSolve - solves polynomial equations numerically
FindRoot - solves all equations numerically
DSolve - solves differential equations analytically
NDSolve - solves differential equations numerically

Once you know what these routines do and how to use them, you have a very powerful set of tools for solving problems in physics. However the hardest part of physics is to set up the mathematical description of the problem, and that you need to practice. This week we work on setting up mechanics problems and then solving them using mathematica.

Problem 1.

(i) A ball is falling vertically through a fluid. Apart from gravity, a drag force F_d acts on the ball. The drag force opposes the motion and increases in proportion to the speed ( $\vec{F_d} = - k \vec{v}$, where k is a drag coefficient that depends on the fluid). Find and plot the time dependence of the position and velocity of a 100g ball which is released from rest at t=0, in a fluid with drag coefficient k=0.02. Choose a time range which shows the terminal velocity of the ball.

(ii) Now consider that the same ball in the same fluid is given an initial velocity v_x = 10m/s in the horizontal direction. Use mathematica to find the motion in the x and y directions. Plot the displacement in the x direction as a function of time and plot the trajectory(parametric plot) of the ball.

(iii) Consider a cannon on a 500m hill. Assuming that the cannon fires 10kg cannonballs horizontally with initial velocity 500m/s, find the range of the cannon for a drag coefficient of k=.002. Compare this to the range in the absence of drag.

Problem 2.

You lift a box of mass m=30kg vertically a height of h meters. However you decide that you want your little brother to lift the other 30 boxes that must be lifted to the same height. Since he is weaker than you, you kindly put an inclined plane(at angle $\theta$to the horizontal) in place to assist him.

(i) If there is no friction, use mathematica to evaluate the integral $\int \vec{F}.\vec{dr}$ and hence prove that you and he do the same amount of work per box.

(ii) Now consider adding friction $f = \mu N$, where $\mu$ is the dynamics friction coefficient and N is the force normal to the inclined plane. How much additional work does your brother do (per box) due to the friction on the inclined plane? (Solve this problem analytically using Mathematica)