Physics 321 -- Spring 2003

 

Homework #5, due Wednesday Feb. 19

 

Read Chapter 3, sections 3.1 – 3.6 and Appendix C.

 

1.         (3) Marion & Thornton: problem 3-1.

 

 

2.         (3) Marion & Thornton, problem 3-2.

 

 

3.         (2) Marion & Thornton, problem 3-3.

 

 

4.         (3) Marion & Thornton, problem 3-13.  Calculate the time derivatives of the x(t) given in the problem, and plug them into the damped harmonic oscillator equation (3.35) with the condition w02 = b2.  This will give you a very simple differential equation for y(t).

 

 

5.         (4) Consider a mass m lying on a frictionless table, attached to one end of a horizontal spring of force constant k.  The other end of the spring is fixed.  If the mass is displaced a distance A along the direction of the spring and then released, it follows simple harmonic motion: x(t) = Acos(wt), where w2=k/m.

 

a) Now hang the spring vertically and measure x downward.  In equilibrium, the mass rests at position x = x0 = mg/k.  Starting from Newton’s 2nd Law, show that if the mass is displaced from equilibrium a distance A and then released, it will follow simple harmonic motion about the position x0.  Hint: You will derive an inhomogeneous differential equation for x(t).  There are two equivalent approaches to solving it: 1) Define a new variable y(t)=x(t)-x0, or

2) Try a solution of the form: x(t) = (solution to homogeneous equation) + C.

 

            b) Starting from equilibrium at t=0, apply a constant downward force F0 to the mass for all times t>0.  Derive an equation for the displacement as a function of time, x(t).

 

 

6.         (3) Marion & Thornton, problem 3-9.

            This is the hardest problem on this homework set.  Do the previous problem before you try this one!  From part (b) of the previous problem, you know the displacement x(t) for 0<t<t0.  For t>t0, the force is gone and you need a solution to the homogeneous equation.  Since x(t) and xdot(t) are continuous functions of time, the initial conditions of the solution for t>t0 , x(t0) and xdot(t0), must match the final values of your solution for 0<t<t0.

 

7.         (4) Marion & Thornton, problem 3-22.

 

 

8.         (3) Marion & Thornton, problem 3-23.  Let’s simplify this problem a bit.  Don’t bother with plotting the two components e-bt and cos(w1t-d) – just plot their product.  You can make nine plots if you want, but I think you’ll learn more if you make only three plots – one for each value of d -- and put three graphs on each plot for the three different values of b.  (Don’t do b = 0.)  There isn’t anything to discuss – just make the plots and look at them!

 

To make the plots, use Mathematica in one of the computer labs on campus.  To plot a function in Mathematica, type in a command like this:

 

                                    Plot[Cos[3x+4]-5*Sin[7x-3], {x, 0, 10}]

 

            After typing in the command, hit Shift(Enter) to execute it.  Notice that built-in functions like Plot, Cos, Sin, Sqrt, and Exp are capitalized, and their arguments are in square brackets.  Notice also that the * for multiplying a constant times a variable or function is optional – I used it in 5*Sin, but didn’t use it in 7x.  The independent variable and its range are in curly brackets.

 

            You can also define your own function, in the following way:

 

                                    MyFunction[x_]:=Cos[3x+4]-5*Sin[7x-3]:

Plot[MyFunction[x], {x,0,10}] 

 

            Notice that the independent variable is followed by “_” in the function definition.

Your defined function can have several parameters, for example:

 

                                    MyFunction2[x_,A_,B_]:=Cos[A*x+B];

                                    Plot[MyFunction2[x,3,4], {x,0,10}]

 

            You can also put several graphs on a single plot by putting a list of functions inside curly brackets {}:

 

                                    Plot[{MyFunction2[x,3,4],MyFunction2[x,7,8]}, {x,0,10}]

 

            You get the idea.