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 attached to one end of a 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(w t), where w²=k/m.

a) Now hang the spring vertically and measure x downward. In equilibrium, I hope you’ll agree that the mass rests at position x = x₀ = mg/k. Starting from Newton’s 2^nd Law, show that if the mass is displaced 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.

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.

8. (3) Marion & Thornton, problem 3-23. This is a bit confusing. Just put one graph on each plot. (Don’t bother with the two components are the comparison with b = 0.) There isn’t much 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.