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iv. Linear

(b) Give a mini-lesson on how to solve second-order differential equationsby working through the following examples. Dont just provide asolution; explain the steps leading up to the solution.

i. y + 5y + 6y= 0

ii. y +y +y = 0

iii. y + 4y + 4y= 0

iv. y 3y 4y= 3e2x

v. y 3y 4y= 2 sin x

4. Harmonic oscillations occur for many different types of systems and it isimportant to recognize when the equations for harmonic motion apply.Three different systems are described below. Each system can be approxi-mately described using the equations for harmonic motion. Break up intothree groupsone group per system. For your groups system, answer thefollowing questions:

(a) What approximations are necessary for this system to exhibit har-monic oscillations?

(b) What is the differential equation that governs the motion of thissystem? Use Newtons second law to arrive at this equation.

(c) What is the solution to the differential equation that you found inpart (b)?

(d) What is the natural frequency of oscillations?

Here are the three systems:

A mass m is tied to a massless spring having a spring constant k.The system oscillates in one dimension along a horizontal fricitonlesssurface.

A particle of mass m is attached to a weightless, extensionless rodto form a pendulum. The length of the rod is L and the systemoscillates in a single plane.

A tube is bent into the shape of a U and is partially filled with aliquid of density . The cross-sectional area of the tube is A andthe length of the tube filled with liquid is L. The liquid is initiallydisplaced so that it is higher on one side of the tube than the other.

Once each group has answered all of the questions, share the results with

the entire class.

5. Consider a mass m attached to a spring of spring constant k . The springis mounted horizontally so that the mass oscillates horizontally on a fric-tionless surface. The spring is attached to the wall on the right and themass is initially moved to the right of its equilibrium position (compressing

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the spring) by a distance s and released. Working individually, determinehow (if at all) the period of the motion would be affected by each of the

changes below. Once you have answered each part on your own, compareyour answers with a classmate.

(a) The spring is replaced with a stiffer spring.

(b) The mass is initially displaced a distances to the left and released.

(c) The mass is replaced with a heavier mass.

(d) The mass is initially displaced a distancer (r < s) to the right andreleased.

6. When you were first introduced to simple harmonic motion, you used theformulamx= kxto find the position of the oscillating mass as a functionof time. This assumes that the origin is defined to be the equilibrium point.

What happens if this is not the case? What would the equation of motionlook like? How would the position of the oscillating mass as a function oftime change?

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