A support beam, within an industrial building, is subjected to vibrations along its length; emanating from two machines situated at opposite ends of the beam. The displacement caused by the vibrations can be modelled by the following equations,

?1 = 3.75 sin (100?? +2π/9)

?2 = 4.42 sin (100?? −2π/5)

a) when both machines are switched on how many seconds does it take for each machine to produce its maximum displacement?

b) at what time does each vibration first reach a displacement of -2mm?

Calculate the density of states function for a systemconsisting of alarge number of independent two-dimensional simple harmonic oscillator.Use an adaptation of the methods developed in"distribution function and densiy of states" for the 3D case

The speed of a water wave is described by v= (gd)^(1/2)  where d is the water depth, assumed to be small compared to the wavelength. Because their speed changes, water waves refract when moving into a region of different depth.

Suppose waves approach the coast from a storm far away to the north–northeast. Demonstrate that the waves move nearly perpendicular to the shoreline when they reach the beach.

Suppose waves approach the coast, carrying energy with uniform density along originally straight wave fronts. Show that the energy reaching the coast is concentrated at the headlands and has lower intensity in the bays.

A square frame made of 4 mm-thick iron rods and an edge
length of 1 m is pushed into a homogeneous magnetic field of B=2 T (T=Vs/m^2) with a constant speed of 15 m/s. Assuming the frame is initially at room temperature (20°C).

a) Why does the frame heat up?

b)What is the temperature of the frame right after the experiment?

c) Now you pull it out of the B-field with the same constant speed, what is the temperature of the frame after it has felt the B-field completely?

Describe why gain saturation in a homogeneously broadened medium occurs

A mass on a spring undergoes simple harmonic motion. At t = 0 its displacement is 1m and its velocity is 1m/s towards the equilibrium position. What single piece of information allows you to determine frequency and amplitude?

A. mass

B. spring constant (k)

C. kinetic energy at t = 0

D. acceleration at t = 0

E. force at t = 0

Consider a one-dimensional chain of identical atoms. The springs between them alternate in strength between values K1 and K2.

a) Find the vibrational frequencies as a function of wave number q. Study the low q limit and find the sound velocity.

b) Discuss the physical meaning of the two branches. Sketch the way the atoms move in both cases!

c) Discuss the dispersion and the normal modes for K1 ≫ K2.

Four identical metallic objects carry the following charges: +1.82, +6.98, -4.95, and -9.10C. The objects are brought simultaneously into contact, so that each touches the others. Then they are separated. (a) What is the final charge on each object? (b) How many electrons (or protons) make up the final charge on each object?

A stage-discharge relationship is an inexpensive tool for obtaining continuous records of river discharge. Using sketches, describe what you understand by a stage-discharge relationship and how it fills the role of a cheap alternative for continuous river flow measurement. What precautions must be taken in the long-term use of stage-discharge relationships?

(c) Describe the dilution gauging method, emphasising the main differences between constant rate injection and gulp injection techniques. Starting from the basic mass balance equation, derive the equation for estimating river discharge using the constant rate injection method. What precautions must be taken when using the dilution gauging method?

Suppose we need to construct a tin can with a fixed volume V cm3 in the shape of a cylinder with radius r cm and height h cm. (Here V should be regarded as a constant. In some sense, your answers should be independent of the exact value of V .) The can is made from 3 pieces of metal: a rectangle for the side and two circles for the top and bottom. Suppose that these must be cut out of a rectangular sheet of metal. Our goal is to find the values of r and h, and the dimensions of this rectangular sheet that minimize its area.

1. Draw a picture of how the rectangle and two circles could be cut out of a larger rectangle. There are multiple ways to do this (I can think of at least 3). Draw as many as you can, solve the problems below for each arrangement and then compare your answers.

2. Label the sides of the rectangle in terms of r and h. Express the rectangle’s area in terms of r and h. Also, note whether there are any assumptions about r and h that you need to make in order for your picture to make sense. (For example, if you draw a circle with diameter 2r inside of a rectangle with side l, then you must have 2r ≤ l.)

3. Use the fact that the can’s volume is V = πr2h to express h in terms of r, and write the rectangle’s area as a function of r. (Or else, you may alternatively solve for r and write the area as a function of h.)

4. Find the value of r (or h) that minimizes the rectangle’s area. What is the correspond- ing value of h (or r), and the dimensions of the rectangle? Your answers will most likely be in terms of V , but the ratio h/r might be a number. What is the minimum area of the rectangle in terms of V ?

5. As mentioned above, you should complete (1)-(4) for as many different arrangements as you can think of. (The math for some might be very simple.) Then compare your answers to find the best way of arranging the 2 circles and rectangle inside the larger rectangle, and the minimum possible area of the rectangle.

6. What if you need to make 2 (or more) cans in the same way. Can you find an arrangement of all the necessary pieces inside a single rectangle that is even more efficient?

a spherical shell of mass 2.0 kg rolls without slipping down a 38 degree slope A) Find the acceleration, the friction force, and the minimum coefficient of friction needed to prevent slipping. B) if the spherical starts from rest at the top how fast is the center of mass moving at the bottom of the slope if the slope is 1.50 m high? PLEASE INCLUDE DIAGRAM

Determine the amplitude of a driven oscillator if the frequency of the driver matches the natural frequency of the oscillator itself. Assume a friction-free system.

A damped oscillating system is driven by a variable force. Which of the following, when increased, will decrease the steady-state amplitude?

1. The magnitude of the maximum driving force
2. The mass of the oscillator
3. The damping coefficient
4. A combination of two. If so, which ones?

A soap bubble is floating in the air. The refractive index of the bubble is n = 1.33. The width of the wall is 115 nm. The light that hits the bubble is reflected on the outer and inner surface and both interfere once they leave. The difference in phase for both for both lights will be wt, where t is the time it takes for the light to go and return through the soap layer (t = d / (c / n)).

a) Draw a picture showing how interference occurs when light is reflected from thin films.

b) The light that most reflects towards the front of the bubble is that where the phase is 2pi. Find the formula for the frequency of light that is most feflected from the front. (At other angles this changes, since d = t / cos (0)). What is this value for the bubble?

c) What happens in other angles? What do the colors of the light reflected by the bubble look like?