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?

I need report about( Lighting pollution) with detail..please use good hand write..

An electron is one of the most fundamental particles in nature. It is everywhere, in all the matter we can see, and it is with electrons that light interacts when it is emitted, absorbed, or scattered in everyday matter. The electron has a more massive cousin called a muon, also with a charge of -1 e, but with a mass of 1.88x10-28 kg. The electron's mass is 9.11x10-31 kg. You can see why a muon is called a "heavy" electron. A muon is also unstable, and left alone at rest in a lab it will turn into an electron, kinetic energy, and two neutrinos with a lifetime of 2.20x10-6 s.

1. Suppose we have a vacuum with two metal plates separated by 2 meters. One of the plates has a hole in it so that a charged particle coming from the other one can pass through into empty space. The plate with a hole is attached to a voltage source of 5,000,000 V and electrons and muons are introduced near the other plate at 0 V. What energies and velocities do the electrons and muons have when they pass through the hole?

2. If an electron or a muon encounters a magnetic field that is perpendicular to the line the particle is moving on, is their path changed? If so how does it depend on whether it is an electron or muon, and if not, why not?

3. Using an accelerator or some other natural phenomenon, suppose that you could observe a muon traveling through your lab at 0.999 times the speed of light. How long would it take to decay?

5. Consider a particle in a two-dimensional, rigid, square box with side a. (a) Find the time independent wave function φ(x,y)describing an arbitrary energy eigenstate. (b)What are the energy eigenvalues and the quantum numbers for the three lowest eigenstates? Draw the energy level diagram

Extra problem: The basis of the l = 1 Hilbert space in the (L², L_z) representation are Y₁₁(θ, φ), Y₁₀(θ, φ), Y_1-1(θ, φ).

(1) Find the matrix expressions of L_x, L_y and L_z. Hint: make use of L₊ and L_-.

(2) Suppose the system is in a normalized state Ψ = c₁Y₁₁ +c₂Y₁₀. Find the possible values and corresponding possibilities when measuring L_z, L², and L_x, respectively.

Suppose a sled passenger with total mass 51 kg is pushed 26 m across the snow (μk = 0.20) at constant velocity by a force directed 37° below the horizontal. (a) Calculate the work of the applied force (in J).

(a)

Calculate the work of the applied force (in J).

(b)

Calculate the work of friction (in J).

(c)

Calculate the total work (in J).

I have the answers for this problem from a key, the answers are 3060, -3060, and 0 for a, b, and c respectively.

I have no clue how to get those answers.