Problem 1 -

Refer to the diagram here -a side view, not to scale. A pipe of radius r connects a lake to a cylindrical tub of diameter d. At rest in the tub is a cube-shaped block of sides s.

Initially, the tub contains no water (but assume that the pipe is already full of water). Then the pipe's valve is opened for a time Δt, so that water pours into the tub. At the end of that time interval, the valve is closed; the flow stops. At that moment, the block is 80.0% immersed and the force exerted on it by the bottom of the tub has decreased by 5.00%. The water level in the lake has not changed to any measureable extent.

The block is then removed from the tub, dried, and transported to the apparatus shown here, which has been at rest, under the conditions shown:

A₁ and A₂ are the areas of mass-less platforms resting on top of cylindrical columns of oil (of density ρ_oil).

The initial column heights (measured from the common level shown) are h_1.i and h_2.i.

Column 1 is open to the atmosphere. Column 2 contains helium (an ideal gas) at an initial pressure P_2.i and a constant temperature.

Now the block is carefully placed onto the platform A₁, and the entire apparatus adjusts to a new equilibrium position (where everything again comes to rest)..

a. Find Δt. (How long was the valve open?)

b. What is the final height, h_2.f, of column 2?

Show how to get an algebraic solution for each. The known values: r, d, s, H, ρ_water, A₁, A₂, ρ_oil, h_1.i, h_2.i, P_2.i, P_atm, g.

Problem 2 -

Refer to the diagram (side view, not to scale). A hemisphere of glass is resting on its circular face at the bottom of a glass aquarium that is filled with water. The water surface is open to the air; air also surrounds the aquarium.

A red or green laser can be directed vertically upward through the bottom of the aquarium, entering the hemisphere a horizontal distance x from the center of its circular face, as shown.

Keep four significant digits in your final answers.

The data:

n_air.all = 1.000

n_water.red = 1.324                n_glass.red = 1.510

n_water.green = 1.337             n_glass.green = 1.525

R = 20.00 cm       W = 43.00 cm     D = 23.00 cm

a. For what range of x-values (within 0 ≤ x ≤ R) will the red laser beam emerge into air directly from water?

b. For what range of x-values (within 0 ≤ x ≤ R) will the red laser beam emerge into air from the aquarium's bottom?

c. For what range of x-values (within 0 ≤ x ≤ R) will the red laser beam emerge into air from the aquarium's side?

d. For what range of x-values (within 0 ≤ x ≤ R) will the green laser beam emerge into air from the aquarium's side?

Problem 3 -

For a long time as a younger adult, a certain physics instructor's eyes each had the following focal range:

NP = 12.0 cm      FP = 35.0 cm

a. What was the Refractive Power (in Diopters) of the glasses he wore to allow him standard far viewing?

b. How close could he hold a book to his eyes (while wearing these glasses) and still focus on the words in the book?

Then time passed. Alas. Now his focal range is this:

NP = 17.0 cm      FP = 25.0 cm

c. What is the Refractive Power (in Diopters) of the glasses he now needs so that he still has standard far viewing?

d. How close can he hold a book to his eyes (while wearing these glasses) and still focus on the words in the book?

 (After you complete result d, notice that it's greater than the standard near-viewing distance. This is why he had to switch to bi-focals-two different sets of lenses: one for distance viewing, the other for close viewing.)

Problem 4 -

A certain student has identical eyes, each with a near point of 69.0 cm and a standard far point. She wears eyeglasses to correct her near vision. One day, she accidentally drops her glasses and the lenses separate from the frames.

a. What would either lens by itself have as a "power rating" ("Nx") if someone tried to sell it as a magnifying glass?

b. Explain how a person with 25-cm near point could gain detail (attain some angular magnification) by using this lens as a magnifying glass to view a tiny object. Use a drawing (with labels) to help your explanation.

c. For a person with the 25-cm near point, what are the maximum and minimum object distances he could use and still see more detail than with his naked eye? And what's the maximum angular magnification he can achieve?

d. If the student uses her own eyeglass lens as a magnifier, what's the standard ("comfort") angular magnification she achieves when viewing a tiny object?

e. Explain fully, with a labeled diagram and accompanying calculations for all distances, how she can use her second lens to double the amount of detail she can see (compared to d, above)-still with "comfort" viewing via the first lens right next to her eye. (An inverted view is OK-she just wants to see twice the detail.)

f. Suppose the student now attempts to use her two eyeglass lenses as a standard two-lens telescope (where the lenses have coinciding focal points, and with her eye next to the eyepiece). If she views wild animal that is 207 m away (and cannot be approached), what angular magnification does she achieve?

g. Explain fully, with a labeled diagram and accompanying calculations for all distances, how she could use her two eyeglass lenses to achieve an angular magnification of +3.00. (Hint: She'll need someone else to hold the lenses, because she won't have her eye near to either lens; this is not anything like the standard telescope setup.).