III Duluth Physics Olympiad

April 23rd, 2016
University of Minnesota, Duluth

Part I: Theoretical

Problem 1:

Astronaut Mark Watney is traveling in a small capsule and needs to reach the Hermes spacecraft. The spacecraft and the capsule are travelling on perpendicular courses, each with velocity v = 5 m/s. At the initial moment of time (t = 0), the distance between the capsule and point A is 3.2 km, while the distance between Hermes and point A is 1.8 km. At any time Watney can eject himself from the capsule with velocity 3 m/s in the direction perpendicular to the capsule’s course. At what time t does he need to press the eject button to reach Hermes?

figure pic3-1.png

Problem 2:

A cylindrical vessel contains a large number of spherical beads made from some unknown material and a small float placed on top of them (see figure 1). An experimenter starts pouring water into the vessel. The water can freely pass between the beads. The plot (see figure 2) shows the distance h between the float and the bottom of the vessel (in centimeters), as a function of volume V of water that was poured into the vessel (in liters). The total mass of all beads is m = 2.1 kg. What is the density of the beads material?

figure pic3-2.png figure pic3-3.png

Problem 3:

A metal wire is used to make the arrangement shown in the figure below. The diameter of the outer ring is D. Two rings of diameter D ⁄ 2 are placed inside it; then two rings of diameter D ⁄ 4 are placed inside those. At the places where they touch, the rings are in electric contact. The electric resistance per unit length of the wire material is λ (Ohm/m).
a. Find the electric resistance R between points A and B.
b. Wire rings of progressively smaller diameter are continued to be added to the arrangement according to the same rule (rings of D ⁄ 8 are inserted into the D ⁄ 4 rings, and so on), to infinity. What is the electric resistance between points A and B of the resultant arrangement?

figure pic3-4.png

Problem 4:

A room has the shape of a cylinder with radius r. A rubber ball is thrown from the center of the room’s floor at an angle α, with some velocity. The ball bounces three times against the walls and ceiling (as shown) and, after time T, falls back to the center of the floor. What was the initial speed of the ball? You may neglect the air resistance and may consider that no energy is lost during the collisions with walls and ceiling.

figure pic3-5.png

Part II: Experimental

Problem 5:

a. Determine the mass m and volume V of a small rock.
Equipment: The object (a pebble), an assortment of large and small paper clips, a glass of water, string, scissors.
The mass of a large paper clip is 6.7 grams, the mass of a small paper clip is 0.93 grams.

b. Determine the mass of a wooden ruler.
Equipment: stand, ruler, string, scissors, modeling clay, a glass of water.

The density of water is ρ = 1000 kg/m3. Pens, pencils, paper, and rulers can be used for writing your lab report but not for measurements.

Problem 6:

Determine the internal diameter of a syringe needle. (Be careful with the needle!)

Equipment: syringe, needle, ruler, stand, pieces of marking tape, cup with water, stopwatch (you may use your own), container for collecting water.