Consider a completely miscible two-component system whose overall composition is X, at a temperature where liquid and gas phases coexist. The composition of the gas phase at this temperature is Xa and the composition of the liquid phase is Xb. Prove the lever rule, which says that the proportion of liquid to gas is (X -Xa)/(Xb -X). Interpret this rule graphically on a phase diagram.

The Plutonium isotope 239Pu has a half-life of 24,000 years and decays by the emission of a 5.2 MeV alpha particle. Plutonium is not especially dangerous if handled because the activity is low, and the alpha radiation doesn’t penetrate the skin. However, there can be serious health concerns if the tiniest particles are inhaled and lodge deep in the lungs. This could happen following any kind of fire or explosion that disperses Plutonium as dust.

a) (3 pts.) Soot particles are roughly 1 μm in diameter, the density of Plutonium is 19,800 kg/m^3 , and its atomic mass is 244 amu. Determine the number of atoms in a 1 μm diameter particle of 239Pu.

b) (1 pt.) Determine the activity (in Bq) of this particle.

c) (2 pts.) Suppose the particle stays lodged in the lungs for 20 years. How much energy is released by emitted alpha particles over this 20-year period?

(Hint: The equation involving �892 tells you how many atoms are remaining, not how many have decayed. The total energy is the number of decayed particles times the energy of an emitted alpha particle.)

We want to determine the dose delivered to the lungs by this particle, averaged over this 20-year period. You can assume all the alpha particles are stopped within a 25 µm-radius sphere around the particle.

d) (2 pts.) What is the mass of the sphere in which the alpha particles deliver all their energy? Assume the tissue density is the same as water, 1000 kg/m^3.

e) (2 pts.) Determine the dose (in Sv) delivered to the lungs by this particle, averaged over this 20-year period. The RBE for alpha particles is 20.

A 515-Hz tuning fork is struck and placed next to a tube with a movable piston, creating a tube with a variable length. The piston is slid down the pipe and resonance is reached when the piston is 112.5 cm from the open end. The next resonance is reached when the piston is 82.5 cm from the open end. Round all answers to one decimal place.

Hint (a)

1. What is the speed of sound in the tube?
   v=v=  m/s
2. How far from the open end will the piston cause the next mode of resonance?
   L=L=  cm

(a) Calculate the speed of an electron that is in the n = 1 orbit of a hydrogen atom, and give your answer v as a fraction of the speed of light in empty space c, for example, v = 0.5 if the answer were v = c/2 = 1.50 × 108 m/s. (It isn’t.)

(b) How many nanometers would be the wavelength of the photon emitted when the electron in a hydrogen atom jumps from the n = 3 orbit to the n = 2 orbit? This is the Hα line, and its light is scarlet, the color of fresh human blood.

(c) How many nanometers would be the wavelength of the photon emitted when the electron in a hydrogen atom jumps from the n = 2 orbit to the n = 1 orbit?

(d) How many nanometers would be the wavelength of a photon that would have the minimum amount of energy needed to ionize any hydrogen atom? (Hint: Electromagnetic radiation with this wavelength or shorter is called extreme ultraviolet radiation.

(e) How many electron-volts (eV) would the electron in part (7)(d) need to have?

List out (calculate) the initial and final velocities, the initial and final momentums, the impulse, and the amount of force required for the change described to occur for the five listed situations: A car with a mass of 4.0 x 103 kg starts at rest and accelerates to 40 km/h in one minute (1). It then slams on its breaks and slows to 5 km/h in 5 seconds (2). It then comes to a complete stop in 1 second (3). Then it reverses to a speed of 7 km/h in 10 seconds (4). Lastly, it comes to a stop in 3 seconds (5).

A cord is wrapped around a pulley of diameter 0.320 m with the free end tied to a weight. That weight is allowed to descend at constant acceleration. A stopwatch is started at the instant when the weight reaches a speed of 0.023 m/s. At a time of t = 1.50 s, the weight has dropped an additional 0.080 m. Calculate the magnitude of the centripetal acceleration of any point on the circumference of the pulley at t = 4.0 s.

Calculate the magnitude of the tangential acceleration of any point on the circumference of the pulley.

Block 1 of mass m₁ slides along a frictionless floor and into a one-dimensional elastic collision with stationary block 2 of mass m₂ = 5m₁. Prior to the collision, the center of mass of the two-block system had a speed of 8.40 m/s.

a) what is the speed of the center of mass after the collision?

b)What is the speed if the block 2 after the collisions?

To study the Earth, some aliens have put a spaceprobe in the solar system. It has an elliptical orbit with a semi-major axis of 2 AU. The closest distance the probe gets to the Sun is 1 AU.

a. What is the farthest the probe gets from the sun?
b. What is the sidereal period of the probe (in years)? What is its synodic period?
c. At what point will the probe be moving the slowest? The fastest?
d. What is the eccentricity, e, of the orbit?

Two cars collide at an icy intersection and stick together afterward. The first car has a mass of 1450 kg and is approaching at 8.5 m/s due south. The second car has a mass of 850 kg and is approaching at 16.5 m/s due west.

(a) Calculate the magnitude of the final velocity, in meters per second, of the cars.

(b) Calculate the direction of the final velocity, in degrees south of west, of the cars.

(c) What is the change in kinetic energy, in joules, for the collision? (This energy goes into deformation of the cars.)

Consider a block of copper.

a. Find the density of mobile charges in a piece of copper, assuming each atom contributes two free electrons.

b. Calculate the average electron velocity in a copper wire 1.00mm in diameter which carries a current of 1.00A.

c. What is the force of attraction between two such wires, 1.00cm apart?

d. If you could somehow remove the stationary positive ions, what would the

electrical repulsion force be?

e. How many times greater is the electrical force in part (d) than the

magnetic force in part (c)?

The electron in a hydrogen atom starts in the n = 7 level. Determine all the possible wavelengths that could be emitted if the electron ends in the first excited state.

Determine the final temperature and composition of the mix of 20 g of ice at -40°C and 30g of water at 80°C.

A ballistic pendulum consists of a large heavy mass, M on the end of a very light rod of length L. The rod is free to pivot at the top and the mass is attached at the bottom, so it hangs like a pendulum. A second mass, m, is fired horizontally at speed v straight into the large mass, and they stick together. The pendulum swings to a maximum angle ?m.

a)Show that you can measure v by measuring ?m, that is obtain an expression for v in terms of M, m, L, g and ? m.

b)Look up the mass and speed of a bullet, and figure out how mass

ive M should be if L is about 0.25 m and we want a maximum angular displacement no greater than 45?.

A block of mass 2 kg and a block of mass 3 kg are sliding on a frictionless surface. They collide and stick together. Before the collision the first block was travelling at 5 m/s in the positive x direction. After the collision the two blocks are travelling at 6 m/s in the negative x direction. What was the x component of the velocity of the second block before the collision?

An electron and a 0.0220 kg bullet each have a velocity of magnitude 490 m/s, accurate to within 0.0100%. Within what lower limit could we determine the position of each object along the direction of the velocity? (Give the lower limit for the electron in mm and that for the bullet in m

(b)

What If? Within what lower limit could we determine the position of each object along the direction of the velocity if the electron and the bullet were both relativistic, traveling at 0.350c measured with the same accuracy? (Give the lower limit for the electron in nm and that for the bullet in m )