1) A string trimmer is a tool for cutting grass and weeds; it utilizes a length of nylon "string" that rotates about an axis perpendicular to one end of the string. The string rotates at an angular speed of 47 rev/s, and its tip has a tangential speed of 56 m/s. What is the length of the rotating string?

2) A planet orbits a star, in a year of length 4.62 x 10⁷ s, in a nearly circular orbit of radius 3.87 x 10¹¹ m. With respect to the star, determine (a) the angular speed of the planet, (b) the tangential speed of the planet, and (c) the magnitude of the planet's centripetal acceleration.

3) A star has a mass of 1.56 x 10³⁰ kg and is moving in a circular orbit about the center of its galaxy. The radius of the orbit is 3.9 x 10⁴ light-years (1 light-year = 9.5 x 10¹⁵ m), and the angular speed of the star is 2.3 x 10^-15 rad/s. (a) Determine the tangential speed of the star. (b) What is the magnitude of the net force that acts on the star to keep it moving around the center of the galaxy?

Over a spatial continuum, it is easy to see why some topological solitons like vortices and monopoles have to be stable. For similar reasons, Skyrmions also have to be stable, with a conserved topological density. The reason is nontrivial homotopy.

Surprisingly, in some phases, but not all phases, the analog of topological solitons, or at least what can be interpreted as them, also emerge over lattice models. Why is that? There is no nontrivial homotopy over a lattice. Why are there some phases of the XY-model with deconfined vortices and antivortices? Why are deconfined monopoles present in some 3D lattice models?

A small solid sphere of mass M₀, of radius R₀, and of uniform density ρ₀ is placed in a large bowl containing water. It floats and the level of the water in the dish is L. Given the information below, determine the possible effects on the water level L, (R-Rises, F-Falls, U-Unchanged), when that sphere is replaced by a new solid sphere of uniform density.
Read it to me
R F U R or U F or U R or F or U  The new sphere has density ρ = ρ₀ and radius R > R₀
R F U R or U F or U R or F or U  The new sphere has radius R = R₀ and density ρ > ρ₀
R F U R or U F or U R or F or U  The new sphere has density ρ < ρ₀ and mass M = M₀
R F U R or U F or U R or F or U  The new sphere has radius R > R₀ and density ρ < ρ₀
R F U R or U F or U R or F or U  The new sphere has mass M = M₀ and radius R < R₀
R F U R or U F or U R or F or U  The new sphere has mass M < M₀ and density ρ = ρ₀

Microwave communication is important for modern information technologies. Your cellphone uses at least three microwave communication technologies: the mobile signal, the WiFi signal, and the GPS location service. Google for the frequency bands for these three technologies, and briefly explain why we can have all three functions running on a single cellphone simultaneously without them interfering with each other.

b) Determine the force the sun exerts on a kilogram of water on the earth’s surface at a point nearest the sun and at a point farthest from the sun. (c) Do the same for the force exerted by the moon. (d) Explain why the tides are associated with the motion of the moon.

a) Kinetic theory. The speed of sound in the air is 330 m/s under standard conditions of temperature and pressure (273 K and 1 atm). Since the size of a molecule is much smaller than the average distance between the molecules, this number provides an estimate of the order of magnitude of the molecular media velocity. Consider a cubic meter of air and concentrate it on a N2 molecule that travels in the x direction at 330 m / s. Its mass is equal to 28 grams / 6.02 x 10 ^ 23.
1) Calculate the amount of movement imparted to the wall when it collides in an elastic way with it. Why is twice the amount of movement that it has?

2) How long does it take to do the 2-meter round trip between collisions against the same wall? Note that collisions between molecules (ideal gas) are not taken into account.

3) What is the average force exerted by a molecule on the wall?

4) If it is known that 6.02 x 10 ^ 23 N2 molecules (i.e. 28 grams) occupy 22.4 liters in standard conditions of temperature and pressure. How many molecules are in 1m ^ 3?

5) Assume that one third of all specific molecules average the force on the wall perpendicular to the x direction. Calculate the pressure (= force / 1m ^ 2) on the wall.

NOTE: It is expected to obtain a pressure of 1 atm (= 10 ^ 5N / m ^ 2). The calculations give a lower pressure. This is because the speed of sound underestimates the effective molecular speed by a factor of (1.4 / 3) ^ 1/2.

A 50.0-kg woman stands at the rim of a horizontal turntable having a moment of inertia of 420 kg · m² and a radius of 2.00 m. The turntable is initially at rest and is free to rotate about a frictionless vertical axle through its center. The woman then starts walking around the rim clockwise (as viewed from above the system) at a constant speed of 1.50 m/s relative to the Earth.

(a) In what direction does the turntable rotate?

counterclockwise clockwise    

With what angular speed does the turntable rotate?
rad/s

(b) How much work does the woman do to set herself and the turntable into motion?
J

A hockey puck of mass 0.44 kg is shot west at 2.50 m/s strikes a second puck, initially at rest, of mass 0.50 kg. As a result of the collision, the first puck is deflected at an angle of 31° north of west, with a speed of 1.30 m/s. What is the speed of the second puck after the collision?

A uniform 5-m long ladder weighing 80 N  leans against a frictionless vertical wall. The foot of the ladder is 1 m  from the wall. What is the minimum coefficient of static friction between the ladder and the floor necessary for the ladder not to slip?

Here we will look at an example of subatomic elastic collisions. High-speed neutrons are produced in a nuclear reactor during nuclear fission processes. Before a neutron can trigger additional fissions, it has to be slowed down by collisions with nuclei of a material called the moderator. In some reactors the moderator consists of carbon in the form of graphite. The masses of nuclei and subatomic particles are measured in units called atomic mass units, abbreviated u, where 1u=1.66×10−27kg. Suppose a neutron (mass 1.0 u) traveling at 2.6×107m/s makes an elastic head-on collision with a carbon nucleus (mass 12 u) that is initially at rest. What are the velocities after the collision? If the neutron's kinetic energy is reduced to 3649 of its initial value in a single collision, what is the mass of the moderator nucleus? Express your answer in atomic mass units as an integer.

White light (ranging in wavelengths from 380 to 750 nm) is incident on a metal with work function W_o = 2.68 eV.

1. For what range of wavelengths (from l_min to l_max) will NO electrons be emitted?

a) Imin=

b) Imax=

In a series of experiments, a gas of exotic molecules containing exactly one electron

each is investigated. It is found that:

• When the gas is cold (so all the molecules are in their ground states) and illuminated

with light, it absorbs strongly at wavelengths 103.40nm, 137.87nm and 248.16nm.

• When the gas is cold and illuminated with light of short wavelength, the electrons are

ejected from the molecules as long as the wavelengths is less than 88.63nm.

• When the gas is heated, it emits light with 6 different wavelengths.

(a) If we chose to set the potential energy outside of the molecule to be zero, how far

below zero energy is the ground state of the molecule?

(b) How many discrete energy level does the molecule have?

(c) What wavelengths of light does the heated gas emit?

(d) A free electron with kinetic energy of 2eV is captured by the atom (which is originally ionized) and goes directly into the ground state. What is the wavelength of the emitted photon?

Hint: Electrons can jump from one energy level to another by emitting/absorbing a photon whose energy is equal to the difference in energies between the two states.

A disk with a c value of 1/2, a mass of 3 kg, and radius of 0.29 meters, rolls without slipping down an incline with has a length of 6 meters and angle of 30 degrees. At the top of the incline the disk is spinning at 28 rad/s. How fast is the disk moving (the center of mass) at the bottom of the incline in m/s?

A cylinder containing ideal gas is sealed by a piston that is above the gas. The piston is a cylindrical object, with a weight of 36.0 N, which can slide up or down in the cylinder without friction. The inner radius of the cylinder, and the radius of the piston, is 7.00 cm. The top of the piston is exposed to the atmosphere, and the atmospheric pressure is 101.3 kPa. The cylinder has a height of 30.0 cm, and, when the temperature of the gas is 20°C, the bottom of the piston is 14.0 cm above the bottom of the cylinder. (a) Determine the pressure of the gas in the cylinder. kPa (b) Find the number of moles of ideal gas in the cylinder. moles (c) Heat is added, gradually raising the temperature of the gas to 165°C. Calculate the distance between the bottom of the cylinder and the bottom of the piston when the piston comes to its new equilibrium position.