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P HYSICS D EPARTMENT, S YRACUSE U NIVERSITY PHY215 - Recitation 1/1 Julian Georg January 17, 2017 1 U NIT C ONVERSITONS Convert the following to SI units: 8 in 66 ft/s 65 mi/h 3h 2 days 1yr 14 in2 215 ft/s2 43.87 kg/ft3 180 nm 83 km2 12 mm 1 2 F ISSIONABLE In the fall of 2002, a group of scientists at Los Alamos National Laboratory determined that the critical mass of neptunium-237 is about 60 kg. The critical mass of a fissionable material is the minimum amount that must be brought together to start a chain reaction. This element has a density of 19.5 g/cm3 . What would be the radius of a sphere of this material that has a critical mass? 3 E PSILON E RIDANI A parsec is a unit to measure astronomical distances. 1 parsec is equal to 3.26 lightyears. A lightyear is the distance light travels in one year. It is equal to 9.46 × 1012 km. The 11th closest star to our solar system is Epsilon Eridani. It is 3.22 parsecs away. What is its distance from the sun in meters and how many seconds does its light take to get to us? 2 4 A GE How many years older will you be 1 gigasecond from now? (Assume a 365-day year.) 5 N EWTON Let’s assume you do not remember the dimensions of the gravitational constant G. But you do remember that the force of gravity in the Newtonian limit is given by FG = G m1 m2 . r2 (5.1) What are the dimensions of G? 3 6 QFT In natural units (~ = c = k B = 1) the only dimension is mass. For example, energy has dimensions of mass according to Einstein’s Mass-Energy-Equivalence E = mc 2 . Length and time have units of inverse mass. (a) Integrals must produce numbers. Therefore, the action S is dimensionless and is given by Z S = d4 x L . (6.1) What are the dimensions of the Lagrangian density L ? d4 x is the volume element of 4dimensional space-time. (b) Let’s say the Lagrangian density is given by 1 L = ∂µ φ∂µ φ + i ψ̄γµ ∂µ ψ + m 2 φ2 + m f ψψ̄ 2 What are the dimensions of φ and ψ? γ is dimensionless and ∂µ is defined as (6.2) ∂ ∂x µ . 4