Phys 263/Amath 261
Assignment 8
SOLUTIONS
1. (a) The speed of the second ship is dened on the rst ship by u= x t (1)
where x is the measured length of the 2nd ship. This is length contracted by x = 1
MIDTERM
Phys 263 / Amath 261 June 16, 2008: 4:30 to 6:30pm
1. For two particles with mass m1 and m2 and positions r1 = (2t2 , 3t, 4) r2 = (1 + t2 , 0, 4t2 ) (a) Find the center of mass coordinates. (b
Phys 263/Amath 261
Assignment 6
SOLUTIONS
1. (a) To set up this problem, use the gure, where the mass m is a distance r from the center of the ring. Then the potential is given by dM Ga = d (1) b b wh
Phys 263/Amath 261
Assignment 6
SOLUTIONS
1. For the orbit r = ke (a) We use the path equation to determine the force law. First note, d1 d e e = = d r d k k and 2 e 2 d2 1 = = d2 r k r then from the
Phys 263/Amath 261
Assignment 5
SOLUTIONS
1. First, consider the center of mass momentum, which reacts to two forces: the tension on the rope (by the hand), and gravity: P = T M g. Now, the mass of th
Phys 263/Amath 261
Assignment 4
SOLUTIONS
1. (a) Look at the equilibrium position, and equate mg = kx k= mg 1000 9.8 = = 98000 x 0.1 N/m (1)
(b) The undamped oscillation period is T= (c) At critical d
Phys 263/Amath 261
Assignment 3
SOLUTIONS
1. The equation of motion of the oscillator is x + 4x = 0
2 where o = k/m = 4. The general solution, from class, is
(1)
x = A sin(0 t + ) which we can also w
Phys 263/Amath 261
Assignment 2
SOLUTIONS
1. The law of conservation of mass does not hold in quantum mechanics and special relativity. 2. The Force dierential equation is dened by F = ma = m in one d
Phys. 263
Assignment 1
SOLUTIONS
1. (a) A + B + C = (5, 1, 3) (b) B C = (1, 4, 5) so that A (B C) = 1 + 8 + 15 = 24. (c) A B = (8, 8, 8) so that C (A B) = 8 + 8 + 8 = 24. (d) A B C = A (B C) = (2, 2,
1
Review: Vector Analysis
REFERENCES: Arya Chapter 5.
1.1
Vectors
A scalar has only a magnitude. A vector has both a magnitude and a direction. In this course, vectors will be represented in two ways
2.5
Nuclear Binding Energy
Clear evidence for the mass-energy equivalence relation is given in the study of mass defects in atomic nuclei. The mass defect is the decit in the mass of a nucleus, compar
1.6.2
Proper time
Consider an object moving with a velocity u relative to our rest frame S . Assume two events occur with the object, separated by some time dt . In the S frame, dt = ds c (116)
since
1.6
Four-dimensional Space
The Lorentz transformation treats the xi with i = 1, 2, 3 as equivalent variables. Lets introduce time as simply a forth coordinate: x0 = ct (85)
This four-dimensional space
2. Consider a process where an event at P causes an event at Q. Lets choose the separation coordinates to be x = x2 x1 > 0 and t = t2 t1 > 0, and V = x/t is the speed of the signal. Then t = t2 t1 = (
1.3
Lorentz Transformation
Since Galilean transformations are inconsistent with Einsteins postulate of the speed of light, we must modify them. Consider our two inertial frames S and S , and let the a
example: Time dilation. Consider two observers S and S . Observer S is stationary and observes S is moving away with a constant velocity u. S sends a beam of light towards a mirror L away, and receive
1
Special Relativity
Recall the Galilean transformation of Newtonian physics: this relates the the coordinates x, y, z, t in one inertial reference frame S to the coordinates x , y , z , t in a dieren
1.2
Gausss law and Poissions Equation
M r r2
Consider a point mass M , with gravitational eld at a distance r of g = G (29)
We can dene a quantity of ux through a sphere of radius r with this point ma
1
Gravitation
mM r r2
Newtons law of universal gravitation F = G (1)
where G = 6.6726 0.0008 1011 Nm2 kg2 . This equation applies strictly only to point particles. If one or both of the particles is r
1.2
Planetary motion
REFERENCE: Arya, Section 7.7 to 7.9
1.3
Rutherford Scattering
The other very important problem involving inverse-square forces is the scattering of charged particles in a Coulomb
REFERENCE: Arya, Sections 7.5 to 7.7 EXAMPLES: Arya 7.1 and 7.2 Review the derivation of the following important equations: 1. The angle as a function of radial distance (r) =
2
(L/r2 )dr E V (r)
L2
1
Central Force Motion
REFERENCE: Arya, Sections 7.1 to 7.4 Review the derivation of the following important equations: 1. The eective single particle equation: r = F (r) r 2. The force (system of) eq
example: A ball of mass m and kinetic energy E is in an elastic collision with a second ball of mass 4m initially at rest. The two balls depart in directions making an angle of 120 degrees with each o
2
Elastic collisions
We now apply the conservation laws to the interaction of two particles, in particular collisions. In general, two types of collisions are possible: Elastic and Inelastic. By denit
1.3
Angular Momentum
L = r p = r mr = r mv
The angular momentum of a single particle is dened as (46)
We can extend this denition to a systems of N particles
N N
L=
k=1
(rk pk ) =
k=1
(rk mk r)
(47)
example: A chain of uniform mass density , length b, and mass M (where = M/b) hangs from both ends. At time t = 0, the ends are adjacent, but one is released. Find the tension in the chain at the xed
1
Systems of particles
REFERENCE: Arya, Chapter 8. The ideas of Newtonian mechanics and the conservation theorems can be straightforwardly extended to systems of N particles.
1.1
Center of Mass
Consid
1.7
2D oscillations
Consider the motion of a particle with two degrees of freedom. Take the restoring force to be proportional to the distance of the particle from a force center located at the origin
example: The equation of motion for a certain driven damped oscillator is x + 3x + 2x = 10 cos t and initially the particle is at rest at the origin. Find the subsequent motion. solution: Compare this