Hi - I'm having some difficulties with a simulation I'm writing. I have a second order ODE with three arguments of derivative information y(1),y(2),y(3) - each changing wrt to the independent variable t.
Currently, in my system of ODEs, I have a variable that I am treating as constant...
I've written a program to show the trajectory of a rocket around the Earth with initial conditions that can be manipulated to fix a circular or elliptical orbit.
We start with Newton's 2nd law and use the equation a=F/m in freespace (so m can divide out) to get a 2nd order ODE and solve by...
Orbital Speed of a Star - Please Help!
Homework Statement
Calculate \Omega(r) and v(r) for the following density models:
(a) all the mass M is at the center of the galaxy;
(b) a constant density adding up to a mass M(R0) at the Sun’s orbit and no mass beyond.
Homework Equations...
Homework Statement
If a quasar varies on a time scale of four months, how does the maximum size of the emitting region compare with the Schwarzschild radius for a 109 MSun black hole?
Homework Equations
The Attempt at a Solution
Calculating the S radius of the black hole is...
Homework Statement
Calculate \Omega(r) and v(r) for the following density models:
(a) all the mass M is at the center of the galaxy;
(b) a constant density adding up to a mass M(R0) at the Sun’s orbit and no mass beyond.
Homework Equations
M(r) = v(r)2r/G
The Attempt at a...
Homework Statement
I'm to prove that the central pressure of a particular star is
Pc = (15/16\pi)(GM2/R4)
I am given the density as a function of r
\rho(r) = \rho0[1 - (r/R)2]
In part a I found the Mass by integrating the equation of hydrostatic equilibrium and found
M =...
Homework Statement
Consider Comet Halley. At a particular instant in time, its position and velocity
are given below, in units of AU and AU/yr relative to the centre of the Sun.
(x,y,z) = 0.331060, -0.455488, 0.166180)
(vx,vy,vz) = (-9.01154, -7.02645, -1.30645)
There are a number...
Homework Statement
We observe a binary system in which the two stars are 1 and 2 arc sec, respectively, from the center of mass. The system is 10pc from us. The period is 33 yr. What are the masses of the two stars, assuming that i=90 degrees?
Homework Equations
The Attempt at a...
Homework Statement
A grandfather clock has a pendulum length of 0.7 m and a mass bob of 0.4 kg. A mass
of 2 kg falls 0.8 m in seven days, providing the energy necessary to keep the amplitude
(from equilibrium) of the pendulum oscillation steady at 0.03 rad. What is the Q of the
system...
Homework Statement
A grandfather clock has a pendulum length of 0.7 m and a mass bob of 0.4 kg. A mass
of 2 kg falls 0.8 m in seven days, providing the energy necessary to keep the amplitude
(from equilibrium) of the pendulum oscillation steady at 0.03 rad. What is the Q of the
system...
Homework Statement
Consider an atom of mass m bonded to the surface of a much larger immobile body by
electromagnetic forces. The force binding the atom to the surface has the expression
F = eacosz + bsinz + dtanz
where a, b and d are constants and z is positive upwards. The...
1. Homework Statement
A block of mass m=1.62 kg slides down a frictionless surface (see below). The block
is released at a height h=3.91 m from the bottom of the loop. Assume the bottom of the
ramp indicated between the dashed lines in the diagram is a segment of a circle with R = 1.5m...
Homework Statement
A block of mass m=1.62 kg slides down a frictionless surface (see below). The block
is released at a height h=3.91 m from the bottom of the loop. Assume the bottom of the
ramp indicated between the dashed lines in the diagram is a segment of a circle with R = 1.5m...
Homework Statement
Use dimensional analysis to find a formula for the free-fall time τ of an astrophysical ball of gas. In this system, the only force is gravity, therefore the only quantities are Newton’s constant, G, and the mass and the radius of the sphere, M and R respectively...
Homework Statement
At the top of the NH3 clouds in Saturn’s atmosphere, the temperature To = 110 K and the pressure Po is about 0.5 bar = 0.5 atm. Below this level at a radius of ro=60268 km, the atmosphere is convective and the temperature changes with radius at a constant rate of dT/dr =...
Homework Statement
A lamina occupies the region inside the circle x^2 + y^2 = 2y but outside the circle x^2 + y^2 = 1. Find the center of mass if the density at any point is inversely proportional to its distance from the origin.
Homework Equations
Xcm = double integral of y*f(x,y)...
Homework Statement
Consider the Sun to be a sphere of hydrogen gas of uniform density equal to its
average density (1440 kg/m3).
a) Integrate the equation of hydrostatic equilibrium for dP/dr from the Sun’s radius to the
core to estimate the central pressure in Pa and in atmospheres...
Homework Statement
Calculate and tabulate the fractions of the total mass of the Earth which would be
accounted for by O, Na, Mg, Al, Si, S, Ca and Fe under the following hypothese.
i) All the elements above are fully oxidized, except that no S is accreted and the
Fe is entirely metallic...
Homework Statement
I am given a table with the abundances of atoms of different elements per 1000 Si atoms.
The question wants me to tabulate the fraction of mass of the earth that each element, and different oxides contribute. But to do this I need to know how many Si atoms there are so I...
I'm working on a double integral and need to change from cartesian to polar coords.
The double integral in cartesian coords is
Double integral from 0 to a (in dy) and from -sqrt(a^2 - y^2) in dx (NOTE minus sign before sqrt)
the little minus sign is troubling. How would I convert this...
Homework Statement
a) A lamp has two bulbs of a type with an average lifetime of 1000 hours. Assuming that we can modelthe probability of failure of these bulbs by an exponential density function \mu = 1000, find the probability that both of the lamps bulbs will fail within 100 hours.
b)...
I know that escape velocity is given by v^2 = 2GM/r
My question is what is the approximate escape speed needed to completely escape the moon & earth's gravity.
Is it the sum of their individual Escape velocities? or is it one equation, with the radius' added and masses added?
Thanks