Archive for March, 2011

ABS or experience?

You might ask yourself is ABS breaking system more efficient than a normal system, or is it doing a better job comparing to an experienced driver.  The answer is NO.  As you see in the following plot, an experienced driver without ABS stops the auto faster than an amateur driver with ABS.

It should be expected since ABS avoids locking by reducing the break power.

Reference: Fundamentals of Physics

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Flare diagram from GOES x-ray data. The largeest peak is the second X-class flare in the current solar cycle (24).

Flare diagram from the GOES x-ray data. The largest peak is the second X-class flare in the current solar cycle (24).

Explosive events on the solar surface release energy in various forms, e.g., accelerated particles, bulk mass motion, and radiated emission in various spectral ranges.  A solar flare is an explosion on the Sun that happens when energy stored in twisted magnetic fields (usually above sunspots) is suddenly released.  Flares produce a burst of radiation across the electromagnetic spectrum, from radio waves to x-rays and gamma-rays. In a typical flare, the brightness increases for several minutes (the flash phase), followed by a slow decay (the decay phase) which lasts up to an hour.  Formation of a flare is usually accompanied by a significant re-arrangement of the magnetic field configuration.

The maximum temperature in the core of a flare event can reach to 10 million Kelvin ! It causes bursts of radiation in gamma and x-ray, extreme ultraviolet, and microwaves. The physical process is the bremsstrahlung of electrons with energies 10-100 keV (an electron with 100 keV energy travels with 1/3 of speed of light).

Scientists classify solar flares according to their x-ray brightness in the wavelength range 1 to 8 Angstroms. There are 3 categories: X-class flares are big;  they are major events that can trigger planet-wide radio blackouts and long-lasting radiation storms. M-class flares are medium-sized; they can cause brief radio blackouts that affect Earth’s polar regions. Minor radiation storms sometimes follow an M-class flare. Compared to X- and M-class events, C-class flares are small with few noticeable consequences here on Earth. Large flares may be visible in white light as well !

Each category for x-ray flares has nine subdivisions ranging from, e.g., C1 to C9, M1 to M9, and X1 to X9. In this figure, the three indicated flares registered (from left to right) X2, M5, and X6. The X6 flare triggered a radiation storm around Earth nicknamed the Bastille Day event.


Class                  Peak (W/m2)  between 1 and 8 Angstroms


B                           I < 10-6


C                          10-6 < = I < 10-5


M                         10-5 < = I < 10-4


X                          I > = 10-4


The image below shows a large flare as recorded with EIT telescope of the SOHO spacecraft in previous solar cycle. We expect to see more solar flares in the coming 4-5 years.

X28 flare in EIT 195 -- The Sun unleashed a powerful flare on 4 November 2003 that could be the most powerful ever witnessed and probably as strong as anything detected since satellites were able to record these events n the mid-1970s. The still and video clip from the Extreme ultraviolet Imager in the 195A emission line captured the event. The two strongest flares on record, in 1989 and 2001, were rated at X20. This one was stronger scientists say. But because it saturated the X-ray detector aboard NOAA's GOES satellite that monitors the Sun, it is not possible to tell exactly how large it was. The consensus by scientists put it somewhere around X28.

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In a set of posts, I show the solution of a few simple 1D and 2D problems in physics. The numerical solutions and plots all are
done using Python. As we go on, the problems will be more and more difficult. Although, I did all that just for fun!

As the simplest problem, I discuss a mass and spring system in one dimension with friction. As you see in the below plot, the system oscillates and the amplitude of the oscillation decreases gradually. That is due to loss of the energy (work of friction force). The rate of decrease of the amplitude is determined by the friction coefficient. I would like to note that although this problem has analytical solution (Haliday, Resnik, Walker, chapters 7 and 8), I used a numerical scheme to solve it (see the Python code below).

Variation of the amplitude of oscillations.

The next plot shows the variation of the potential, kinetic, and total energies. As expected, when e.g. the kinetic energy is at zero, the potential energy has a maxima.

Note that I only used the simplest form of the drag force (-kv). In reality, one may consider terms of higher order. Alternatively, one might take the friction coefficient, k, as a function of velocity. And here is the code. I didn’t manage to put it in a way it should look like. In case you have an idea, let me know.

Program to solve 1D mass and spring system in presence of friction


F = dp/dt = – kx


import pylab
import numpy
import scipy

from pylab import *
from numpy import *
from scipy import *

# initial condition, constants
h0 = .50     # initial deviation from equilibrium in m
v0 = 1.0     # m/s
t0 = 0.0     # s

# constants
time = 10.        # duration in sec
dt = 1.0e-3       # time step in sec
nstep = int(time/dt) # number of time steps

k = 10.0         # spring constant in N/m
m = 1.0          # mass in kg
k1_drag = 0.5    # kg/s

eta = 0.8        # v_in/v_out in an inellastic encounter
Energy = 0.5 * m * pow(v0,2) + 0.5 * k * pow(h0,2)

# array definitions
h = zeros((nstep,), dtype=float32)  # height in meters
v = zeros((nstep,), dtype=float32)  # velocity in m/sec
a = zeros((nstep,), dtype=float32)  # acceleration in m/sec^2
t = zeros((nstep,), dtype=float32)  # time in sec
K = zeros((nstep,), dtype=float32)  # kinetic energy
V = zeros((nstep,), dtype=float32)  # potential energy
En = zeros((nstep,), dtype=float32) # total energy

#———– initial conditions ———
i = 0
v[0] = v0
h[0] = h0
K[0] = 0.5 * m * pow(v0,2)
V[0] = 0.5 * k * pow(h0,2)
t[0] = t0
En[0] = K[0] + V[0]

for i in arange(1,nstep):#———– main loop ———

a[i-1] = -(k/m) * h[i-1] – (k1_drag/m) * v[i-1]
v[i] = v[i-1] + a[i-1] * dt
dE = – k1_drag * abs(v[i-1]) * abs(v[i-1]) * dt # drag
Energy = Energy + dE
h[i] = h[i-1] + v[i-1]*dt
t[i] = t[i-1] + dt
K[i] = 0.5 * m * pow(v[i],2)
V[i] = 0.5 * k * pow(h[i],2)
En[i] = K[i] + V[i]

tz = [0, time]
hz = [0, 0]
plot(t, h, tz, hz,’k’)
xlabel(‘Time (s)’, fontsize=18)
ylabel(‘Amplitude (m)’, fontsize=18)
tex = r’$F\,=\,-kx\, -\, k_dv$’
text(5, .4, tex, fontsize=24)
savefig(‘spring_oscil.png’, dpi=100)
plot(t, K, t, V, t, En, ‘r’, tz, hz,’k’)
xlabel(‘Time (s)’, fontsize=18)
ylabel(‘Energy (J)’, fontsize=18)
legend((‘Kinetic’, ‘Potential’, ‘Total’),’upper right’, shadow=True)
savefig(‘spring_energy.png’, dpi=100)

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