Twisted One 151's Weblog

Physics Friday 150

twistedone151 — Fri, 07 Jan 2011 09:30:22 +0000

Consider a skater of mass m on a smooth skating rink (so that we may ignore friction in this problem) with a cylindrical pillar, of radius R on the rink. A rope (of negligible mass compared to the skater) has one end fastened to the column, and extends straight out tangentially from the column for a length L. If the skater grabs the end of the rope while having a velocity v perpendicular to the rope, and then spirals inward, the rope winding onto the column. If the rope remains straight and taut throughout the spiral, what will the skater’s speed be upon reaching the pillar? In what direction?

Treating the rope as massless, we see that the part of the rope wound onto the column doesn’t move, and the point where it winds on has motion with no component along the length. Thus, we see the column does no work on the rope, and so the rope can do no work on the skater. Thus, the kinetic energy of the skater is unchanged, and so is his speed; he collides with the column with a speed v. Since the rope does no work on the skater, we see the force it exerts on the skater must be perpendicular to the skater’s path at all times (so that F·ds=0), and since the rope exerts force only parallel to its length, we see the skater’s velocity (tangent to his path) must always be perpendicular to the rope. Thus, when he reaches the column, the rope is fully wound, and so tangent to the column. Therefore, the skater’s velocity is normal to the column at impact; he hits it straight on. (Specifically, the skater’s path is the involute of the circle).

Filed under: Math/Science

Monday Math 149

twistedone151 — Mon, 03 Jan 2011 13:06:53 +0000

What is the probability that two independently randomly chosen integers are mutually prime (have no common factor greater than 1)? The probability for four random integers? For n integers in general?

The probability that a random integer is divisible by a given prime p is (since every pth integer is a multiple of p); thus, the problem of two independent random integers are both divisible by p is . Thus, the probability that they do not have p as a common factor is .
Thus, for the two random integers to be mutually prime, they must have no primes as a common factor; thus, the probability is the product of the above for all primes,

Now, recalling from here that , thus the probability is
,
and since we found here that , we see that the probability that two random integers are mutually prime is , which is approximately 60.8%.
Similarly, the probability of four integers being all divisible by a prime p is , so the probability that they do not all have p as a common factor is . Analogous to the above, the probability that the greatest common factor of all four integers is one is
,
and since (see here); we see that the probability for four integers is , which is approximately 92.4%.
Generalizing, we see that for n≥2 independent random integers, the probability that the greatest common factor of all n integers is 1 is given by , which increases toward unity as n gets larger
(since decreases asymptotically toward 1 for increasing real s>1).

Filed under: Math/Science

Physics Friday 149

twistedone151 — Fri, 31 Dec 2010 10:01:33 +0000

For a block sliding down an inclined surface with angle of incline θ and with coefficient of kinetic friction μ. If we consider the forces, we see that the block can slide down the incline with constant velocity when the net forces are zero. Balancing forces perpendicular to the plane, the normal force is thus . The kinetic friction is thus , while the component of gravity parallel to the plane is thus . We see that these cancel when
, or when .

Now, suppose we have a large incline with , and we start our block sliding with a velocity v₀ in a horizontal direction; that is to say, along the plane in a direction perpendicular to the direction of the slope. What, then, will be the speed a long time later?

Let v be the total speed of the block, and let v_y be the velocity component in the downslope direction. Now, we have two forces parallel to the plane; the friction force, with magnitude

directed opposite to the motion of the block, and the component of gravity, also of magnitude , directed downslope.
The former produces an acceleration of magnitude opposite the direction of motion, while the gravity gives the same acceleration in the downslope direction; thus
, and , so that
, and thus
, for C some constant.
Now, at time t=0, our motion is purely horizontal, and so v_y=0, v=v₀, and so
C=v₀.
Now, after a long time, the horizontal component of the velocity will be effectively zero, so that the velocity is entirely downslope. Then v=v_y, and so becomes
,
and so the speed after a long time is
, half the initial speed; now in the downslope direction.

Filed under: Math/Science

Monday Math 148

twistedone151 — Mon, 27 Dec 2010 09:34:39 +0000

Find .

One could integrate over y by using
; however, when you put in the limits, the resulting integral over x becomes very difficult.

Note that the region of integration, the region bounded by the parabolas and , has twofold rotational symmetry about the origin. Thus, we consider the change in variables , which represents axes rotated by 180°. We then have
,
and if we rename u and v as x and y, respectively, we see
; for our integrand , we have , which combined with the rotational symmetry of the region of integration, means that the integral is zero.

Filed under: Math/Science

Physics Friday 148

twistedone151 — Fri, 24 Dec 2010 09:04:24 +0000

A thin bimetallic strip of thickness d is straight at a temperature T₀, with length L₀. The two metals have coefficients of linear thermal expansion α₁ and α₂, with α₂>α₁. If the temperature is raised to a temperature T not significantly greater than T₀, what is the angle θ through which the strip bends?

We should expect the curvature to be uniform, leaving the strip to form a circular arc of angle θ. If the radius of curvature for the inner half of the strip is r, then (with θ in radians), the length of that metal is L₁=rθ.
We can ignore any expansion in the direction of the thickness, so the length of the other metal is
L₂=(r+d)θ.
Taking the difference, we see
L₂-L₁=dθ,
so .

Now, the formula for thermal expansion is
(see this previous post),
so for the inner side,
,
and solving for L₁,
.
Analogously,
.
Thus, the difference is
,
and so the angle θ is
.

Filed under: Uncategorized

Total Lunar Eclipse Monday Night

twistedone151 — Mon, 20 Dec 2010 17:00:26 +0000

Update Sunday: This post was supposed to be pre-scheduled for tomorrow (Monday), but I hit the wrong button.

Here is NASA’s page on the total lunar eclipse tonight, which should be visible throughout North America. The moment of greatest eclipse will be at 08:16:57 UT (that’s 11:16 PM for my fellow Alaskans and me); the penumbral eclipse begins at 05:29 UT and ends at 11:04 UT

Filed under: Math/Science

Monday Math 147

twistedone151 — Mon, 20 Dec 2010 10:58:02 +0000

Given two non-zero complex numbers z₁ and z₂ such that , show that the arguments of z₁ and z₂ differ by π/2.

Recall that for any complex number z, , thus

Now, recall that for any complex number z, , and since , we see that
, and so
,
and we proved last week that z₁ and z₂ have arguments differing by π/2 (and thus give perpendicular vectors) if and only if .

Consider the geometric interpretation. Given a parallelogram with sides given by the vectors corresponding to z₁ and z₂, then z₁+z₂ and z₁-z₂ correspond to the diagonals; thus, the above is equivalent to stating that if the diagonals of a parallelogram are of equal length, then the parallelogram is a rectangle.

Filed under: Math/Science

Go and Read

twistedone151 — Mon, 20 Dec 2010 05:12:21 +0000

Go and read this:
“Finally, A Hamas Leader Admits That Israel Killed Mostly Combatants In Gaza” by Alan M. Dershowitz

The key point:

The Israel Defense Force put the total number of known combatants killed at 709 and the number of known civilian deaths at 295, with 162 (mostly men of fighting age) “unknown.” Such a ratio, if true, would be far better than that achieved by any other nation in a comparable conflict. Not surprisingly, Israel’s enemies initially disputed this ratio and claimed that the number of combatants killed was far lower and the number of civilians far higher. The United Nations, the Goldstone Report, various “human rights” organizations and many in the media automatically rejected Israel’s documented figures, preferring the distorted numbers offered by Hamas’ and other Palestinian sources.

But a statement recently made by a Hamas leader confirms that Israel was correct in claiming that approximately 700 combatants were killed.

Filed under: Politics

Micro-credit

twistedone151 — Mon, 20 Dec 2010 03:56:16 +0000

The BBC News reports on the epidemic of suicides and stress-related deaths in southeast India due to pressure arising from inability to repay micro-loans. According to the article, the government of the Indian state of Andhra Pradesh is reporting over eighty suicides in the past few months after defaulting on loans from micro-finance companies. Further, banks have stopped lending to the micro-credit institutions due to fears that large portions of the money loaned may be unrecoverable, in a situation similar to the subprime mortgage crisis.
So, it seems that micro-credit hasn’t lived up to all the talk about how it would help the poor improve their lives

Filed under: Politics

More from Mexico

twistedone151 — Mon, 20 Dec 2010 02:51:16 +0000

Continuing in the vein of this post, another Huffington Post report from Mexico: Mexico Mom Killed Demanding Justice For Slain Teen Daughter.

The brutal killing of activist Marisela Escobedo Ortiz stunned people across Mexico, and a group of women angrily demanding justice gathered outside the state prosecutors’ office in this border city, where the victim’s daughter was killed.

Escobedo’s slaying “shows that in Mexico, it is the victim who suffers,” anti-crime activist Alejandro Marti said.

The uproar resulted in the suspension of three state judges who had ordered the freeing of the main suspect in the slaying of Escobedo Ortiz’s daughter – the same man who was identified as a chief suspect in the mother’s death.

Filed under: Uncategorized

Prison Break (no, not the TV show)

twistedone151 — Mon, 20 Dec 2010 02:05:59 +0000

141 inmates have escaped a Mexican prison near the border with Laredo, Texas; authorities suspect that prison employees helped. According to the Huffington Post:

The public safety department of Tamaulipas state, where the prison is located near the border with Laredo, Texas, said 141 inmates got out through a service entrance used by vehicles, “presumably with the assistance of the prison staff.”

The department said the prison’s director could not be located, adding that he and other officials were under investigation.

Eighty-three of the prisoners were being held for trial or had been convicted of crimes like theft, assault and other state offenses, while 58 were being held on federal charges, which include weapons possession and drug trafficking.

Filed under: Uncategorized

Here’s one for the holidays

twistedone151 — Mon, 20 Dec 2010 00:42:24 +0000

Christmas trees depress religious minorities, LiveScience reports.

When people who did not celebrate Christmas or who did not identify as Christian filled out surveys about their moods while in the same room as a small Christmas tree, they reported less self-assurance and fewer positive feelings than if they hadn’t been reminded of the holiday, according to a new study.

The university students didn’t know the study was about Christmas, said study researcher Michael Schmitt, a social psychologist at Simon Fraser University in British Columbia, Canada. Nonetheless, he said, the presence of the tree caused non-celebrators and non-Christians to feel subtly excluded.

“Simply having this 12-inch Christmas tree in the room with them made them feel less included in the university as a whole, which to me is a pretty powerful effect from one 12-inch Christmas tree in one psychology lab,” Schmitt told LiveScience.

Filed under: Math/Science, Politics

Religious Freedom, or the Lack Thereof

twistedone151 — Sun, 19 Dec 2010 23:36:24 +0000

Aasia Bibi, a Christian mother of five, awaiting execution in Pakistani jail for blaspheming the prophet Muhammad:

Sent Nov. 11, the nun’s message told the story of a Catholic farm worker, Aasia Bibi, convicted of violating antiquated blasphemy law propped up by Pakistan’s Islamist political parties. Allegedly, when some women workers pressured Aasia to renounce her Christian faith and accept Islam in the summer of 2009, Aasia responded that Jesus had died on the cross for the sins of humanity and she asked them what Muhammad, the prophet of Islam, had done for them. Her crime, blaspheming the prophet, carries a mandatory death sentence.

Filed under: Politics

Another Tech Bubble?

twistedone151 — Sun, 19 Dec 2010 22:22:04 +0000

That’s the question Dan Lyons asks at The Daily Beast. With Facebook valuated at $56 billion (based on shares trading last Friday), and Twitter at a $3.7 billion valuation without having yet generated a dime of revenue, amongst other such examples, it seems to me that he has a point.

Filed under: Politics

More on past predictions

twistedone151 — Sun, 19 Dec 2010 21:39:35 +0000

NPR looks back at predictions about the year 2011 made back in 1931. The bit that stood out to me was a few sentences from the predictions of sociologist William Ogburn:
“The role of government is bound to grow. Technicians and special interest groups will leave only a shell of democracy.”

Filed under: Uncategorized