Recent content by Almanzo

I take it that the charges come from infinity. The work required to bring them in from infinity is the sum of the electrostatic energies of the pairs of charges. With 8 charges there are 8*7/2 = 28 such pairs. These can be viewed as the 12 sides of the cube, its 12 surface diagonals and its 4...

The twin paradox involves a set of twins, or rather a set of two watches set to an equal time and traveling at different velocities. After some interval they are supposed to read different times. Now the time measured by a watch is its proper time, whatever route it has traveled; there is no...

I am unsure about the source of the confusion. There are three things to be considered: 1. To calculate the time needed to fall from a certain height in Earth gravity. 2. To calculate the length of a line segment in spacetime. 3. To calculate the radius of a circle, if a chord and...

I have sometimes feared that, by a chain of events which steadily become more improbable, I would live forever, though in steadily declining health. Think about it. To observe oneself as being alive, one doesn't need much of one's faculties. The idea is nightmarish enough to keep one awake at...

This is a question to which different people will give different answers. For example, the theory of Bohm would imply that the electron does trace a kind of orbit, but a very intricate one, which isn't at all like a circle or an ellipse. The Many Worlds theory would have the electron trace...

In space, a circle touching the bullet's track on the inside would have a much larger radius (hence less curvature) than a circle touching the ball's track on the inside. However, the problem is to be solved in spacetime. Now, the bullet takes less time to complete its track than the ball (which...

Prime numbers have exactly two (positive integer) divisors: 1, and the number itself. after all, N = 1 * N, and N = N * 1. As far as I know, some patterns have been found which generate only prime numbers, but no pattern has been found which generates all of them. In general, to see if some...

Infinite expressions, like 1 + 1/2 + 1/4 + 1/8 + ... , may or may not refer to existing mathemathical objects. If one wants to reason about such an object, one must first ascertain its existence. (For a finite expression, such as 3 * 1/5; the existence of the object is usually "prewired" in the...

Oh, yes. Gn(0) = (-1/n)*(6/n3)*e0=(-6/n4). Not zero at all. But Gn(x) does tend to zero if x increases without bound. So, the integral of gn(x) from 0 to infinity would be (+6/n4). Strangely, because someone had earlier suggested zero as the answer, I accepted a result of zero without...

Last night, I have thought about it some more. x3/(ex-1) = (x3/ex)*1/(1-e-x) x3/(ex-1) = (x3/ex)*(1 + e-x + e-2x + e-3x + e-4x + ...) x3/(ex-1) = x3 * (e-x + e-2x + e-3x + e-4x + ...) Let Gn(x) = (-1/n)*(x3 + 3x2/n + 6x/n2 + 6/n3)*e-nx gn(x) = dGn(x)/dx = (x3 + 3x2/n + 6x/n2 +6/n3)*e-nx...

This is an improper integral, because the integrand x3/(ex-1) is not defined at the edge of the integration domain, x=0. e0 - 1 = 1 - 1 = 0. 03/0 = 0/0 is undefined. So one must first ascertain whether it converges. To find a primitive function, one might substitute ex = u, or x = elog(u)...

I think it might be well to define what you mean by T, w, N, x, and so on, because that would make the problem easier to understand. The way I understand it, there is an inclined plane, and theta is the angle between the plane and the horizontal. Mass m1 is lying on the plane, while mass mass...

The kinetic energy (KE) would not be helpful here, becase chemical energy is being converted into kinetic energy when the gun is fired. Momentum, however, is helpful. Think of the system (man plus pistol plus bullet) splitting in two. The two parts (man plus pistol) and (bullet) will have...

After 10 seconds, the flare will have a vertical speed (vertical component of velocity) of 10 s * 9.81 m/s2 = 98. 1, m/s. (Its vertical speed was initially zero, as it was fired in a horizontal direction. Unless they mean horizontal relative to the ascending helicopter. I assume they don't mean...

I assume that it is the volume of the sphere which you wish to calculate. In polar coordinates, the volume element is dr * (r * dtheta) * {r * sin(theta) * dphi}. Or r2 *dr * sin(theta) * dtheta * dphi. This must be integrated for r between 0 (the midpoint) and R (the surface), for theta...