Math Is Fun Forum

that

Area of Circle = f(r), and Circumference of Circle = f'(r)

AND

volume of sphere = g(r) and surface area of sphere = g'(r)

is this a coincidence?

here's a physics problem i can't figure out for the life of me,

You are given3 stationary particles in 3d space, their x,y,z coordinates, and masses. You are given a 4th particle with a given mass. You must find the location to place the 4th particle such that the gravitational forces balance and the object remains stationary.

Sounds simple enough, but working it out has proved challenging.

Consider the location of each of the three particles as vectors A,B, C, and call the forth vector D. The formula for gravitational force is

(with r being the distance between the two, m[sub]1[/sub] and m[sub]2[/sub] the mass of the two particles, and G the gravitational constant, all except r are known in this problem)

likewise, the force of attraction between vectors A and D would be given by:

where • denotes the dot product.

Thats fine, but if we're going to be adding these vectors together we need to know their directions. Since the force vector from D to A has the same direction as the vector (A-D), we could normalize this vector:

and then multiplying it by the magnitude of the force, to obtain the force vector. Doing this gives us:

This is just the force between A and D, if we consider all three we get:

which is what i might call a hideous equation, and i don't think this can be solved by any elementary methods.

So, any other ideas?

curses, I'm confounded.

okay,  currently learning about angular momentum, which, in the case of a single particle, is given as

L = r x p, where r is the position vector of the particle, and p =  momentum of the particle = mv, where m is the mass of the particle and v is  the velocity vector of the particle.

Good so far.

but my book soon defines the angular momentum of a system of particles to be:

L = ∑ L[sub]i[/sub] where L[sub]i[/sub] is the angular momentum of the i'th particle.

still okay.

now they show that dL/dt  = net Torque of the system

L = ∑ L[sub]i[/sub] = ∑ m (r[sub]i[/sub]) x (v[sub]i[/sub])

differentiating with respect to t yields:

dL/dt =  ∑ m(r[sub]i[/sub])x(dv[sub]i[/sub] /dt) + m(v[sub]i[/sub])x(dr[sub]i[/sub]/dt)

now we know dr[sub]i[/sub]/dt is the change in the position vector over dt, which is the same as the velocity vector v[sub]i[/sub], substituting this gives us:

dL/dt =  ∑ m(r[sub]i[/sub])x(dv[sub]i[/sub] /dt) + m(v[sub]i[/sub])x(v[sub]i[/sub])

by definition of cross product, (v[sub]i[/sub])x(v[sub]i[/sub]) is zero, furthermore, we can replace dv[sub]i[/sub]/dt with a[sub]i[/sub], the acceleration of the i'th particle, so we obtain:

dL/dt =  ∑ m(r[sub]i[/sub])x(a[sub]i[/sub])   + 0

which we can also write as

dL/dt = ∑ (r[sub]i[/sub])x(m*a[sub]i[/sub])  and sustituting F[sub]net,i[/sub] for m*a[sub]i[/sub], we obtain

dL/dt = ∑ (r[sub]i[/sub])x(F[sub]net,i[/sub])  but this is exactly how we define the net torque on the system about the axis of rotation O where O is at the tale of r, the position vector.

And so we get dL/dt = net torque, or the rate of change in the angular momentum of the system about O equals the net torque of the system about O.

That all makes perfect sense, until my book gave this word of caution, which i quote word for word here (comments by me are made in brackets []:

this equation [net torque = dL/dt] is analogous to Fnet = dP/dt [newtons law for linear momentum] but requires an extra caution: torques and the system's angular momentum must be measured relative to the same origin.[makes sense!] If the center of mass of the system is not accelerating relative to an inertial frame, that origin can be any point. [...um] However, if the center of mass of the system is accelerating, the origin can be only at the center of mass. [WHAT? WHY?] As an example, consider a wheel as the system of particles. If the wheel is rotating about an axis, that is fixed relative to the ground, then the origin for applying the equation  [net torque = dL/dt]  can be any point that is stationary relative to the ground. However, if the wheel is rotating about an axis that is accelerating (such as when the wheel rolls down a ramp) then the origin can only be at the wheels center of mass.

and they've lost me with that. I see no reason why we can't, for instance, apply the same reasoning to find the net torque about a stationary origin, of a system of particles that falls freely. I can't see how an acceleration of the systems center of mass, crashes the logic behind the formula dL/dt = net torque.

a thousand pounds of chocolate to whoever can explain this to me.