- #1
olgerm
Gold Member
- 533
- 34
I have thought about an experiment which to demostrate gravitomagntism: spinning massive cylinder creates gravitomagnetic field above it. Oscillating pendulum above the cylinder departs from its trajectory because of "gravito-Lorents" force.
To calculate magnetic force above the cylinder I use formula:
[itex]\vec B=\frac{G*4*π}{c^2}*\frac{\vec{v}×\vec{l}*m}{[\vec{l}]^3}[/itex]
where:
B is magnetic field in some point A.
G is gravitational constant.
c is speed of light.
v is speed of moving pointmass at point C.
l is vector [itex]l=\vec{AC}[/itex]
m is mass of pointmass .
To get gravitomagnetic field by whole cylinder integrate B over volume, because every point of cylinder may be seen as pointmass.
for circle:
[itex]\vec B(x, y, h, r)=\int_0^{2*π} (\frac{G*4*π*ρ}{c^2}*\frac{\vec{v}×\vec{l}*r}{[\vec{l}]^3} *dα)[/itex]
for disc:
[itex]\vec B(x, y, h, R)=\int_0^R (\int_0^{2*π} (\frac{G*4*π*ρ}{c^2}*\frac{\vec{v}×\vec{l}}{[\vec{l}]^3}*r* dα)dr)[/itex]
And finally for cylinder:
[itex]\vec B(x, y, R, H, H_0)=\int_{H_0}^{H+H_0} (\int_0^R (\int_0^{2*π} (\frac{G*4*π*ρ}{c^2}*\frac{\vec{v}×\vec{l}}{[\vec{l}]^3}*r* dα)dr)dh)[/itex]
where:
R is cylinder radius.
H is cylinder height.
H0 is (vertical) distance between pendulum and cylinder.
[itex]\vec{v}=ω*(Sin(a)*r, Cos(a)*r, 0)[/itex]
[itex]\vec{l}=(x - Cos(a)*r, y - Sin(a)*r, h)[/itex]
where ω is angular velocity of cylinder.
so [itex]\vec v×\vec l = (h*r*Cos(a),h*r*Sin(a), -r*x*Cos(a) - r*y*h*Sin(a) + r^2)[/itex] and
[itex]|\vec{l}|^3=(h^2+ (x - r*Cos(a))^2+ (y - r*Sin(a))^2)^{3/2}[/itex]
[itex]\vec B(x, y, R, H, H_0)=\int_{H_0}^{H+H_0} (\int_0^R (\int_0^{2*π} (\frac{G*4*π*ρ}{c^2}*\frac{(h*r*Cos(a),h*r*Sin(a), -r*x*Cos(a) - r*y*Sin(a) + r^2)*r*ω}{(h^2 + (x - r*Cos(a))^2 + (y - r*Sin(a))^2)^{3/2}}* dα)dr)dh)[/itex]
[itex]\vec B(x, y, R, H, H_0)=\frac{G*4*π*ρ*ω}{c^2}*\int_{H_0}^{H+H_0} (\int_0^R (\int_0^{2*π} (\frac{(h*r*Cos(a),h*r*Sin(a), r-x*Cos(a) - y*Sin(a))*r^2}{(x^2+y^2+r^2+h^2-2*r(x*Cos(a)+y*Sin(a)))^{3/2}}* dα)dr)dh)[/itex]
[itex]\vec B(x, y, R, H, H_0)=\frac{G*4*π*ρ*ω}{c^2}*\int_{H_0}^{H+H_0}(\int_0^R((\frac{h*π*r*x*(\sqrt{h^4*(x^2+y^2-r^2)^2+2*h^2*(r^2+x^2+y^2)}-(h^2+r^2+x^2+y^2))}{(x^2+y^2)*\sqrt{h^4+(x^2+y^2-r^2)^2+2*h^2*(r^2+x^2+y^2)}},\frac{h*π*r*y*(\sqrt{h^4+(x^2+y^2-r^2)^2+2*h^2*(r^2+x^2+y^2)}-(h^2+r^2+x^2+y^2))}{(x^2+y^2)*\sqrt{h^4 + (x^2+y^2-r^2)^2+2*h^2*(r^2+x^2+y^2)}},?)*dr)dh)[/itex]
Is my equation correct?
Can anyone simplify that equation for me?
How would be best to observe/measure gravitomagnetic field ,with pendulum changing it´s trajectory (because of gravitomagnetic field´s vertical component), pendulum changing it´s oscillation period(because of gravitomagnetic field´s horizontal component), or with two cylinders pulling/pushing each other like normal magnets?
May this experiment work?
Has such experiment ever been done before?
To calculate magnetic force above the cylinder I use formula:
[itex]\vec B=\frac{G*4*π}{c^2}*\frac{\vec{v}×\vec{l}*m}{[\vec{l}]^3}[/itex]
where:
B is magnetic field in some point A.
G is gravitational constant.
c is speed of light.
v is speed of moving pointmass at point C.
l is vector [itex]l=\vec{AC}[/itex]
m is mass of pointmass .
To get gravitomagnetic field by whole cylinder integrate B over volume, because every point of cylinder may be seen as pointmass.
for circle:
[itex]\vec B(x, y, h, r)=\int_0^{2*π} (\frac{G*4*π*ρ}{c^2}*\frac{\vec{v}×\vec{l}*r}{[\vec{l}]^3} *dα)[/itex]
for disc:
[itex]\vec B(x, y, h, R)=\int_0^R (\int_0^{2*π} (\frac{G*4*π*ρ}{c^2}*\frac{\vec{v}×\vec{l}}{[\vec{l}]^3}*r* dα)dr)[/itex]
And finally for cylinder:
[itex]\vec B(x, y, R, H, H_0)=\int_{H_0}^{H+H_0} (\int_0^R (\int_0^{2*π} (\frac{G*4*π*ρ}{c^2}*\frac{\vec{v}×\vec{l}}{[\vec{l}]^3}*r* dα)dr)dh)[/itex]
where:
R is cylinder radius.
H is cylinder height.
H0 is (vertical) distance between pendulum and cylinder.
[itex]\vec{v}=ω*(Sin(a)*r, Cos(a)*r, 0)[/itex]
[itex]\vec{l}=(x - Cos(a)*r, y - Sin(a)*r, h)[/itex]
where ω is angular velocity of cylinder.
so [itex]\vec v×\vec l = (h*r*Cos(a),h*r*Sin(a), -r*x*Cos(a) - r*y*h*Sin(a) + r^2)[/itex] and
[itex]|\vec{l}|^3=(h^2+ (x - r*Cos(a))^2+ (y - r*Sin(a))^2)^{3/2}[/itex]
[itex]\vec B(x, y, R, H, H_0)=\int_{H_0}^{H+H_0} (\int_0^R (\int_0^{2*π} (\frac{G*4*π*ρ}{c^2}*\frac{(h*r*Cos(a),h*r*Sin(a), -r*x*Cos(a) - r*y*Sin(a) + r^2)*r*ω}{(h^2 + (x - r*Cos(a))^2 + (y - r*Sin(a))^2)^{3/2}}* dα)dr)dh)[/itex]
[itex]\vec B(x, y, R, H, H_0)=\frac{G*4*π*ρ*ω}{c^2}*\int_{H_0}^{H+H_0} (\int_0^R (\int_0^{2*π} (\frac{(h*r*Cos(a),h*r*Sin(a), r-x*Cos(a) - y*Sin(a))*r^2}{(x^2+y^2+r^2+h^2-2*r(x*Cos(a)+y*Sin(a)))^{3/2}}* dα)dr)dh)[/itex]
[itex]\vec B(x, y, R, H, H_0)=\frac{G*4*π*ρ*ω}{c^2}*\int_{H_0}^{H+H_0}(\int_0^R((\frac{h*π*r*x*(\sqrt{h^4*(x^2+y^2-r^2)^2+2*h^2*(r^2+x^2+y^2)}-(h^2+r^2+x^2+y^2))}{(x^2+y^2)*\sqrt{h^4+(x^2+y^2-r^2)^2+2*h^2*(r^2+x^2+y^2)}},\frac{h*π*r*y*(\sqrt{h^4+(x^2+y^2-r^2)^2+2*h^2*(r^2+x^2+y^2)}-(h^2+r^2+x^2+y^2))}{(x^2+y^2)*\sqrt{h^4 + (x^2+y^2-r^2)^2+2*h^2*(r^2+x^2+y^2)}},?)*dr)dh)[/itex]
Is my equation correct?
Can anyone simplify that equation for me?
How would be best to observe/measure gravitomagnetic field ,with pendulum changing it´s trajectory (because of gravitomagnetic field´s vertical component), pendulum changing it´s oscillation period(because of gravitomagnetic field´s horizontal component), or with two cylinders pulling/pushing each other like normal magnets?
May this experiment work?
Has such experiment ever been done before?