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The gravitational potential experienced by a point on the $z$-axis due to a hemisphere

Identifying functions on the unit disk with functions on the upper hemisphereHow do I calculate the area of an ellipse projected onto a hemisphere?Triple Integration of gravitational potentialFind the potential at the center of the sphere.Trying to find the mass of sphere using density per unit areaUsing spherical coordinates find the limits of integration of the region inside a sphere with center $(a,0,0)$ and radius $a$Find flux through a sphereThe integral for gravitational potential of a uniform planetEvaluate the integral by changing to spherical coordinates as belowContinuity of potential function at an interior point

0

$begingroup$

A solid hemisphere of uniform density $rho$ occupies the region $$x^2+y^2+z^2le a^2,qquad zle0.$$ Find the gravitational potential due to the hemisphere at the point $(0,0,s)$ where $sgt0$. A uniform rod of density $m$ per unit length lies on the $z$-axis between $(0,0,c)$ and $(0,0,d)$ where $dgt cgt0$. Show that the force exerted on the rod by the hemisphere is $$psi(c)-psi(d)$$ where $$psi(lambda)=frac2pi Gmrho3left(fraca^3+lambda^3-left(a^2+lambda^2right)^3/2lambdaright).$$

So to determine the potential at $(0,0,s)$, I evaluated the following integral over the region $R$:

$$iiint_Rfracrho Gsqrtx^2+y^2+(z-s)^2,dx,dy,dz$$

Using a spherical substitution, the integral becomes:

$$int_r=0^aint_theta=fracpi2^piint_phi = 0^2 pifracr^2sin(theta)sqrtr^2+s^2-2rscos(theta)dphi dtheta dr$$

Which evaluates to
$$varphi(s)=fracpirho G3left(frac2a^3+3a^2s+s^3-sqrt(s^2+a^2)^3sright)$$

As for the second part, my understanding is that all we have to do is evaluate

$$mint_c^dvarphi^'(s),ds=mvarphi(d)-mvarphi(c)$$

My question is: is there anything wrong with my $varphi$ or is my method from the second half wrong?

Thanks!

multivariable-calculus vector-fields

share|cite|improve this question

edited Mar 17 at 1:43

Robert Howard

2,2583935

asked Mar 16 at 23:49

SulSul

332114

$endgroup$

add a comment | 

0

$begingroup$

A solid hemisphere of uniform density $rho$ occupies the region $$x^2+y^2+z^2le a^2,qquad zle0.$$ Find the gravitational potential due to the hemisphere at the point $(0,0,s)$ where $sgt0$. A uniform rod of density $m$ per unit length lies on the $z$-axis between $(0,0,c)$ and $(0,0,d)$ where $dgt cgt0$. Show that the force exerted on the rod by the hemisphere is $$psi(c)-psi(d)$$ where $$psi(lambda)=frac2pi Gmrho3left(fraca^3+lambda^3-left(a^2+lambda^2right)^3/2lambdaright).$$

So to determine the potential at $(0,0,s)$, I evaluated the following integral over the region $R$:

$$iiint_Rfracrho Gsqrtx^2+y^2+(z-s)^2,dx,dy,dz$$

Using a spherical substitution, the integral becomes:

$$int_r=0^aint_theta=fracpi2^piint_phi = 0^2 pifracr^2sin(theta)sqrtr^2+s^2-2rscos(theta)dphi dtheta dr$$

Which evaluates to
$$varphi(s)=fracpirho G3left(frac2a^3+3a^2s+s^3-sqrt(s^2+a^2)^3sright)$$

As for the second part, my understanding is that all we have to do is evaluate

$$mint_c^dvarphi^'(s),ds=mvarphi(d)-mvarphi(c)$$

My question is: is there anything wrong with my $varphi$ or is my method from the second half wrong?

Thanks!

multivariable-calculus vector-fields

share|cite|improve this question

edited Mar 17 at 1:43

Robert Howard

2,2583935

asked Mar 16 at 23:49

SulSul

332114

$endgroup$

add a comment | 

$begingroup$

A solid hemisphere of uniform density $rho$ occupies the region $$x^2+y^2+z^2le a^2,qquad zle0.$$ Find the gravitational potential due to the hemisphere at the point $(0,0,s)$ where $sgt0$. A uniform rod of density $m$ per unit length lies on the $z$-axis between $(0,0,c)$ and $(0,0,d)$ where $dgt cgt0$. Show that the force exerted on the rod by the hemisphere is $$psi(c)-psi(d)$$ where $$psi(lambda)=frac2pi Gmrho3left(fraca^3+lambda^3-left(a^2+lambda^2right)^3/2lambdaright).$$

So to determine the potential at $(0,0,s)$, I evaluated the following integral over the region $R$:

$$iiint_Rfracrho Gsqrtx^2+y^2+(z-s)^2,dx,dy,dz$$

Using a spherical substitution, the integral becomes:

$$int_r=0^aint_theta=fracpi2^piint_phi = 0^2 pifracr^2sin(theta)sqrtr^2+s^2-2rscos(theta)dphi dtheta dr$$

Which evaluates to
$$varphi(s)=fracpirho G3left(frac2a^3+3a^2s+s^3-sqrt(s^2+a^2)^3sright)$$

As for the second part, my understanding is that all we have to do is evaluate

$$mint_c^dvarphi^'(s),ds=mvarphi(d)-mvarphi(c)$$

My question is: is there anything wrong with my $varphi$ or is my method from the second half wrong?

Thanks!

multivariable-calculus vector-fields

share|cite|improve this question

edited Mar 17 at 1:43

Robert Howard

2,2583935

asked Mar 16 at 23:49

SulSul

332114

$endgroup$

share|cite|improve this question

edited Mar 17 at 1:43

Robert Howard

2,2583935

asked Mar 16 at 23:49

SulSul

332114

share|cite|improve this question

edited Mar 17 at 1:43

Robert Howard

2,2583935

asked Mar 16 at 23:49

SulSul

332114

edited Mar 17 at 1:43

Robert Howard

2,2583935

edited Mar 17 at 1:43

Robert Howard

2,2583935

asked Mar 16 at 23:49

SulSul

332114

asked Mar 16 at 23:49

SulSul

332114