A point mass m 1 is held in place at the origin, and another point mass m 2 is f

Question

A point mass m1 is held in place at the origin, and another point mass m2 is free to move a distance r away at a point P having coordinates x, y, and z. The gravitational potential energy of these masses is found to be U(r) =-G(m1m2) /r , where G is the gravitational constant.
(a) Show that the components of the force on m2 due to m1 are

Fx = -Gm1m2x/( x2+y2 +z2 )3/2

Fy= -Gm1m2y/( x2+y2 +z2 )3/2

Fz= -Gm1m2z/( x2+y2 +z2 )3/2

(b) Show that the magnitude of the force on m2 is F= G(m1m2) /r2. Does m1 attract or repel m2? How do you know?

Explanation / Answer

(a) Distance of the given point, r = (x^2 + y^2 + z^2)1/2 Fx = F cos      = (- G m1 m2 / r^2) ( x / r)      = - G m1 m2 x / r^3      = - G m1 m2 x / [ (x^2 + y^2 + z^2)1/2 ] ^ 3      = - G m1 m2 x / (x^2 + y^2 + z^2)3/2 Fy = F cos      = (- G m1 m2 / r^2) ( y / r)      = - G m1 m2 y / r^3      = - G m1 m2 y / [ (x^2 + y^2 + z^2)1/2 ] ^ 3      = - G m1 m2 y / (x^2 + y^2 + z^2)3/2 Fz = F cos      = (- G m1 m2 / r^2) ( z / r)      = - G m1 m2 z / r^3      = - G m1 m2 z / [ (x^2 + y^2 + z^2)1/2 ] ^ 3      = - G m1 m2 z / (x^2 + y^2 + z^2)3/2 (b) Net force, F = (F1^2 + F2^2 + F3^2 ] =   G m1 m2 ( x^2 + y^2 + z^2 ) / (x^2 + y^2 + z^2)3/2 = G m1 m2 / (x^2 + y^2 + z^2)1/2 = G m1 m2 / r^2 There is attraction force on m2 by m1, since the gravitational force is always attractive force. Fy = F cos      = (- G m1 m2 / r^2) ( y / r)      = - G m1 m2 y / r^3      = - G m1 m2 y / [ (x^2 + y^2 + z^2)1/2 ] ^ 3      = - G m1 m2 y / (x^2 + y^2 + z^2)3/2 Fz = F cos      = (- G m1 m2 / r^2) ( z / r)      = - G m1 m2 z / r^3      = - G m1 m2 z / [ (x^2 + y^2 + z^2)1/2 ] ^ 3      = - G m1 m2 z / (x^2 + y^2 + z^2)3/2 (b) Net force, F = (F1^2 + F2^2 + F3^2 ] =   G m1 m2 ( x^2 + y^2 + z^2 ) / (x^2 + y^2 + z^2)3/2 = G m1 m2 / (x^2 + y^2 + z^2)1/2 = G m1 m2 / r^2 There is attraction force on m2 by m1, since the gravitational force is always attractive force. Fy = F cos      = (- G m1 m2 / r^2) ( y / r)      = - G m1 m2 y / r^3      = - G m1 m2 y / [ (x^2 + y^2 + z^2)1/2 ] ^ 3      = - G m1 m2 y / (x^2 + y^2 + z^2)3/2 Fz = F cos      = (- G m1 m2 / r^2) ( z / r)      = - G m1 m2 z / r^3      = - G m1 m2 z / [ (x^2 + y^2 + z^2)1/2 ] ^ 3      = - G m1 m2 z / (x^2 + y^2 + z^2)3/2 (b) Net force, F = (F1^2 + F2^2 + F3^2 ] =   G m1 m2 ( x^2 + y^2 + z^2 ) / (x^2 + y^2 + z^2)3/2 = G m1 m2 / (x^2 + y^2 + z^2)1/2 = G m1 m2 / r^2 There is attraction force on m2 by m1, since the gravitational force is always attractive force. Fz = F cos      = (- G m1 m2 / r^2) ( z / r)      = - G m1 m2 z / r^3      = - G m1 m2 z / [ (x^2 + y^2 + z^2)1/2 ] ^ 3      = - G m1 m2 z / (x^2 + y^2 + z^2)3/2 (b) Net force, F = (F1^2 + F2^2 + F3^2 ] =   G m1 m2 ( x^2 + y^2 + z^2 ) / (x^2 + y^2 + z^2)3/2 = G m1 m2 / (x^2 + y^2 + z^2)1/2 = G m1 m2 / r^2 There is attraction force on m2 by m1, since the gravitational force is always attractive force. Fz = F cos      = (- G m1 m2 / r^2) ( z / r)      = - G m1 m2 z / r^3      = - G m1 m2 z / [ (x^2 + y^2 + z^2)1/2 ] ^ 3      = - G m1 m2 z / (x^2 + y^2 + z^2)3/2 (b) Net force, F = (F1^2 + F2^2 + F3^2 ] =   G m1 m2 ( x^2 + y^2 + z^2 ) / (x^2 + y^2 + z^2)3/2 = G m1 m2 / (x^2 + y^2 + z^2)1/2 = G m1 m2 / r^2 = G m1 m2 / r^2 There is attraction force on m2 by m1, since the gravitational force is always attractive force.

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