Suppose a hole is drilled through the center of the earth and a bowling ball of mass m is dropped into the hole, as shown below. Construct a mathematical model that describes the motion og the ball. At time t, let r denote the distance from the center of the earth to the mass m, M denote the mass of the earth, Mr denote the mass of that portion of the earth within a sphere of radius r, and ? denote the constant density of the earth.

3. Suppose a hole is drilled through the center of the earth and a bowling ball of mass m is dropped into the hole, as shown below. Construct a mathematical model that describes the motion 0g the ball. At time t, let 1" denote the distance from the center of the earth to the mass m, M denote the mass of the earth, MT denote the mass of that portion of the earth within a sphere of radius 'r, and 6 denote the constant density of the earth. 2 + (GM ) r = o R3 Let a = GM 13 11 r+ ar = 0 O Let me r= e at r = r Auxiliary equation NOV 2 at at + ae x + a = 0 ( t ) -Vait r =Ae pit + Be We know it O e = Cost + 2 sint - Jati e = Cos (- Sat ) + isin (-Sat ) Jati = Cos ( Jat ) - 2 sin ( rat) e = Cos ( fat ) + 2 sin ( rat ). r = A [Gos ( at ) - 2sin ( fat ) ) + B [ Cos ( fat ) + i sin ( fat )) = ( A + B ) Cos ( sat ) + ( - A + $ ) i sin( Jat ) Let . A + B = C, ( - A + B ) i = C, r (t ) = C, GosK/GM) t + &Sib R 3 GM) t R 3