ver m( t ) = up USEFUL INFORMATION COM? solve E Chain rule: - du = dt L = Iw , where I is the moment of inertia of a rigid object and w its angular velocity Isphere = = Mr2, Vsphere = 3 r is the radius of the sphere, M is the mass of the solid sphere. Mass density p = Change in momentum for a rocket with exhaust velocity Vex is dP = mdu + Vexdm Cartesian coordinates dV = dxdydz Cylindrical coordinates dV = rdrdqdz Spherical coordinates dV = r2 sin 0 drd0dy, z = r cos 0 sin 20 = 2 sin 0 cos 0 1 - cos 20 sin20 = Trigonometry 2 1 + cos 20 cos2 0 = 2 dx Integrals = Inx XPage 3 Problem 3. Center of mass for two inverted hemispheres. (20 points) N A. A uniform solid hemisphere with radius R and mass M sits with its flat side on the xy-plane, and is centered on the z-axis. Since the z-axis is a rotational symmetry axis for this solid, the center of mass lies on the z-axis, and XCM = yCM = 0. Show that the z-coordinate of the center of mass for this solid object is given by ZCM = (Use spherical coordinates for z and dV to perform the necessary integral ) (12 points) R- radius M. mass P= MR MJmkiv close in / z dm dm = pav B. The hemisphere in part (A) is turned upside down and placed on top of a uniform solid cylinder with same mass 3M, base radius R, and height H = 2R. Using the N result for the center of mass of two combined objects as R = _ MIR1 +M2R2, determine M1 +M2 the location of the z-coordinate of the center of mass for this combined system of the cylinder and hemisphere. (8 points) , =3, K=RH= ZR D = M.R, +M2R2. M, R + 3 M, R. M . + M L M, + 3M