20.8. EXERCISES 387

21. Let the angle between the z axis and the sides of a right circular cone be α . Alsoassume the height of this cone is h. Find the z coordinate of the center of mass of thiscone in terms of α and h assuming the density is constant.

22. Let the angle between the z axis and the sides of a right circular cone be α . Alsoassume the height of this cone is h. Assuming the density is σ = 1, find the momentof inertia about the z axis in terms of α and h.

23. Let R denote the part of the solid ball, x2 +y2 + z2 ≤ R2 which lies in the first octant.That is x,y,z≥ 0. Find the coordinates of the center of mass if the density is constant.Your answer for one of the coordinates for the center of mass should be (3/8)R.

24. Show that in general for L angular momentum,

dLdt

= Γ

where Γ is the total torque,Γ≡∑ri×F i

where F i is the force on the ith point mass.