91Ó°ÊÓ

A uniform circular disc of radius \(a\) and centre \(O\) is cut in half along a diameter. Show that the centre of mass of one of the halves is at a distance \(4 a / 3 \pi\) from \(O\). A uniform circular cone of height \(a\) has a plane base of radius \(a\). The cone is cut in half along a plane passing through its axis. Find the distances of the centre of mass of one of the halves from its plane faces.

Prove that the centre of mass of a uniform solid right circular cone, of height \(h\) and semi-vertical angle \(\alpha\) is at a distance \(\frac{3}{4} h\) from its vertex. A frustum is cut from the cone by a plane parallel to the base at a distance \(\frac{1}{2} h\). from the vertex. Show that the distance of the centre of mass of this frustum from its larger plane end is \(11 \mathrm{~h} / 56\). This frustum is placed with its curved surface in contact with a horizontal table. Show that equilibrium is not possible unless \(45 \cos ^{2} \alpha \geqslant 28\).

A uniform solid cone has a base radius \(r\) and height \(4 r .\) lt rests with its plane face on an inclined plane which is rough enough to prevent sliding. The cone will topple when the inclination of the plane to the horizontal is greater than: (a) \(45^{\circ}\) (b) arctan \(\frac{1}{4}\) (c) \(\arctan \frac{3}{4}\) (d) \(90^{\circ}\) (e) \(\arctan \frac{1}{2}\).

The centre of gravity of a uniform lamina in the form of a quadrilateral coincides with the centre of gravity of four particles of equal weight placed at the vertices of the quadrilateral.

Show that the centre of mass of a uniform solid right circular cone of height \(h\) is at a distance \(\frac{1}{4} h\) from its base. From a uniform solid right circular cylinder, of radius \(r\) and height \(h\), a right circular cone is bored out. The base of the cone coincides with one end of the cylinder and the vertex \(\mathrm{O}\) is at the centre of the other end. Show that the centre of mass of the remainder of the cylinder is at a distance \(3 h / 8\) from \(O\). The bored-out cylinder is placed with \(\mathrm{O}\) uppermost on a horizontal plane which is rough enough to prevent slipping: the plane is then gradually tilted. Show that the cylinder topples when the inclination of the plane to the horizontal exceeds \(\tan ^{-1}(8 r / 5 h)\)