PYQs

1) Show that the moment of inertia of an elliptic area of mass \(M\) and semi-axis \(a\) and \(b\) about a semi-diameter of length \(r\) is \(\dfrac{1}{4} M \dfrac{a^{2} b^{2}}{r^{2}}\). Further, prove that the moment of inertia about a tangent is \(\dfrac{5 M}{4} p^{2}\), where \(p\) is the perpendicular distance from the centre of the ellipse to the tangent.

[2017, 10M]

2) Calculate the moment of inertia of a solid uniform hemisphere \(x^{2}+y^{2}+z^{2}=a^{2}\), \(z \geq 0\) with mass \(m\) about the \(OZ-axis\).

[2015, 10M]

3) For solid spheres \(A\), \(B\), \(C\) and \(D\), each of mass \(m\) and radius \(a\), are placed with their centers on the four corners of a square of side \(b\). Calculate the moment of inertia of the system about a diagonal of the square.

[2013, 10M]

4) A pendulum consists of a rod of length $2a$ and mass \(m\); to one end of spherical bob of radius \(\dfrac{a}{3}\) and mass \(15m\) is attached. Find the moment of inertia of the pendulum:

(i) About an axis through the other end of the rod and at right angle to the rod.

(ii) About a parallel axis through the centre of mass of the pendulum. [Given: the centre of mass of the pendulum is \(\dfrac{a}{12}\) above the centre of the sphere].

[2012, 30M]

5) Let \(a\) be the radius of the base of a right circular cone of height \(h\) and mass \(M\). Find the moment of inertia of that right circular cone about a line through the vertex perpendicular to the axis.

[2011, 12M]

6) A uniform lamina is bounded by a parabolic are of latus rectum \(4a\) and a double ordinate at a distance \(b\) from the vertex. If \(b =\dfrac{a}{3}(7+4 \sqrt{7})\), show that two of the principal axis at the end of a latus rectum are the tangent and normal there.

[2010, 12M]

7) The flat surface of a hemisphere of radius \(r\) is cemented to one flat surface of a cylinder of the same radius and of the same material. If the length of the cylinder be \(l\) and the total mass be \(m\), show that the moment of inertia of the combination about the axis of the cylinder is given by: \(m r^{2} \dfrac{\left(\dfrac{l}{2}+\dfrac{4}{15} r\right)}{\left(l+\dfrac{2 r}{3}\right)}\).

[2009, 12M]

8) A rectangular plate swings in a vertical plane about one of its corners. If its period is one second, find the length of its diagonal.

[2005, 12M]

9) A solid body of density \(\rho\) is in the shape of the solid formed by the revolution of the Cardioid \(r=a(1+\cos \theta)\) about the initial line Show that the moment of inertia about the straight line through the pole and perpendicular to the initial line is \(\dfrac{352}{105} \pi \rho a^{2}\).

[2003, 12M]

10) Find the moment of inertia of a circular wire about:
(i) a diameter and
(ii) a line through the centre and perpendicular to its plane.

[2002, 12M]

11) Determine the moment of inertia of a uniform hemisphere about its axis of symmetry and about an axis perpendicular to the axis of symmetry and though centre of the base.

[2001, 12M]