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 ${\displaystyle \frac{1}{4}}M{\displaystyle \frac{a^2{b}^2}{r^2}}$. Further, prove that the moment of inertia about a tangent is ${\displaystyle \frac{5M}{4}}{p}^2$, 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 $x^2+y^2+z^2=a^2$, $z\ge 0$ 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 ${\displaystyle \frac{a}{3}}$ 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 ${\displaystyle \frac{a}{12}}$ 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 $b={\displaystyle \frac{a}{3}}(7+4\sqrt{7})$, 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: $m{r}^2{\displaystyle \frac{({\displaystyle \frac{l}{2}}+{\displaystyle \frac{4}{15}}r)}{(l+{\displaystyle \frac{2r}{3}})}}$.

[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 ${\displaystyle \frac{352}{105}}\pi \rho a^2$.

[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]