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 14Ma2b2r2. Further, prove that the moment of inertia about a tangent is 5M4p2, 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 x2+y2+z2=a2, z≥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 a3 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 a12 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=a3(7+4√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: mr2(l2+415r)(l+2r3).

[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 ρ is in the shape of the solid formed by the revolution of the Cardioid r=a(1+cosθ) about the initial line Show that the moment of inertia about the straight line through the pole and perpendicular to the initial line is 352105πρa2.

[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]