1998 Paper 3 Q9

Year: 1998
Paper: 3
Question Number: 9

Course: zNo longer examinable
Section: Moments of inertia

Difficulty: 1700.0 Banger: 1484.0

moments of inertia cone integration parallel axis theorem perpendicular axis theorem small oscillations compound pendulum rigid body

Problem

A uniform right circular cone of mass $m$ has base of radius $a$ and perpendicular height $h$ from base to apex. Show that its moment of inertia about its axis is ${3\over 10} ma^2$, and calculate its moment of inertia about an axis through its apex parallel to its base. \newline[{\em Any theorems used should be stated clearly.}] The cone is now suspended from its apex and allowed to perform small oscillations. Show that their period is $$ 2\pi\sqrt{ 4h^2 + a^2\over 5gh} \,. $$ \newline[{\em You may assume that the centre of mass of the cone is a distance ${3\over 4}h$ from its apex.}]

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Difficulty Rating: 1700.0

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Banger Rating: 1484.0

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Problem source

A uniform right circular cone of mass $m$ has 
base of radius $a$ and perpendicular
height $h$ from base to apex. 
Show that its moment of inertia about its axis is ${3\over 10} ma^2$, 
and calculate its moment of inertia about an axis through
its apex parallel to its base.
\newline[{\em Any theorems used should be stated clearly.}]
The cone is now suspended from its apex and allowed
to perform small oscillations. Show that their
period is
$$
2\pi\sqrt{ 4h^2 + a^2\over 5gh} \,.
$$
\newline[{\em You may assume that the centre of mass of the cone
is a distance ${3\over 4}h$ from its apex.}]

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