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14. You should already be familiar with the mathematical treatment of an
ideal pendulum, in which the pendulum bob is modeled as a point
mass on the end of a rigid rod of negligible mass. In this problem you
will consider the behaviour of more complex types of pendulum. You
will be given all the information you need in the sections below.
For a general pendulum of any shape and size the period P is given by
p= 2*pie*root(I/gM(LCM))
where g is the acceleration due to gravity, M is the total mass of the
pendulum, LCM is the effective length of the pendulum, defined as the
distance from the pivot to the centre of mass, and I is the moment of
inertia around the pivot point. For a point mass m fixed at a distance
r from the pivot I = mr^2, while for a uniform rod of mass m and length
r attached to the pivot at one end I = 1/3mr^2
. For more complex objects
the total moment of inertia can be calculated by adding together values
for the component parts.
(c) Now consider the case of a real pendulum, with a bob of mass
Mb (which you may treat as a point mass) attached to the pivot
using a uniform rod of mass Mr and length L, and find the period
in this case. Show that the result for a real pendulum reduces to
the results for an ideal pendulum and a rod pendulum by taking
appropriate limits.
ideal pendulum, in which the pendulum bob is modeled as a point
mass on the end of a rigid rod of negligible mass. In this problem you
will consider the behaviour of more complex types of pendulum. You
will be given all the information you need in the sections below.
For a general pendulum of any shape and size the period P is given by
p= 2*pie*root(I/gM(LCM))
where g is the acceleration due to gravity, M is the total mass of the
pendulum, LCM is the effective length of the pendulum, defined as the
distance from the pivot to the centre of mass, and I is the moment of
inertia around the pivot point. For a point mass m fixed at a distance
r from the pivot I = mr^2, while for a uniform rod of mass m and length
r attached to the pivot at one end I = 1/3mr^2
. For more complex objects
the total moment of inertia can be calculated by adding together values
for the component parts.
(c) Now consider the case of a real pendulum, with a bob of mass
Mb (which you may treat as a point mass) attached to the pivot
using a uniform rod of mass Mr and length L, and find the period
in this case. Show that the result for a real pendulum reduces to
the results for an ideal pendulum and a rod pendulum by taking
appropriate limits.