Three identical circular rings, each of mass m and radius r, are joined together to form a physical pendulum. They can oscillate about a pivot point (denoted by the triangle) in two different configurations as shown. vertical horizontal 000 = (1) Express the moments of inertia of each configuration about their pivot points in terms of m and r. [Hint: (1) The moment of inertia of a ring about an axis perpendicular to its plane and passing through the center of mass is Iring mr². (2) The moment of inertia about an axis parallel to an axis passing through the center of mass is obtained via the parallel axis theorem: I = ICM + md², where d is the (perpendicular) distance between the axes. (3) You can express the total moment of inertia with respect to the pivot as the sum of the individual moments of inertia with respect to the pivot, i.e., Ipivot = Σ; Ipivot j.] 1.5 seconds, what is the period = (2) If the period of oscillation of the vertical configuration is Tv Th of oscillation of the horizontal configuration?
Angular Momentum
The momentum of an object is given by multiplying its mass and velocity. Momentum is a property of any object that moves with mass. The only difference between angular momentum and linear momentum is that angular momentum deals with moving or spinning objects. A moving particle's linear momentum can be thought of as a measure of its linear motion. The force is proportional to the rate of change of linear momentum. Angular momentum is always directly proportional to mass. In rotational motion, the concept of angular momentum is often used. Since it is a conserved quantity—the total angular momentum of a closed system remains constant—it is a significant quantity in physics. To understand the concept of angular momentum first we need to understand a rigid body and its movement, a position vector that is used to specify the position of particles in space. A rigid body possesses motion it may be linear or rotational. Rotational motion plays important role in angular momentum.
Moment of a Force
The idea of moments is an important concept in physics. It arises from the fact that distance often plays an important part in the interaction of, or in determining the impact of forces on bodies. Moments are often described by their order [first, second, or higher order] based on the power to which the distance has to be raised to understand the phenomenon. Of particular note are the second-order moment of mass (Moment of Inertia) and moments of force.
![Three identical circular rings, each of mass m and radius
r, are joined together to form a physical pendulum. They
can oscillate about a pivot point (denoted by the triangle)
in two different configurations as shown.
vertical
8
horizontal
OOC
=
(1) Express the moments of inertia of each configuration about their pivot points in terms of m
and r. [Hint: (1) The moment of inertia of a ring about an axis perpendicular to its plane
and passing through the center of mass is Iring
mr². (2) The moment of inertia about
an axis parallel to an axis passing through the center of mass is obtained via the parallel
ICM+md2, where d is the (perpendicular) distance between the axes. (3)
You can express the total moment of inertia with respect to the pivot as the sum of the
individual moments of inertia with respect to the pivot, i.e., Ipivot = Σ; Ipivot,j]
axis theorem: I =
(2) If the period of oscillation of the vertical configuration is Tv
Th of oscillation of the horizontal configuration?
= 1.5 seconds, what is the period](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4d409dbe-0069-4bdd-acc1-9391e01545d6%2F5e21ee3f-fc3a-4b63-9f55-6a753c9e0a19%2Fcz72asd_processed.jpeg&w=3840&q=75)
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