Definition of second moment of area (moment of inertia) and radius of gyration
Moment of inertia :
In classical mechanics, moment of inertia, also called mass moment of inertia, rotational inertia, polar moment of inertia of mass, or the angular mass (SI units kg·m2, US units lbm ft2), is a property of a distribution of mass in space that measures its resistance to rotational acceleration about an axis. Newton’s first law, which describes the inertia of a body in linear motion, can be extended to the inertia of a body rotating about an axis using the moment of inertia. That is, an object rotating at constant angular velocity will remain rotating unless acted upon by an external torque. In this way, the moment of inertia plays the same role in rotational dynamics as mass does in linear dynamics, describing the relationship between angular momentum and angular velocity, torque and angular acceleration. The symbols I and sometimes J are usually used to refer to the moment of inertia or polar moment of inertia.
The moment of the inertia force on a particle around an axis multiplies the mass of the particle by the square of its distance to the axis, and forms a parameter called the moment of inertia. The moment of inertia of individual particles sum to define the moment of inertia of a body rotating about an axis. For rigid bodies- moving in a plane, such as a compound pendulum, the moment of inertia is a scalar, but for movement in three dimensions, such as a spinning top, the moment of inertia becomes a matrix, also called a tensor.
(Above figure shows-another view of a tightrope walker who uses the moment of inertia of the long rod to help maintain balance. This is Samuel Dixon crossing the Niagara river in 1890.)
Moment of inertia appears in Newton’s second law for the rotation of a rigid body, which states that the torque necessary to accelerate rotation is proportional to the moment of inertia of the body. Thus, the greater the moment of inertia the greater the torque needed for the same acceleration. Many systems use masses with large moment of inertia to maintain a rotational velocity and resist small variations in applied torque. For example, the long pole held by a tight-rope walker maintains a zero angular velocity resisting the small torque applied by the walker to maintain balance. Another example is the rotating mass of a flywheel which maintains a constant angular velocity resisting the torque variations in a machine.
The moment of inertia of an object is defined by the distribution of mass around an axis. It depends not only on the total mass of the object, but also on the square of the perpendicular distance from the axis to each element of mass. This means the moment of inertia increases rapidly as masses are distributed more distant from the axis. For example, consider two wheels that have the same mass, one that is the size of a bicycle wheel and one that is half that size. The larger wheel has four times the moment of inertia even though it is only twice the diameter.
Moment of inertia around a fixed axis is a scalar, however the rotation of a body in space can occur around the three coordinate axes. In this case, the moment of inertia associated with the three coordinate axes define a matrix of scalars called the inertia matrix, also known as the inertia tensor.
(Above figure shows-A flywheel is a wheel with a large moment of inertia used to smooth out motion in machines. This example is in a Russian museum.)
Radius of Gyration :
The radius of gyration of a body about an axis of rotation is defined as the radial distance of a point from the axis of rotation at which,if the whole mass of the body is assumed to be concentrated.Its moment of inertia about the given axis would be the same as with its actual distribution of mass.
If M is the mass of the body,its moment of inertia I in terms of its radius of gyration ‘k’ can be written as :
or I=M*(K SQUARED)
or k=square root of (I/M)
THE TOTAL MASS OF A ROTATING BODY MAY BE SUPPOSED TO BE CONCENTRATED AT A RADIAL DISTANCE ‘k’ FROM THE AXIS OF ROTATION SO FAR AS THE MOMENT OF INERTIA OF THE BODY ABOUT THAT AXIS IS CONCERNED.
NOTE: IT CANNOT BE SUPPOSED TO BE CONCENTRATED AT THE ‘CENTER OF MASS’ OF THE BODY.
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