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Units and Dimensions/Moment of Inertia

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A physical quantity is a quantity that can be measured directly or indirectly. Mass of a body, length of an object, time of an event, velocity, acceleration, momentum, etc., are physical quantities. The quantities that cannot be measured, like fatigue, hate, and affection, cannot be called physical quantities. Measurement means comparing a physical quantity with another homogeneous quantity of the same kind taken as a standard to determine how often the given standard is contained in the given physical quantity.

Units and Dimensions

To measure a physical quantity, a standard is needed which is acceptable internationally. The

standard should be some convenient, definite, and easily reproducible quantity of the same kind in which the physical quantity is expressed. This standard is called a unit. Thus, a unit is a standard quantity with which a physical quantity of the same kind is compared for measuring it.

For the measurement of a physical quantity, two things are required:

(i) The unit in which the physical quantity is expressed.

(ii) The number (or numerical value) of times the given unit is contained in the physical quantity.

There are seven fundamental quantities that are independent of each other, namely mass, length, time, temperature, luminous intensity, electric current, and the quantity of substance. These quantities are called fundamental physical quantities, and the units of these physical quantities are called fundamental units. These quantities are called fundamental or base quantities because they all are independent of each other, and all other quantities in mechanics can be expressed in terms of the fundamental quantities. Any physical quantity derived from the fundamental physical quantities by multiplying or dividing them is called a derived physical quantity, and its units are called the derived unit. For example, the unit of speed is length/time since speed is given by the distance traveled in the given time. Since the physical quantity speed can be derived from the fundamental quantities of length and time, speed is a derived physical quantity, and its unit is the derived unit.

The number of times a fundamental quantity is contained in the given derived physical quantity is known as the dimension of the fundamental quantity. For example, the area of an object is expressed as [L] x [L] = [L2] = [M0L2T0].  Thus, units and dimensions are critical in understanding physical quantities. The relation between the units of a physical quantity and the quantity is called a dimensional equation.

Moment of Inertia

The term moment of inertia refers to the quantity that describes how a body resists angular acceleration and is calculated by multiplying each particle’s mass by its square of the distance from the axis of rotation. Alternatively, it can be explained as a quantity determining the amount of torque required for a particular angular acceleration in a rotational axis. The SI unit for the moment of inertia is kgm2. Moment of Inertia is often specified with respect to a certain axis of rotation. It primarily depends on how mass is distributed around a rotational axis. The moment of inertia varies with respect to the axis chosen. 

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