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Welcome to our Physics lesson on Moment of Inertia in Different Systems of Rotational Motion, this is the third lesson of our suite of physics lessons covering the topic of Dynamics of Rotational Motion, you can find links to the other lessons within this tutorial and access additional physics learning resources below this lesson.
The main equation of moment of inertia I = m × r2 can take other forms based on the structure and shape of the rotating object involved. The formulae for all cases involved are found by integration. However, here it is not the case to deal so much with the method how these equations are obtained. We will simply provide the formulae for some common situations without losing time with the procedure.
Moment of inertia of a bar of mass m and length L when rotating around its centre as shown in the figure, is:
Moment of inertia of a bar of mass m and length L when rotating around its end as shown in the figure, is:
Moment of inertia of a cylinder or disc (a disc is a cylinder with a very short height compared to its base radius) of mass m and base radius R when rotating around its axis of symmetry, which passes along the height as shown in the figure, is:
Moment of inertia of a cylinder or disc of mass m, height h and base radius R when rotating around its central diameter as shown in the figure, is:
Moment of inertia of a ring of mass m and radius R when rotating around its axis of symmetry, which passes along the centre as shown in the figure, is:
Moment of inertia of a ring of mass m and radius R when rotating around its diameter as shown in the figure, is:
Moment of inertia of a sphere of mass m and radius R when rotating around its central diameter as shown in the figure, is:
Moment of inertia of a spherical shell of mass m and radius R when rotating around its central diameter as shown in the figure, is:
Calculate the moment of inertia for the following objects:
a. Since the diameter of cross-sectional area is much smaller than the length of wire, we consider it as a long bar instead of a cylinder. Therefore, we neglect the wire's thickness and focus only on its length, i.e. we have L = 12 m and m = 2 kg. Therefore, applying the formula of moment of inertia for a long bar rotating around its centre
we obtain after substitutions,
b. In this case, we can take the wooden rod as a cylinder since its length is not much bigger than its radius. We have R = d / 2 = 12 cm / 2 = 6 cm = 0.06 m, h = 80 cm = 0.8 m and m = 400 g = 0.4 kg. Therefore, applying the equation
we obtain after substitutions,
c. Moment of Inertia of a hollow sphere is calculated by the equation
where m = 600 g = 0.6 kg and R = 20 cm = 0.2 m. Substituting these values in the above equation, we obtain:
You have reached the end of Physics lesson 7.2.3 Moment of Inertia in Different Systems of Rotational Motion. There are 8 lessons in this physics tutorial covering Dynamics of Rotational Motion, you can access all the lessons from this tutorial below.
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