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Chapter 10: Problem 71

Consider the rotational inertia of a thin, flat object about an axis perpendicular to the plane of the object. Show that this is equal to the sum of the rotational inertias about two perpendicular axes in the plane of the object, passing through the given axis. (This is called the perpendicular-axis theorem.)

Short Answer

Expert verified
The perpendicular-axis theorem, \( I_{z} = I_{x} + I_{y} \), states that the moment of inertia of a flat object about an axis perpendicular to its plane is equal to the sum of the moments of inertia of the object about two perpendicular axes in its plane. This can be shown by expressing each moment of inertia in terms of tiny mass elements and their distances to the axis of rotation, and applying the Pythagorean theorem.

Step by step solution

01

Define the rotational inertia for axes in plane

The rotational inertia, or moment of inertia, of the object around an axis in its plane is given by summing the product of each tiny mass's squared distance from the axis and the tiny mass itself. We can represent this mathematically as: \( I_{x} = \int y^{2} dm \) and \( I_{y} = \int x^{2} dm \) for the x and y axes respectively.
02

Define the rotational inertia for the perpendicular axis

Similarly, we can define the moment of inertia about an axis perpendicular to the plane of the object and passing through the same point. This is represented by: \( I_{z} = \int r^{2} dm \), where r is the distance from the perpendicular axis to the tiny mass.
03

Express r in terms of x and y

Remember the Pythagorean theorem? It states that in any right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In this case, we can use it to express r in terms of x and y: \( r^{2} = x^{2} + y^{2} \)
04

Substitute r into the equation defined for \( I_{z} \)

We can now substitute the expression of r into \( I_{z} \). We have: \( I_{z} = \int (x^{2} + y^{2}) dm \) which can further be broken down to \( I_{z} = \int x^{2} dm + \int y^{2} dm \)
05

Prove the perpendicular-axis theorem

Looking at the last equation closely, we see that \( I_{z} = I_{x} + I_{y} \). This is essentially the perpendicular-axis theorem, which states that the moment of inertia of a flat object about an axis perpendicular to the plane of the object is equal to the sum of the moments of inertia of the object about two perpendicular axes in its plane.

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