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Chapter 16: Problem 4

In the integral for the moment \(M_{x}\) of a thin plate, why does \(y\) appear in the integrand?

Short Answer

Expert verified
Answer: The \(y\) value in the integrand represents the distance of each mass element \(\delta m\) from the x-axis, and it is crucial for determining the moment of the thin plate. Multiplying the \(y\) value by \(\delta m\) gives the individual moment contribution of that mass element about the x-axis. Including the \(y\) value in the integrand allows the integral to capture the combined effect of all the individual moments, resulting in the total moment \(M_x\) of the thin plate.

Step by step solution

01

Defining Moment and Center of Mass

The moment of an object is a measure of the distribution of mass within the object with respect to an axis. The moment about the x-axis (\(M_x\)) refers to the distribution of mass along the x-axis. The center of mass of an object, which can be considered the average position of all the mass, plays an important role in determining its moments. In the context of a thin plate, the center of mass is often the point where the object would balance perfectly.
02

Integral Formula for \(M_x\)

An integral is used to find the moment \(M_x\) of a thin plate. The integral formula is given by: \[M_x = \int_A y \cdot \delta m\] where \(A\) is the area of the thin plate, \(y\) is the distance from the x-axis, and \(\delta m\) is an infinitesimal mass element on the plate. This formula sums up the product of each mass element and its corresponding distance from the x-axis.
03

The Role of \(y\) in the Integrand

The \(y\) term in this integrand plays a crucial role in determining the moment of the thin plate. The distance of each mass element \(\delta m\) from the x-axis is represented by \(y\), and multiplying it by \(\delta m\) gives the individual moment contribution of that mass element about the x-axis. Therefore, the \(y\) term is necessary to account for the varying distances of the mass elements from the x-axis. Including the \(y\) value in the integrand allows us to capture the combined effect of all these individual moments in the integral, which is then equal to the total moment \(M_x\) of the thin plate.

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