Problem 4 - Calculus Early Transcendentals

Question

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

Step-by-Step Solution

Answer

Answer: The y term is present in the integrand of the moment for a thin plate because it represents the perpendicular distance between the axis of rotation (parallel to the x-axis) and the small rectangular mass element on the plate with an area dA. This distance is necessary to consider because it determines the contribution of the mass element to the overall moment around the x-axis, helping to quantify the moment's rotational tendency around the x-axis, considering the mass distribution and distance from the axis of rotation.

Step 1: Understand the concept of the moment of a thin plate

Moment, denoted as \(M_x\) or \(M_y\), is a measure of the plate's resistance to rotation. In other words, it can be seen as the turning force or torque acting around a specific axis. The moment depends on the mass distribution, and in some cases, the distance of mass from the axis of rotation is taken into account.

Step 2: Consider the mass distribution and the axis of rotation

In the case of a thin plate, the mass distribution is typically considered to be uniform. When calculating the moment \(M_x\), the axis of rotation is considered to be a line parallel to the x-axis; similarly, for \(M_y\), the axis of rotation is parallel to the y-axis.

Step 3: Express the moment of a thin plate in terms of an integral

With the understanding of mass distribution and axis of rotation, the moment of a thin plate can be expressed as an integral over the area of the plate. Let the mass density be represented by \(\delta(x, y)\) in gm/cm^2. For moment \(M_x\): $$M_x = \int\int_A y\delta(x, y) dA$$ For moment \(M_y\): $$M_y = \int\int_A x\delta(x, y) dA$$

Step 4: Explain the presence of \(y\) in the integrand of moment \(M_x\)

In the integrand of the moment \(M_x\), \(y\) appears because it represents the perpendicular distance between the axis of rotation (parallel to the x-axis) and the small rectangular mass element on the plate, which has an area \(dA\). This distance is necessary to consider because it determines the contribution of the mass element to the overall moment around the x-axis. Similarly, in the case of \(M_y\), \(x\) appears in the integrand for the same reason. Hence, the \(y\) term is present in the integrand because it helps to quantify the moment's rotational tendency around the x-axis, considering the mass distribution and distance from the axis of rotation.