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8.3 Applications to Physics and Engineering

8.3 Applications to Physics and Engineering. In this section, we will discuss only one application of integral calculus to physics and engineering and this topic is: The Center of Mass of Planar Lamina.

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8.3 Applications to Physics and Engineering

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  1. 8.3 Applications to Physicsand Engineering In this section, we will discuss only one application of integral calculus to physics and engineering and this topic is: The Center of Mass of Planar Lamina Consider a thin flat plate of material with uniform density called a planar laminar. We think of center of mass as its balancing point.

  2. Centroid of a Plane Region Consider a flat plate with uniform density that occupies a region R of the plane A. y Moment of R about the y-axis: Moment of R about the x-axis: R x a b Mass of Plate = (Density)(Area) Centroid of R: Note: If a lamina has the shape of a region that has an axis of symmetry, then the center of mass must lie on that axis.

  3. B.R lies between on the interval [a, b] where y Mass of R x a b Centroid of R:

  4. Examples: 1) Find the centroid of the region bounded by the curves. Solutions: y x 1 e D I D I

  5. 2) Find the center of mass of a semicircular plate of radius r. y By principle of symmetry, center of mass must lie on the y-axis. x (-r, 0) (r, 0)

  6. 3) Find the centroid bounded by the given curves. Solutions: Points of intersection y (1, 3) x (-2,0)

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