The measure of the linear density at a point of a rod varies directly as the third power of the measure of the distance of the point from one end. The length of the rod is 8 meters and the linear density is 2 kg/m at the center. Find the total mass and the center of mass of the rod.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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2 and 3
Evaluate the following integrals:
1. Using the Pappus Theorem and the fact that the volume of a right circular cone with radius
r and height h is equal to
find the centroid of the region bounded by y = 3- x,
y-axis, and a-axis.
2. The measure of the linear density at a point of a rod varies directly as the third power of
the measure of the distance of the point from one end. The length of the rod is 8 meters
and the linear density is 2 kg/m at the center. Find the total mass and the center of mass
of the rod.
3. Introduce an appropriate coordinate system and find the coordinate of the given simple
homogeneous plane region.
2 units
1unit
Transcribed Image Text:Evaluate the following integrals: 1. Using the Pappus Theorem and the fact that the volume of a right circular cone with radius r and height h is equal to find the centroid of the region bounded by y = 3- x, y-axis, and a-axis. 2. The measure of the linear density at a point of a rod varies directly as the third power of the measure of the distance of the point from one end. The length of the rod is 8 meters and the linear density is 2 kg/m at the center. Find the total mass and the center of mass of the rod. 3. Introduce an appropriate coordinate system and find the coordinate of the given simple homogeneous plane region. 2 units 1unit
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