Chapter 10: Problem 37
A thin, flat, uniform metal triangle lies in the \(x y\) plane with its corners at \((1,0,0),(0,1,0),\) and the origin. Its surface density (mass/area) is \(\sigma=24\). (Distances and masses are measured in unspecified units, and the number 24 was chosen to make the answer come out nicely.) (a) Find the triangle's inertia tensor I. (b) What are its principal moments and the corresponding axes?
Short Answer
Step by step solution
Understanding the Triangle's Shape and Properties
Calculating the Area of the Triangle
Establishing the Mass Distribution
Forming the Inertia Tensor Basis
Calculating the Inertia Tensor Elements
Finding \(I_{xx}\) and \(I_{yy}\)
Calculating \(I_{xy}\) and Principal Moments
Identifying Principal Axes
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Principal Moments
The inertia tensor for a 2D object like our metal triangle is a matrix that embodies how mass is distributed about various axes in the plane. The principal moments emerge from this tensor when it is diagonalized, which means transforming the tensor into a form where all off-diagonal elements are zero.
- The diagonal elements then represent the principal moments.
- These values signify the object's inertia about specific, orthogonal axes called principal axes.
- For a uniformly dense right triangle, symmetry helps simplify calculating these moments.
Surface Density
Understanding surface density is crucial for several reasons:
- It allows for the straightforward calculation of total mass when the area is known.
- It's essential for determining how the mass contributes to inertia.
- Knowing surface density simplifies evaluating how objects behave under physical forces.
Uniform Metal Triangle
The triangle we are considering is defined as having:
- Vertices at known coordinates forming a right triangle.
- Fixed surface density provided in the problem.
- Symmetry that aids in simplifying calculations.
This is particularly advantageous in theoretical problems like ours, where uniformity simplifies assumptions, allowing a focus on core concepts like inertia tensors and principal axes.