CE 2010 Handout 8 Sections 43-44 Exam 1 Learning Objectives

Ce 2010 Handout 8 Sections 43 44 Exam 1learning Objectives E

Explain the cross product. Apply the cross product to determine 3D moments. Cross Product Method for finding _____________ using position vectors and force vectors. A crossed with B gets C → A x B = C. Magnitude of C is _____________________. Direction of C is found using the right hand rule. C is ___________________ to the plane containing A and B. Cross Product Laws of Operation include non-commutativity, with A x B ≠ B x A; scalar multiplication distributing as a(A x B) = A x aB = (A x B)a; and distributivity: A x (B + D) = A x B + A x D.

Moments with Vectors: _________________________, where MO is the moment about point or axis O, r is the position vector from O through any point that F pass through, and F is a force. Magnitude of MO is rFsinϕ = Frsinϕ = Fd. Direction of MO is ____________________ to the plane containing r and F.

Crossing Cartesian Unit Vectors: i x i = _____, i x j = _____, i x k = _____, j x i = _____, j x j = _____, j x k = _____, k x i = _____, k x j = _____, k x k = _____.

Using the cross product to find moments, MO = r x F = (rxi + ryj + rzk) x (Fxi + Fyj + Fzk) results in MO = (ryFz - rzFy)i + (rzFx - rxFz)j + (rxFy - ryFx)k. For a moment application, MO = r x F = (ryFz - rzFy)i + (rzFx - rxFz)j + (rxFy - ryFx)k. The same result can be found by calculating the determinant of a 3 x 3 matrix.

Method for Finding the Determinant: Ex #1 Determine the moment of the force F at point A about point P; express the result as a Cartesian vector. Varignon’s Theorem applies when multiple forces are applied at the __________ _____________. The assignment also involves watching videos produced by PBS and NOVA on materials research, and writing summaries, describing a selected material system, and preparing PowerPoint slides about it.

Paper For Above instruction

The cross product is a fundamental vector operation used extensively in engineering mechanics and physics to analyze three-dimensional forces and moments. It allows the calculation of a vector that is perpendicular to the plane defined by two other vectors, providing critical information about directions of rotational effects, such as torque and moments in rigid body dynamics. Understanding the properties and applications of the cross product facilitates accurate and efficient analysis of structures, mechanical systems, and material behaviors in three-dimensional space.

At its core, the cross product of two vectors A and B results in a third vector C, symbolized as C = A x B. The magnitude of this vector, |C|, is determined by the product of the magnitudes of A and B and the sine of the angle θ between them: |C| = |A||B|sinθ. The direction of vector C is perpendicular to both A and B, which can be ascertained using the right-hand rule: if the fingers of the right hand point from A toward B, then the thumb indicates the direction of C. This directional property implies that the cross product vector is orthogonal to the plane containing the original vectors, making it invaluable in calculating moments and torques in three-dimensional space.

The laws of the cross product highlight its non-commutative nature; that is, A x B ≠ B x A. Instead, A x B = – B x A, emphasizing the importance of order in calculations. Scalar multiplication distributes over the cross product, described mathematically as a(A x B) = (aA) x B = A x (aB) = a(A x B). The operation is also distributive over vector addition: A x (B + D) = A x B + A x D. These properties streamline the calculation of complex moment and force interactions in space, making the cross product a critical tool for engineers and physicists.

Moments, or torques, are a measure of a force's tendency to cause a body to rotate about a point or axis. In vector form, the moment MO about point or axis O is given by the cross product of the position vector r (from O to the point of force application) and the force vector F: MO = r x F. The magnitude of MO depends on the perpendicular distance d from the point O to the line of action of the force: |MO| = Fr sinϕ = Fd, where ϕ is the angle between r and F. The direction of the moment vector is perpendicular to the plane containing r and F, again determined via the right-hand rule. This ensures moments are vector quantities with well-defined directions, essential for analyzing rotational effects in structures, machinery, and materials.

The Cartesian unit vectors i, j, and k facilitate calculations of cross products in three dimensions. Their fundamental relationships are: i x i = 0, i x j = k, i x k = –j, j x i = –k, j x j = 0, j x k = i, k x i = j, k x j = –i, and k x k = 0. These establish a basis for deriving the cross product of any two vectors expressed in component form. For example, for vectors A and B as A = Axi + Ayj + Azk and B = Bxi + Byj + Bzk, the cross product is computed by expanding the determinant of a 3 x 3 matrix whose first row contains the unit vectors, second row the components of A, and the third row the components of B. This calculation yields the vector components for the resulting moment or force vector.

Applying this to moments, such as MO = r x F, involves substituting the components of the position vector r and the force vector F and computing the resulting vector. This approach is efficient and straightforward, especially when combined with the determinant method, which offers a systematic way to handle multiple components and orientations in 3D space. In practical engineering scenarios, such as determining the moment of a force at a point or analyzing the effect of wind on structures, this process provides precise and reliable results.

In the context of real-world engineering problems, Varignon's theorem simplifies the calculation of moments when multiple forces act concurrently on a body at the same point. It states that the moment of a resultant force equals the sum of the moments of the individual forces about the same point, provided all forces are concurrent. This principle is integral when analyzing complex systems, such as structures subjected to multiple loads, as it enables breaking down complex force systems into simpler components for easier calculation and interpretation.

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