Meriam, Kraige & Bolton — Statics of Particles: Problem 2_28

⚡ Mecademy AIENG정역학 · ch2  Problem Statement 2/28 The force of magnitude F acts along the edge of the triangular plate. Determine the moment of F about point O. Problem 2/28 ● Calculation Process 1. Geometric Analysis and Force Vector Formulation: From the figure, the triangular plate has a base of length along the -axis and a height along the -axis. Let the vertices be at , , and . The force acts along the hypotenuse from point to point . The vector representing the line of action from to is: The magnitude of this vector is the length of the hypotenuse: The force vector can be expressed in terms of its magnitude and the unit vector along : 2. Determine Moment using Vector Cross Product: The moment of a force about a point is given by , where is a position vector from to any point on the line of action of the force. Let's use point : Calculating the cross product: bx hyO(0,0)A(0,h)B(b,0)F AB AB = AB(b−0)i+(0−h)j=bi−hj L= b+h 22 FF AB F=F⋅ = L AB (bi− b+h 22 F hj) OM = O r×Fr O A(0,h) r = A 0i+hj M = O r × A F=(hj)× (bi−hj)[ b+h 22 F ] M = O [h⋅ b+h 22 F b(j×i)−h⋅h(j×j)] Since and : 3. Alternative Scalar Method (Perpendicular Distance): The magnitude of the moment is , where is the perpendicular distance from to the line of action. The area of the triangle can be calculated in two ways: 1. Using base and height : 2. Using the hypotenuse as the base and as the height: Equating the two: Thus, the magnitude of the moment is: From the diagram, the force tends to rotate the plate