Meriam, Kraige & Bolton — Statics of Particles: Problem 2_54

⚡ Mecademy AIENG정역학 · ch2  Problem Statement 2/54 The indicated force–couple system is applied to a small shaft at the center of the plate. Replace this system by a single force and specify the coordinate of the point on the x-axis through which the line of action of this resultant force passes. Problem 2/54 ● Step 1: Identify the given force and couple vectors at point From the problem diagram, we define a right-handed coordinate system where the -axis is vertical (perpendicular to the plate). The force and the couple are applied at the origin . Resultant Force (): The force is directed along the negative -axis. Resultant Couple (): The moment is shown as a clockwise arrow in the -plane (rotating from the axis toward the axis). According to the right-hand rule, a clockwise moment about the -axis is negative. ● Step 2: Determine the equivalent single force system To replace the force-couple system at with a single force , we must place this force at a point such that its moment about is equal to the original couple . The problem specifies that the point lies on the -axis, so its coordinates are . 1. Moment Equation: 2. Substitute vectors: 3. Perform cross product: Recalling that , we have: O z FM O RF=6 kNy R=F=−6 kN j=−6000j N M O M=400 N⋅m xy+y+x z M = O −400k N⋅m OR P(x,y)OM O Px(x,0,0) M = O r×R −400k=(xi)×(−6000j) i×j=k −400k=−6000x(i×j) −400k=−6000xk ● Step 3: Solve for the -coordinate 1. Equate scalar components: 2. Calculate : 3. Convert to millimeters (if preferred)