Engineering Mechanics: Statics

Learn how to treat forces as vectors, resolve them into u and v components, and form the resultant by head-to-tail addition, using the cosine and sine laws for vector triangles.

Apply parallelogram law and vector triangle to determine F and theta, given a 500 N resultant along the positive y axis and a 700 N force at 15 degrees.

Calculate the magnitude and direction of the resultant force from two forces using a parallelogram, vector triangle, and law of cosines and sines measured counterclockwise from the positive x axis.

Learn to find the resultant of coplanar forces by resolving them into perpendicular components and summing x and y components using Cartesian vector notation with i and j.

Express three forces as Cartesian vectors using i and j notation, compute F1, F2, and F3 components from given magnitudes and angles via similar triangles and sine–cosine relationships.

Resolve multiple forces into x and y components with i and j, sum to obtain the force resultant, and determine its magnitude and direction using the Pythagorean theorem and tangent.

Resolve each force into x and y components, sum the components to form the resultant, and compute the magnitude with the square root of the sum of the squares (413.204 newtons). Then determine the direction of the resultant, measured counterclockwise from the positive x axis, which is 24.24 degrees.

Resolve the three forces into x and y components to match a 450 N resultant along the positive u axis, giving f1 ≈ 473.6 N at 10.9 degrees.

Learn to represent three dimensional forces in a right handed coordinate system, resolve vectors with the parallelogram law, and derive unit vectors, direction cosines, and coordinate direction angles.

Learn to form a three-dimensional vector from transverse theta and azimuth phi angles, compute a prime in the xy plane, derive a_x, a_y, a_z, and sum components for a resultant.

Compute coordinate direction angles for F1 and F2 and express them as cartesian vectors, with F1 = 53.1 i − 44.5 j + 40 k and F2 = −130 k.

Determine the magnitude and coordinate direction angles of the resultant force using vector components and alpha, beta, gamma. Sketch the resultant and interpret its x, y, z components.

Solve for the magnitude and direction angles alpha, beta, gamma of f1 in a three-force statics problem, using vector components and a -350 k resultant in the bracket.

Explain position vectors that locate points from the origin, define r = x i + y j + z k in three dimensional space, and compute r_b − r_a.

Apply position vectors to determine coordinates for points A and B, the AB vector, its magnitude, and the direction angles alpha, beta, and gamma.

Learn to express a line force using position vectors, derive unit vectors, and compute Cartesian force components, magnitude, and coordinate direction angles for a cable-rigged pole example.

Learn to determine the magnitude and coordinate direction angles of the resultant force at point A using position vectors, AB and AC, and unit vectors, with a step-by-step example.

Explore the dot product, the scalar (inner) product, and its use to find the angle between vectors and the parallel and perpendicular components via the projection.

Apply the dot product to determine the angle between cables AB and AC using vector coordinates and unit vectors, then compute the magnitude of F1's projection on AC.

Calculate the angle between BA and BC via dot product of direction vectors, then determine the 3 kN force projection along BC using unit vectors and vector projections.

Solve exam 1 problem by finding the angle between cable AC and the strut, and its projected force along the strut via unit vectors and dot products.

Express the two forces in vector form, then compute their resultant, and determine its magnitude and coordinate direction angles.

Determine EF1 and theta so the resultant force is vertical with 800 N by resolving all rope forces into x and y components using the 3-4-5 triangle relationships.

Analyze particle equilibrium using sum of forces equals zero and free body diagrams, then apply springs, cables, and normal forces to 2D problems.

Compute the two-dimensional equilibrium for a lift sling lifting a 500 kg container, derive AB and AC tensions as functions of theta, and find the shortest length about 1.72 m.

Engineering mechanics statics: use two-dimensional free-body diagrams and spring forces to find the stretches of three springs attached to a 2 kg block in equilibrium.

Solve a 2d equilibrium problem of a cable pulley system to determine the position x and the tension for a 100 kg sack, using a free-body diagram and geometry.

Master three-dimensional equilibrium by resolving forces into x, y, and z components and summing them to zero, solving for F1, F2, and F3.

Solve the three-dimensional equilibrium for a bucket held by cables ad, bd, and cd under a 20 lb weight, using a free body diagram and position vectors.

Define the scalar moment as force times the perpendicular distance to a point, use the right-hand rule for counterclockwise positive and clockwise negative rotations in two-dimensional examples like wrenches.

Calculate 2d moments about a point by using the perpendicular distance to the line of action and summing f d for each force, with counterclockwise positive.

Compute the moment about point B for three forces on a beam using f times d, identify perpendicular distances, decompose angled forces into components, and apply positive counterclockwise sign for rotation.

Compute the resultant moment about point O from three forces by using perpendicular distances, applying counterclockwise as positive, and summing the individual moments.

Compute the resultant moment about point O by summing the moments of two angled forces using components, distances (six feet) and the counterclockwise convention, preparing for vector cross-product in 3D.

Explore the cross product of vectors, its perpendicular direction to the plane containing A and B, the right-hand rule, and determinant method to compute A × B in 3-D.

Compute vector moments with m = r × f about a point, using any point on the force line of action. Apply transmissibility and three-dimensional cross products.

Calculate the moment about point A by crossing two position vectors AB and AC with a 100 N force, showing both methods yield the same result.

Calculate the moment of a force about the origin using r cross f, forming the position vector and evaluating a determinant to obtain the vector moment and its magnitude.

Determine the moment about point A by forming a 600 N force vector along BC, computing AB, and cross AB with F to obtain moment components in x, y, z.

Compute the moment of a force about a line using r cross f and the line's unit vector to obtain the line-parallel moment via the mixed triple product.

Apply the mixed triple product to find the moment about line AA using r_ac, r_bc, and the force vector f; obtain magnitude 13.8 kN·m and a vector along AA.

Compute the moment of a force about line AB using the unit AB vector and r × F; express as a vector and interpret the sign as rotation about AB.

Explore couple moments from equal opposite parallel forces separated by distance, using scalar formation m = f d and the right-hand rule to yield a counterclockwise moment.

Learn to form couple moments using two methods: sum moments about an arbitrary point and the vector method with rab, ra, and rb; 2d problems rotate about the z axis.

Determine the magnitude of F for two couples acting on a beam so the resultant moment is 450 ft counterclockwise, using scalar moment in a 2D setup and perpendicular distances.

Apply the first vector method by choosing point B, compute moments of the 150 lb and 200 lb couples about B, and obtain 240 lb-ft resultant about the z axis.

Use vector method two to determine the couple moment on the pipe by forming the position vector between the forces, computing r × F, and finding magnitude and rotation direction.

Learn how to simplify multi-force systems into an equivalent single force and a resultant moment about a reference point, enabling easier statics analysis.

Replace the beam loads with an equivalent resultant force and a couple moment about point O, then compute the resultant magnitude and direction and the moment about O.

In this 3d statics example, replace three forces with a single resultant at point O and a coupled moment, yielding -200 i + 700 j - 600 k N.

Learn to replace a multi-force system with a single resultant force by matching moments about a reference point and locating its line of action on the post from point B.

Compute the 3D equivalent by summing forces to a -215 kN resultant, then locate its line of action to reproduce the original moment about the origin.

Explore rigid body equilibrium in two dimensions by analyzing supports and free body diagrams, including cables, pins, rollers, and smooth surfaces, with emphasis on translating forces and solving for reactions.

Learn how pins and supports prevent translation and rotation, draw clear free-body diagrams, and apply sum of forces and moments to solve 2D rigid body equilibrium problems.

Solve 2D rigid-body equilibrium for a cantilever beam at fixed support A using a free-body diagram and sum of forces and moments to find fx, fy, and Ma.

Determine reactions at supports B and A for a bent rod with a collar and smooth surface, using a free-body diagram, x-y equilibrium, and moment about A to find Ma.

Draw a free-body diagram of the beam, apply moments about A, and solve for the vertical load F and reaction at A with a 50 kilonewton tension in BC.

Master 3d rigid body equilibrium by analyzing cable, smooth surfaces, rollers, ball-and-socket joints, journal bearings, pins, hinges, and fixed supports, focusing on forces and moments that prevent translation and rotation.

Explore three-dimensional equilibrium by forming six equations from force components and moments about the x, y, and z axes. Use vectors to compute moments and solve up to six unknowns.

This three-dimensional statics example demonstrates solving a ball-and-socket supported structure with two cables, using a free-body diagram and moments to find tensions and reactions.

learn to solve systems of equations with matrices from equilibrium equations by forming ax=b and using a inverse b on a calculator, maintaining a consistent unknown order.

Determine the reaction components at the collar and roller for a square rod statics problem by drawing the free body diagram and applying force and moment equilibrium.

this example solves a 3d equilibrium with three smooth journal bearings, determining f2 so bearing c's y reaction is zero, using a 3d free‑body diagram and moment equations about A.

Apply vector methods to find the moment about a specified axis for a 50-pound plant weight, using cross products and a unit axis vector, yielding the magnitude.

Construct a free body diagram of the balloon under 800 N uplift and three rope tensions. Solve a three-equation equilibrium system in vector form to keep the balloon in place.

A statics problem uses free-body diagrams and moment balance to find spring B stiffness for a horizontal beam under an 800 N load at C, yielding 2.5 kN/m.

Learn how to analyze trusses using the method of joints, identify two-force members, and detect zero-force members through free-body diagrams and 2D equilibrium.

Apply the method of joints to a two-truss problem, first finding support reactions from a free-body diagram, then compute each member's force and identify tension or compression.

Apply the method of joints to a ten-member truss to find each member's force, classify as tension or compression, and identify a zero-force member using joint free-body diagrams.

Apply the method of sections to solve truss problems by cutting through at most three unknown members, using equilibrium equations and moments to find member forces.

Apply the method of sections to a truss, cut to reveal three unknowns, and use the top portion to ignore reactions while solving for tension or compression.

This example uses the method of sections to solve a truss, finding forces in ed, eh, and gh and proving whether each member is in tension or compression.

Learn how frames and machines support loads and transfer forces. Apply free-body diagrams, equilibrium, and moment methods to solve a spring-loaded frame example.

Analyze a frame hoist to determine the load forces in members DB and FB using free-body diagrams, two-force-member rules, and equilibrium about points E and C.

examine a toggle clamp machine problem in engineering mechanics: statics, by drawing free-body diagrams, resolving forces, and determining the clamping force at a using moment methods.

define center of gravity and center of mass, derive x bar for balance using moments, and extend to centroids of areas, volumes, and lines with integration.

Compute the centroid y-bar of the shaded region bounded by y = x and y = 1/100 x^2 using horizontal and vertical rectangle methods with da, y tilde, and integration.

Analyze the centroid of a wire using line centroid formulas, exploiting symmetry to set x bar to zero and compute y bar from y = x^2, yielding (0, 1.823 ft).

Determine the x-bar centroid of the shaded area using a vertical rectangle area-centroid approach with integration, yielding x-bar = 3/8 a.

Learn to locate centroids of composite bodies by dividing shapes into simple regions and applying line and area centroid formulas, avoiding integration.

Determine the centroid of a homogeneous rod by splitting into three regions, using line centroid equations, and summing x, y, z tilde components to find x-bar, y-bar, z-bar.

Locate the area centroid by computing x bar and y bar for the shaded region from a reference origin, decomposing into triangles, squares, a quarter circle, and a half circle.

Learn to analyze distributed loads on beams, locate the resultant force at the centroid via area under the curve, and model w(x) as a pressure times beam width.

Convert a distributed load into a single resultant force by using the area under the curve and the centroid locations, then apply moments about point A to find its position.

Replace the distributed soil loads on the slab with a single 3,900-lb resultant, locating its line of action 11.27 ft from point O using centroids and moment summation.