Engineering Mechanics: Statics

• Engineering Mechanics is the application of mechanics to solve problems involving engineering elements.
• Mechanics is the study of forces and their effects.

Statics

• Statics deals with bodies at rest or in equilibrium.
• Equilibrium is the state where the net force and net moment acting on a body are zero.
• Forces are vector quantities, possessing both magnitude and direction.
• Common force types include: weight, tension, compression, friction, and applied forces.
• Moments (or torques) are the turning effect of a force about a point.
• Moments are calculated as the product of the force magnitude and the perpendicular distance from the line of action of the force to the point (moment arm).
• Free Body Diagrams (FBDs) are essential for solving statics problems.
• An FBD isolates the body of interest and shows all external forces acting on it.
• Supports and connections exert reaction forces on a body.
• Common support types include: roller supports (one reaction force, perpendicular to the surface), pin supports (two reaction forces, horizontal and vertical components), and fixed supports (two reaction forces and a moment).
• Equilibrium equations in 2D are: ΣFx = 0, ΣFy = 0, and ΣM = 0.
• ΣFx represents the sum of all horizontal forces.
• ΣFy represents the sum of all vertical forces.
• ΣM represents the sum of all moments about a point.
• The choice of the point about which to sum moments can significantly simplify the problem.
• Trusses are structures composed of members connected at joints, designed to support loads.
• Truss members are assumed to be pin-connected, meaning they can only transmit axial forces (tension or compression).
• The method of joints involves analyzing the forces acting at each joint of the truss.
• Apply the equilibrium equations (ΣFx = 0 and ΣFy = 0) at each joint to solve for the unknown member forces.
• The method of sections involves cutting through the truss and analyzing a section of it.
• Apply the equilibrium equations (ΣFx = 0, ΣFy = 0, and ΣM = 0) to the section to solve for the unknown member forces.
• Frames and machines are structures composed of members connected by pins or hinges, designed to support loads.
• Unlike trusses, frame members can carry axial forces, shear forces, and bending moments.
• To analyze frames and machines, draw FBDs of individual members and apply equilibrium equations to each member.
• Friction is a force that opposes motion between two surfaces in contact.
• Static friction prevents motion from starting.
• Kinetic friction acts when motion is occurring.
• The maximum static friction force is proportional to the normal force: Fs,max = μsN, where μs is the coefficient of static friction.
• The kinetic friction force is also proportional to the normal force: Fk = μkN, where μk is the coefficient of kinetic friction.
• The coefficient of static friction is generally greater than the coefficient of kinetic friction (μs > μk).
• The angle of repose is the angle at which an object will start to slide down an inclined plane due to gravity where tan θ = μs.
• Centroids represent the geometric center of an area or volume.
• The centroid of an area is the point where the area could be perfectly balanced.
• The centroid location can be found using integration or by using composite areas.
• For composite areas, the centroid location is calculated by summing the product of each area's centroid location and area, then dividing by the total area.
• Moments of inertia describe a body's resistance to rotational acceleration about a given axis.
• The area moment of inertia is a geometric property of a cross-section.
• For a rectangle, the area moment of inertia about its centroidal axis is (1/12)bh³, where b is the base and h is the height.
• The parallel axis theorem is used to calculate the moment of inertia about an axis that is parallel to the centroidal axis: I = Ic + Ad², where Ic is the moment of inertia about the centroidal axis, A is the area, and d is the distance between the two axes.

Dynamics

• Dynamics deals with bodies in motion and the forces that cause that motion.
• Kinematics describes the motion of bodies without considering the forces causing the motion.
• Kinetics relates the forces acting on a body to its motion.
• Displacement is the change in position of an object.
• Velocity is the rate of change of displacement with respect to time.
• Acceleration is the rate of change of velocity with respect to time.
• For constant acceleration, kinematic equations relate displacement, velocity, acceleration, and time.
• Equations include: v = u + at, s = ut + (1/2)at², v² = u² + 2as, where (u) is the initial velocity, (v) is the final velocity, (a) is the acceleration, (t) is the time, and (s) is the displacement.
• Projectile motion is the motion of an object launched into the air, subject only to gravity.
• The horizontal and vertical components of projectile motion can be analyzed separately.
• Newton's Second Law of Motion states that the net force acting on an object is equal to the product of its mass and acceleration (F = ma).
• Mass is a measure of an object's resistance to acceleration.
• Weight is the force of gravity acting on an object (W = mg), where g is the acceleration due to gravity (approximately 9.81 m/s²).
• Work is done when a force causes displacement.
• Work is calculated as the product of the force and the displacement in the direction of the force (W = Fd cosθ).
• Energy is the capacity to do work.
• Kinetic energy is the energy of motion (KE = (1/2)mv²).
• Potential energy is stored energy due to position or configuration.
• Gravitational potential energy is the energy stored due to an object's height above a reference point (PE = mgh).
• The work-energy principle states that the net work done on an object is equal to the change in its kinetic energy (Wnet = ΔKE).
• Power is the rate at which work is done (P = W/t).
• Momentum is the product of an object's mass and velocity (p = mv).
• Impulse is the change in momentum of an object (J = Δp = FΔt).
• The principle of impulse and momentum states that the impulse acting on an object is equal to the change in its momentum.
• Conservation of momentum states that the total momentum of a closed system remains constant if no external forces act on it.
• Collisions can be elastic (kinetic energy is conserved) or inelastic (kinetic energy is not conserved).
• In a perfectly elastic collision, the relative velocity of separation is equal to the negative of the relative velocity of approach.
• In a perfectly inelastic collision, the objects stick together after the collision.
• Angular displacement is the change in angular position of an object.
• Angular velocity is the rate of change of angular displacement with respect to time.
• Angular acceleration is the rate of change of angular velocity with respect to time.
• Relationships between angular and linear quantities include: v = rω, a = rα, where r is the radius of rotation, ω is the angular velocity, and α is the angular acceleration.
• Moment of inertia (I) describes an object's resistance to rotational acceleration about a given axis.
• Torque (τ) is the rotational equivalent of force (τ = Iα).
• Angular momentum is the product of an object's moment of inertia and angular velocity (L = Iω).
• Conservation of angular momentum states that the total angular momentum of a closed system remains constant if no external torques act on it.