🌊College Physics II – Mechanics, Sound, Oscillations, and Waves Unit 12 – Static Equilibrium and Elasticity

Key Concepts and Definitions

• Static equilibrium occurs when an object is at rest and the net force and net torque acting on it are zero
• A force is a push or pull that can cause an object to change its motion or shape
• Torque, also known as moment of force, is the rotational equivalent of linear force and causes an object to rotate
• Stress is the force per unit area applied to an object, measured in pascals (Pa) or newtons per square meter (N/m²)
• Strain is the relative deformation or change in shape of an object when a force is applied, expressed as the ratio of change in length to original length
• Elastic modulus quantifies a material's resistance to elastic deformation under stress, with different moduli for various types of deformation (Young's modulus, shear modulus, bulk modulus)
• Hooke's law states that the force required to extend or compress a spring is directly proportional to the distance of extension or compression, as long as the elastic limit is not exceeded

Forces and Torques in Equilibrium

• For an object to be in static equilibrium, the vector sum of all forces acting on it must be zero ($\sum \vec{F} = 0$)
• Similarly, the vector sum of all torques about any point must also be zero ($\sum \vec{\tau} = 0$)
  • This condition ensures that there is no net rotational motion
• Forces can be represented as vectors, with both magnitude and direction
• Torque is calculated as the cross product of the position vector (from the axis of rotation to the point of force application) and the force vector ($\vec{\tau} = \vec{r} \times \vec{F}$)
• The magnitude of torque depends on the magnitude of the force and the perpendicular distance from the axis of rotation to the line of action of the force
• Counterclockwise torques are considered positive, while clockwise torques are negative

Conditions for Static Equilibrium

• For an object to be in static equilibrium, two conditions must be satisfied:
  1. The net force acting on the object must be zero in all directions ($\sum \vec{F} = 0$)
  2. The net torque about any point must be zero ($\sum \vec{\tau} = 0$)
• These conditions apply to both two-dimensional and three-dimensional systems
• In 2D, the net force condition can be split into horizontal and vertical components ($\sum F_x = 0$ and $\sum F_y = 0$)
• For 3D systems, the net force condition includes all three components ($\sum F_x = 0$, $\sum F_y = 0$, and $\sum F_z = 0$)
• The net torque condition in 2D requires that the sum of torques about a single point is zero
• In 3D, the net torque condition must be satisfied about any arbitrary axis

Free-Body Diagrams and Problem-Solving

• Free-body diagrams (FBDs) are essential tools for solving static equilibrium problems
• An FBD represents all the forces acting on an object, drawn as vectors with their tails at the object's center
• Steps for solving static equilibrium problems using FBDs:
  1. Identify the object in equilibrium and isolate it from its surroundings
  2. Draw the FBD, including all forces acting on the object
  3. Choose a convenient coordinate system and resolve forces into components if necessary
  4. Apply the conditions for equilibrium ($\sum \vec{F} = 0$ and $\sum \vec{\tau} = 0$)
  5. Solve the resulting equations to find unknown forces or dimensions
• When solving for torques, choose a convenient axis of rotation to simplify calculations
• Be consistent with the sign convention for forces and torques throughout the problem

Applications in Structures and Engineering

• Static equilibrium principles are crucial in the design and analysis of structures and machines
• Bridges, buildings, and cranes must be designed to support loads without collapsing or tipping over
  • Engineers calculate forces and torques to ensure the structure remains in equilibrium under various loading conditions
• Trusses are common structural elements composed of connected beams, forming triangular units
  • The forces in each member of a truss can be determined using the method of joints, applying equilibrium conditions at each joint
• Machines such as levers, pulleys, and gears use the principles of equilibrium to transmit and multiply forces
• Archimedes' principle states that the upward buoyant force on a submerged object is equal to the weight of the displaced fluid, a consequence of pressure and equilibrium

Stress, Strain, and Elastic Moduli

• Stress and strain describe how materials deform under applied forces
• Normal stress is the force per unit area perpendicular to the surface, while shear stress is the force per unit area parallel to the surface
• Normal strain is the change in length divided by the original length, and shear strain is the angular deformation caused by shear stress
• Elastic moduli relate stress and strain for a material:
  • Young's modulus (E) relates normal stress to normal strain
  • Shear modulus (G) relates shear stress to shear strain
  • Bulk modulus (K) relates pressure to volumetric strain
• These moduli are material properties that determine a substance's stiffness and resistance to deformation
• Poisson's ratio (ν) is the ratio of transverse strain to axial strain, describing how a material contracts in the direction perpendicular to the applied stress

Elasticity in Materials

• Elasticity is a material's ability to return to its original shape after being deformed by an applied force
• The elastic limit is the maximum stress a material can withstand without permanent deformation
• Hooke's law, $F = kx$, describes the linear relationship between force and displacement for elastic materials
  • The spring constant (k) is a measure of the material's stiffness
• Stress-strain curves graphically represent a material's elastic behavior
  • The linear portion of the curve corresponds to the elastic region, where deformation is reversible
  • The slope of the linear region is the elastic modulus (Young's modulus for normal stress)
• Beyond the elastic limit, materials enter the plastic region, where deformation is permanent and non-linear
• Ductile materials, such as metals, can undergo significant plastic deformation before fracture, while brittle materials, like ceramics, fracture with little plastic deformation

Practical Examples and Real-World Connections

• Bridges and buildings rely on static equilibrium to maintain stability under loads (weight of vehicles, people, and the structure itself)
• Cranes and lifting equipment use the principles of torque and equilibrium to safely transport heavy loads
• Elastic materials, such as springs and rubber bands, store potential energy when stretched or compressed, which can be used for various applications (spring-powered devices, shock absorbers)
• Stress and strain analysis is essential in material science and engineering to predict how materials will behave under different loading conditions (tension, compression, shear)
• The elastic properties of materials are crucial in the design of products that require flexibility or the ability to return to their original shape (athletic shoes, flexible electronics)
• Biomechanics applies the concepts of stress, strain, and elasticity to understand the behavior of biological tissues (bones, tendons, muscles) under physical loads
• Seismic design in architecture and engineering considers the elastic response of structures to earthquakes and other dynamic loads to minimize damage and ensure safety