Vector Mechanics and Fundamental Physics Principles
Vector Characteristics
A vector is a physical quantity characterized by a point of application, a magnitude (or modulus), a direction, and a sense. Alternatively, it can be defined by a number of independent components. Vectors are essential to describe physical phenomena that cannot be characterized by a single value.
The four main characteristics of a vector are:
- Point of Application
- Magnitude (or Modulus): Determines the size of the vector.
- Direction: Defines the line in space along which the vector acts.
- Sense: Specifies which way along the line of action the vector points.
Types of vectors based on equality criteria:
- Free Vectors: Their initial point (origin) can be set at any point.
- Fixed Vectors: Their initial point (origin) is fixed at a specific point.
- Equipollent Vectors: Vectors with equal magnitude, direction, and sense.
- Sliding Vectors: Equipollent vectors acting along the same line.
- Concurrent Vectors: Vectors sharing the same initial point (origin).
- Unit Vectors: Vectors with a magnitude equal to one.
- Opposing Vectors: Vectors with the same magnitude and direction but opposite sense (also called anti-parallel vectors).
- Collinear Vectors: Vectors acting along the same line of action.
Hooke’s Law
Hooke’s Law states that the unit elongation (ε) of an elastic material is directly proportional to the applied force (F). An elastic body recovers its original shape once the deforming force ceases. In any spring or elastic body, the deforming force is directly proportional to the strain (elongation) produced.
Dynamometer
A dynamometer is an instrument used to measure forces. Typically, a dynamometer’s operation is based on a spring that follows Hooke’s Law, where the deformation is proportional to the applied force. These instruments generally consist of a spring in a cylinder (plastic, cardboard, or metal), usually with two hooks, one at each end. Dynamometers have a scale marked in units of force on the cylinder. By hanging an object and exerting a force on the lower hook, the cylinder’s cursor moves on the outer scale, indicating the force’s value.
Law of Universal Gravitation
The law of universal gravitation states that the force attracting two bodies is directly proportional to the product of their masses and inversely proportional to the square of the distance separating them (F = –G(m1m2)/r2 u).
Newton’s Laws of Dynamics
1st Law (Principle of Inertia)
Every body continues in its state of rest or uniform rectilinear motion unless compelled to change that state by forces impressed upon it.
2nd Law (Law of Force)
The change of motion is proportional to the motive force impressed and is made in the direction of the straight line in which that force is impressed.
3rd Law (Law of Action and Reaction)
To every action, there is always opposed an equal reaction; or the mutual actions of two bodies upon each other are always equal and directed to contrary parts.
Difference Between Mass and Weight
| Mass | Weight |
|---|---|
| Quantity of matter in a body | Force with which the Earth attracts a body |
| Characteristic property of bodies; constant value for each body | Not a characteristic property of bodies; varies with gravity |
| Scalar quantity | Vector quantity |
| SI unit: kilogram (kg) | SI unit: Newton (N) |
| Measured with scales | Measured with a dynamometer |
Principle of Conservation of Momentum
If the resultant of all forces acting on a system is zero, then in any interaction (impact, explosion, etc.), the total momentum before and after the interaction remains constant. Momentum is a quantity linked to the dynamic state of a body and remains unchanged over time in isolated systems.
Energy
Energy is a fundamental concept in physics. All bodies, by being formed of matter-energy, may possess additional energy due to their motion, chemical composition, position, temperature, and other properties. Energy is often defined as the ability to do work.
Potential Energy
Potential energy is the energy that can be assigned to a conservative system or body due to its position or configuration. If a region of space has a field of conservative forces, then the work required to move a mass from a reference point (usually called ground level) to another point is the potential energy of the field at that point. By definition, ground level has zero potential energy.
Types of potential energy:
- Gravitational potential energy: Associated with a body’s position in a gravitational field.
- Electrostatic potential energy (V): Related to the electric field (E = -∇V).
- Elastic potential energy: Associated with the stress field of a deformable body.
Kinetic Energy
Kinetic energy is a fundamental concept in both classical and relativistic mechanics, as well as quantum mechanics. It is a scalar quantity associated with the motion of each particle in a system (Ec = 1/2 * mv2). An interesting property is that kinetic energy is an extensive quantity, meaning the total kinetic energy of a system can be expressed as the sum of the kinetic energies of its disjoint parts. For example, the kinetic energy of a body composed of n particles can be found by summing the kinetic energies of each individual particle.