Understanding Mechanics: Statics, Dynamics, and Kinematics

Posted on Dec 9, 2024 in Physics

The Mechanics

The three branches of mechanics are statics, dynamics, and kinematics.

Objective: Studying Motion

Kinematics describes the motion of bodies using variables such as:

Length: The key variable is the meter.
Time: The fundamental unit is the second.
Mass: The fundamental unit is the kilogram.
Number of substances: The fundamental unit is the mole.
Electric Current: The fundamental unit is the ampere.
Luminous Intensity: The fundamental unit is the candela.
Temperature: The fundamental unit is the kelvin.

Dynamics: Explaining Motion

Dynamics explains why bodies move. A new important dimension is work (mass × length × time). Two important concepts are mass and force.

Mass is a property of matter. It is a scalar quantity.
Force is the interaction between bodies. It is a vector quantity.

Force is the cause of any movement of bodies. The unit of force is the newton. Mass and force cannot exist without each other. When matter interacts with another force field, a force appears.

When two bodies are arranged on a line, they immediately interact. There are two masses, so there are two forces between them. The magnitude of interacting forces is equal. F1 and F2 have the same direction (the line where the two masses are located) but different meanings (the arrowhead).

Newton’s third law states that two bodies attract each other with the same force.

From a mechanical point of view, regarding movement, there are four fundamental forces:

Weight (w): The force with which the Earth attracts us. Weight is a vector (equals mass times gravity).
Normal: When bodies lie on a surface, the surface interacts with the body. The normal force is the result of this interaction.
Friction: When a body is on an inclined surface and remains at rest, it’s because there is an interaction between the surface and the body that prevents movement. This is friction.
Sometimes, a body moves with uniform rectilinear motion. This means that the friction force is equal to the force causing the motion.

There are two types of friction forces:

Static friction and kinetic friction. The friction force is defined as the product of μ (the coefficient of friction) and the normal force.

Tension: When bodies are hanging, the Earth interacts with the body, creating a force called weight. The rope holding the body experiences a force called tension.

Applying the Four Forces

If a body is at rest, the sum of the forces must be zero.

Statics: Bodies in Equilibrium

Statics studies bodies in equilibrium. The sum of the forces equals mass times acceleration.

Electric Field

When work is done, the body accumulates kinetic energy. If the body changes position, the work is associated with potential energy.

Kinetic energy is represented by K = 1/2 × mass × velocity².
Potential energy is represented by U = mass × gravity × height.
All these variables are measured in joules (newton × meter).

If mechanical work is performed, it will always be compressed in time.

The time it takes to perform work, or vice-versa, is related to power.

The speed at which work is done is known as power.

Power is work divided by time.

Work is force times displacement.

Power is equal to force times velocity.

Power has two forms of expression:

Power is measured in watts.
It can also be measured in horsepower (1 hp = 736 or 746 watts).

Work has three forms of expression.

Mechanical energy is defined as kinetic energy plus potential energy. When a particle moves in an environment without friction, mechanical energy is conserved.

Example Problem

A resultant force acts on a 10 kg body at rest on a smooth surface. Calculate the body’s speed when the work done is 320 joules.

Since the problem mentions speed, we need to calculate the change in kinetic energy. The work done is transformed into kinetic energy.

Work = ΔK = K_final – K_initial = 1/2 × mass × velocity_final² – 1/2 × mass × velocity_initial²

Since the initial velocity is zero, the equation simplifies to:

Work = 1/2 × mass × velocity_final²

Solving for velocity_final:

velocity_final = √(2 × Work / mass) = √(2 × 320 J / 10 kg) = 8 m/s

Another Exercise

A stone is thrown vertically upward with an initial kinetic energy of 500 joules. At a given instant, its velocity is 10 m/s. Calculate the particle’s height.

Mechanical energy is conserved: E_A = E_B

U_A + K_A = U_B + K_B

Since the initial height is zero, U_A = 0. Therefore:

K_A = U_B + K_B

500 J = mass × gravity × height_B + 1/2 × mass × velocity_B²

Solving for height_B:

height_B = (500 J – 1/2 × mass × velocity_B²) / (mass × gravity)

Newton’s Laws

The sum of the forces is zero.

Action and reaction forces are equal in magnitude and opposite in direction.

The sum of the forces equals mass times acceleration. This law leads to the concept of momentum: impulse equals the change in momentum.

Statics and Equilibrium

Statics describes the behavior of bodies in equilibrium. There are two fundamental principles: the first and second conditions of equilibrium. Here, we introduce the concept of torque (τ).

Previously, we considered bodies as particles. Now, we consider them in their full extent, but idealized (no deformation or breaking). We define a center of gravity where all forces act.

Torque involves force and motion, which can cause elongation, deformation, or breaking. When a force acts on a body for some time, it creates momentum. When a force does work, it produces a vector.

For example, when something opens or closes, we say that the body has rotated.

Torque is generated when a force causes a body to rotate. The two variables involved are force and the arm.

Torque is the cross product of force and the arm. Torque is a vector, and its units are newton-meters.

The arm of force is a segment drawn perpendicular from the rotation center to the line of action of the force.