Dynamics: Resultant Forces, Hooke's Law, and Mechanical Principles Study Notes

Fundamental Cases of Resultant Force Calculation

The intensity of the resultant force ($F_R$) acting on a body is determined based on the relative direction and sense of the acting forces ($F_1, F_2, \dots, F_{N}$).

• Case 1: Forces Acting in the Same Direction and Same Sense

  • The intensity of the resultant force is obtained by the sum of the intensities of the individual forces.
  • Formula: $F_R = F_1 + F_2$

• Case 2: Forces Acting in the Same Direction but Opposite Senses

  • The intensity of the resultant force is given by the difference between the intensities of the individual forces.
  • Formula: $F_R = F_1 - F_2$

• Case 3: Perpendicular Forces ($90^\circ$ angle)

  • When forces are perpendicular, the intensity of the resultant force is found using the Pythagorean Theorem.
  • Formula: $F_R^2 = F_1^2 + F_2^2$

• Case 4: Oblique Forces (Angle $\theta$)

  • When forces act at an angle other than $0^\circ, 90^\circ, \text{ or } 180^\circ$, the Law of Cosines is applied to find the resultant.
  • Formula: $F_R^2 = F_1^2 + F_2^2 + 2 \cdot F_1 \cdot F_2 \cdot \cos(\theta)$

Definition and Principles of Dynamics

Dynamics is the part of mechanics that studies the motion of bodies and the causes that produce it.

• Force Conceptualization:

  • Force is the physical agent capable of producing acceleration (dynamic effects) and/or deformations (static effects).
  • Dynamics specifically focuses on force as an agent that modifies the vectorial velocity ($\mathbf{V}$) of a body in terms of its magnitude (modulus), direction, and/or sense.

• Force as a Vector Quantity:

  • The effects of a force depend on four variables:
    1. The effect resulting from the intensity (modulus) of the force.
    2. The effect resulting from the sense of the force.
    3. The effect resulting from the direction of the force.
    4. The effect resulting from the point of application of the force.

• Types of Forces:

  1. Contact Forces: Forces that act only when bodies are in physical contact.
  2. Field Forces (Action at a Distance): Forces that act even when bodies are separated by a distance (e.g., gravitational, electric, magnetic forces).

Specific Force Types: Weight and Tension

• Weight Force ($P$):

  • Weight is a field force of attraction that a celestial body (like Earth) exerts on objects at or near its surface.
  • The direction of the weight force is always towards the center of the celestial body.
  • Formula: $P = m \cdot g$
    • Where $m$ is the mass of the body and $g$ is the acceleration due to gravity.

• Tension/Traction Force ($T$):

  • Tension is a force transmitted from one end of a rope, wire, or cable to the other.
  • It is referred to as traction ($T$) or simply tension ($\tau$).

Hooke's Law and Elasticity

In the regime of elastic deformation, the force exerted by a spring is proportional to the displacement of its end from its equilibrium position.

• Formula: $F = k \cdot \Delta x$

  • $F$: Intensity of the deforming force.
  • $k$: Spring constant (coefficient of proportionality). This value depends on the material the spring is made of and its dimensions.
  • $\Delta x$: The deformation (elongation or compression) produced in the spring.

• Units for Spring Constant ($k$): Often expressed in $N/m$ or $N/cm$.

Friction Force (Atrito)

• Definition: Friction ($f_a$) is the resistance force that opposes the relative motion (or tendency of motion) between two surfaces in contact.
• Friction can increase or decrease depending on the nature of the surfaces.

Solved Problems and Exercises

Exercise 1: Billiard Ball Acceleration

• Scenario: A billiard ball with a mass of $0.2\,kg$ is in Uniformly Varied Motion (MRU - Accelerated) due to a constant resultant force of $5\,N$.
• Task: Find the acceleration.
• Calculation: Using $F_R = m \cdot a$
  • $5 = 0.2 \cdot a$
  • $a = \frac{5}{0.2} = 25\,m/s^2$

Exercise 2: Velocity Acquisition

• Scenario: A constant force of intensity $F = 30\,N$ is applied to a body of mass $m = 4.0\,kg$ initially at rest ($V_0 = 0$).
• Task: Determine the velocity acquired by the body after $8.0\,s$.
• Calculation:
  1. Find acceleration: $30 = 4.0 \cdot a \rightarrow a = 7.5\,m/s^2$
  2. Find velocity: $V = V_0 + a \cdot t$
  3. $V = 0 + (7.5 \cdot 8.0) = 60\,m/s$

Exercise 3: Trigonometric Force Components

• Scenario: A force of $10\,N$ is applied to a block at an angle of $60^\circ$ from the horizontal.
• Data provided: $\sin(60^\circ) = 0.87$; $\cos(60^\circ) = 0.5$.
• Task: Find the horizontal ($F_x$) and vertical ($F_y$) components.
• Calculation:
  • $F_x = F \cdot \cos(60^\circ) = 10 \cdot 0.5 = 5\,N$
  • $F_y = F \cdot \sin(60^\circ) = 10 \cdot 0.87 = 8.7\,N$

Exercise 4: Resultant of Perpendicular Forces

• Scenario: Two perpendicular forces of $5\,N$ and $12\,N$ act on a particle.
• Calculation:
  • $F_R^2 = 5^2 + 12^2 = 25 + 144 = 169$
  • $F_R = \sqrt{169} = 13\,N$

Exercise 5: System of Connected Blocks

• Scenario: Two blocks, A and B, with masses $10\,kg$ and $20\,kg$ respectively, are joined by a rope of negligible mass. They are at rest on a horizontal plane without friction. A horizontal force $F = 60\,N$ is applied to block B.
• Tasks: Determine the acceleration of the system and the tension in the rope.
• Calculation for System Acceleration:
  • $F = (m_A + m_B) \cdot a$
  • $60 = (10 + 20) \cdot a \rightarrow 60 = 30 \cdot a$
  • $a = 2\,m/s^2$
• Calculation for Tension ($T$) on Block A:
  • $T = m_A \cdot a$
  • $T = 10 \cdot 2 = 20\,N$

Exercise 6: Fuvest - SP (Truck Weight)

• Scenario: To weigh a truck at a fiscal post, three scales are used simultaneously under each axle/set of wheels. The scales indicate $30,000\,N$, $20,000\,N$, and $10,000\,N$.
• Task: Find the total weight of the truck.
• Calculation: The total weight is the sum of the partial weights.
  • $P_{total} = 30,000 + 20,000 + 10,000 = 60,000\,N$

Exercise 7: Hooke's Law from Graph

• Scenario: A graph represents the force intensity ($F$) as a function of deformation ($\Delta x$). At $F = 60\,N$, the deformation is not explicitly given, but the constant $k$ is calculated.
• Task A: Determine the spring constant ($k$).
  • Given data points from a similar table suggest: $F = 90\,N$, $k = 1200\,N/m$.
  • $90 = 1200 \cdot \Delta x \rightarrow \Delta x = 0.075\,m = 7.5\,cm$
• Task B (UFSM-RS Exercise): A runner uses a rubber band with $k = 300\,N/m$. The maximum elongation reached is $28\,cm$.
• Calculation:
  • $\Delta x = 28\,cm = 0.028\,m$
  • $F = k \cdot \Delta x = 300 \cdot 0.028 = 84\,N$
  • (Option E in the multiple choice: $84$)