Learning Outcomes

Upon completing this lesson, you will be able to:

• Identify and model common forces in dynamic systems, including weight, normal force, friction, and spring forces.
• Differentiate between static and kinetic friction and apply their respective formulas.
• Analyze the kinetics of a particle on an inclined plane by resolving forces into appropriate components.
• Apply the equation of motion $(\Sigma \vec{F} = m\vec{a})$ to solve complex problems involving multiple forces and acceleration.

Topics Covered

• The Equation of Motion Revisited
• Common Forces in Dynamics
• Kinetics on an Inclined Plane
• Problem-Solving with Examples

I. The Equation of Motion Revisited

As established in the previous lesson, the foundation of kinetics is Newton's Second Law, which gives us the equation of motion:

$$ \Sigma \vec{F} = m\vec{a} $$

This vector equation is the link between the causes of motion (the net force, $\Sigma \vec{F}$, from the Free-Body Diagram) and the resulting motion (the acceleration, $\vec{a}$, from the Kinetic Diagram). When solving problems, we break this down into scalar components, most commonly:

$$ \Sigma F_x = ma_x $$

$$ \Sigma F_y = ma_y $$

Mastering kinetics involves correctly identifying all the forces acting on a body.

II. Common Forces in Dynamics

Let's take a deeper look at the specific forces that frequently appear in dynamics problems.

Weight (Gravity): The weight force $(W)$ is the gravitational pull of the Earth on an object. It always acts vertically downward. Its magnitude is calculated as $W = mg$, where $g \approx 9.81 , m/s^2$

Normal Force $(N)$: The normal force is a contact force exerted by a surface on an object. It always acts perpendicular to the surface. Important: The normal force is not always equal to the weight. It is a reactive force that adjusts its magnitude to prevent the object from accelerating through the surface.

Spring Force $(F_s)$: The force exerted by a spring is given by Hooke's Law, $F_s = kx$, where $k$ is the spring stiffness and $x$ is the displacement from its unstretched length. The force always acts in a direction that opposes the displacement.

Friction Force $(f)$: Friction is a contact force that opposes the motion or impending motion between two surfaces. It always acts parallel to the surface.

1. Static Friction $(f_s)$: This force acts when the object is not moving. It has a variable magnitude that matches the applied force up to a maximum value: $f_s \le \mu_s N$, where $\mu_s$ is the coefficient of static friction.
2. Kinetic Friction $(f_k)$: This force acts when the object is sliding. It has a constant magnitude given by: $f_k = \mu_k N$, where $\mu_k$ is the coefficient of kinetic friction. Typically, $\mu_k < \mu_s$.

III. Kinetics on an Inclined Plane

A very common and important application of kinetics is analyzing the motion of an object on a sloped surface (an inclined plane).

A. Choosing the Coordinate System

For these problems, it is highly advantageous to tilt the coordinate system. We typically align:

• The $x$-axis to be parallel to the slope.