Solving Problems with Newton's Second Law

Hello! Welcome to the next lesson in our course.

In our previous lessons on kinematics, we focused on describing the motion of particles—their position, velocity, and acceleration—without considering the causes of that motion. We learned to use both rectangular (x-y) and normal-tangential (n-t) coordinates to analyze motion along straight and curved paths.

Today, we transition from kinematics to kinetics. We will now investigate the why behind the motion. The central goal of this lesson is to achieve the following learning outcome:

Apply Newton's second law (F=ma) to solve for forces and accelerations in particle dynamics problems.

We will connect the forces acting on a particle to the acceleration it experiences. This is one of the most fundamental principles in all of mechanics. We will see how to apply this law systematically using different coordinate systems, building directly on the kinematic skills you've already developed.

1. The Foundation: Newton's Second Law and the Equations of Motion

At the heart of kinetics is Newton's Second Law of Motion. It provides the essential link between the forces acting on a body and the resulting change in its motion.

The equation $\sum \vec{F}=m\vec{a}$ is a vector relationship. This is powerful because it means we can break it down into scalar component equations along the axes of a chosen coordinate system. This is where our work in kinematics pays off.

• In Rectangular (x-y) Coordinates:
  $\sum F_x=m{a}_x$
  $\sum F_y=m{a}_y$
  These equations are perfect for analyzing rectilinear motion or projectile motion, where accelerations are easily expressed along horizontal and vertical axes.

• In Normal-Tangential (n-t) Coordinates:
  $\sum F_t=m{a}_t=m\dot{v}$
  $\sum F_n=m{a}_n=m\frac{v^2}{\rho }$
  This system is ideal for analyzing motion along a known curved path, as it directly relates forces to the change in an object's speed ($F_t$) and its change in direction ($F_n$). The force $\sum F_n$ is often called the centripetal force—the net force required to keep an object moving in a curve.

A Systematic Approach to Solving Problems

To apply these equations correctly and consistently, we will follow a structured problem-solving strategy. This approach organizes the problem and ensures all forces and motions are accounted for.

In essence, the strategy is:

1. Draw a Free-Body Diagram (FBD): You've done this in Statics. Identify and draw all external forces acting on the body of interest (e.g., weight, normal forces, friction, tensions).
2. Draw a Kinetic (or Mass-Acceleration) Diagram: On the other side of an equals sign, draw the body with its acceleration vector, $m\vec{a}$. This helps visualize the "effect" side of the equation.
3. Choose a Coordinate System: Select the system (rectangular or n-t) that best aligns with the motion.
4. Apply the Equations of Motion: Write the $\sum F=ma$ equations for each coordinate direction.
5. Solve: Solve the resulting algebraic equations for the unknowns. You may need to use kinematic relationships from previous lessons if acceleration is not constant or if you need to find velocity or position.

2. Application in Rectangular Coordinates

Let's see this strategy in action for problems best suited to rectangular coordinates, such as blocks on flat or inclined surfaces and systems connected by pulleys. The video below provides excellent worked examples.

As you saw in the video, a key step for inclined plane problems is to rotate your coordinate system. By aligning the x-axis with the incline and the y-axis perpendicular to it, the acceleration vector $\vec{a}$ lies entirely along one axis ($a_x=a,a_y=0$). This simplifies the equations of motion significantly, as the only force you need to resolve into components is the weight vector.

3. Application in Normal-Tangential (n-t) Coordinates

Now, let's turn our attention to objects moving along curved paths. As we saw in the last lesson, the n-t coordinate system is perfectly suited for this. This is especially relevant to your aerospace goals, as it's the natural way to analyze forces during aircraft maneuvers.

Recall the equations of motion in n-t coordinates:

• $\sum F_t=m{a}_t$: The net tangential force causes the particle to speed up or slow down.
• $\sum F_n=m{a}_n=m(v^2/\rho )$: The net normal force (centripetal force) causes the particle to change direction and stay on its curved path.

The following video shows how to apply this framework.

A classic aerospace application of n-t coordinates is analyzing a "level, coordinated turn." In this maneuver, the pilot banks the aircraft so that the lift force not only supports the aircraft's weight but also provides the necessary centripetal force to make it turn.

Conclusion

In this lesson, we made the pivotal step from describing motion to explaining it. By combining Newton's Second Law with our knowledge of kinematics, we can now solve for unknown forces or the resulting accelerations.

Key Takeaways:

• Newton's Second Law, $\sum \vec{F}=m\vec{a}$, is the bridge between kinetics and kinematics.
• The key to solving kinetics problems is a systematic procedure: draw a Free-Body Diagram, choose an appropriate coordinate system, and write the component equations of motion.
• For rectilinear motion, use rectangular coordinates ($\sum F_x=m{a}_x$, $\sum F_y=m{a}_y$). Aligning axes with the direction of acceleration is a crucial strategy.
• For curvilinear motion, normal-tangential coordinates are often superior ($\sum F_t=m\dot{v}$, $\sum F_n=m{v}^2/\rho$), as they directly relate forces to changes in speed and direction.

Next Lesson Preview:

While the $\vec{F}=m\vec{a}$ method is powerful for finding forces and accelerations at a specific instant, it can be cumbersome for problems where you need to relate forces to changes in speed over a distance. In our next lesson, we will introduce an alternative approach derived from Newton's Second Law: the Work-Energy Principle. This method provides a powerful shortcut for solving problems involving forces, velocity, and displacement.