Newton's Third Law and Forces in Classical Mechanics

Exam Tip: To identify a third law pair, ask yourself: "If Object A exerts force $F$ on Object B, what is Object B doing to Object A?" The answer will always be the reaction force.

Identifying Common Forces

To solve motion problems, you must be able to categorize the specific types of forces acting on a system.

Normal and Tension Forces

Normal Force ($F_N$): This is the support force exerted by a surface on an object. It acts perpendicularly (at a $90^\circ$ angle) to the surface where the object rests. If you place a book on a flat table, the table exerts an upward normal force to support the book's weight.

Tension ($T$): This is a pulling force that acts along a stretched, flexible connector, such as a rope, cable, or string. Tension always pulls away from the object it is attached to.

Propulsion and Thrust

How do we move? We rely on propulsion forces, which are the forces that allow us to move around, such as through walking or riding in a car.

A specific type of propulsion is thrust. Thrust is a reaction force that pushes a body forward in response to a backward force. For instance, a rocket engine expels gas downward at high velocity; the reaction to this downward force is the upward thrust that accelerates the rocket.

Analyzing Systems: Free-Body Diagrams

To calculate how an object will move, we use Free-Body Diagrams (FBDs). An FBD is a simplified sketch of an object that shows all the external forces acting upon it.

When drawing an FBD, remember:

1. Represent the object as a single point or a simple shape.
2. Draw each force as a vector (an arrow) pointing away from the object.
3. Only include forces acting on the object, not forces the object exerts on others.
4. Acceleration is determined by the forces acting on the object at this very moment in time, not by forces from the past.

Resolving Forces on an Incline

In many real-world problems, objects are not on flat surfaces but on inclined planes (slopes). In these cases, the force of gravity (weight) does not act perpendicular to the surface, which makes the math more complex. We must use trigonometric identities to resolve the weight into components.

If an object of mass $m$ is on a slope with an angle $\theta$, the weight ($W = mg$) can be split into two perpendicular components:

1. Perpendicular component ($W_{\perp}$): This component presses the object into the surface and is balanced by the normal force.
2. Parallel component ($W_{\parallel}$): This component pulls the object down the slope.

Using trigonometry, we define these components as:

$$\begin{aligned} W_{\perp} &= mg \cdot \cos(\theta) \\ W_{\parallel} &= mg \cdot \sin(\theta) \end{aligned}$$

By resolving the weight this way, you can easily apply Newton's Second Law ($\sum F = ma$) to find the acceleration of an object sliding down a ramp or being pulled up one.