Mechanics: Kinematics and Dynamics

TL;DR

Kinematics describes how things move (position, velocity, acceleration), while dynamics explains why they move (forces). Newton's Laws are the foundation for understanding how forces cause these changes in motion. By combining these ideas, you can predict and explain the motion of almost anything.

1. The Mental Model

Think of motion in two parts: first, describing the journey itself, and second, understanding what pushed or pulled to make it happen. It's like describing a car trip versus explaining why the car sped up or slowed down.

2. The Core Material

Kinematics: Describing Motion

Kinematics is all about the "how" of motion. We use a few key terms to describe it:

• Position ($\vec{x}$ or $\vec{r}$): Where an object is. It's usually measured from an origin (a starting point). For 1D motion, it's just a number; for 2D or 3D, it needs coordinates.
• Displacement ($\Delta\vec{x}$ or $\Delta\vec{r}$): The change in position. It's the straight-line distance and direction from the start to the end point, not the total path traveled.
• Distance: The total length of the path traveled, regardless of direction.
• Velocity ($\vec{v}$): How fast an object's position changes and in what direction. It's displacement over time ($\Delta\vec{x} / \Delta t$). Speed is just the magnitude of velocity (how fast, ignoring direction).
• Acceleration ($\vec{a}$): How fast an object's velocity changes. It's change in velocity over time ($\Delta\vec{v} / \Delta t$). Acceleration means speeding up, slowing down, or changing direction.

For constant acceleration, these equations are super handy:

1. $v = v_0 + at$
2. $\Delta x = v_0 t + \frac{1}{2}at^2$
3. $v^2 = v_0^2 + 2a\Delta x$
4. $\Delta x = \frac{v_0 + v}{2}t$

Here, $v_0$ is initial velocity, $v$ is final velocity, $a$ is constant acceleration, $t$ is time, and $\Delta x$ is displacement. Choose the equation that best suits the information you have and what you need to find.

Dynamics: Explaining Motion

Dynamics addresses the "why" of motion – forces! Sir Isaac Newton's three laws are the pillars here.

• Newton's First Law (Inertia): An object at rest stays at rest, and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force. This means objects resist changes to their current state of motion.
• Newton's Second Law ($\vec{F}_{\text{net}} = m\vec{a}$): The net force acting on an object is equal to the product of its mass ($m$) and acceleration ($\vec{a}$). This is the workhorse equation of dynamics. A larger net force causes a larger acceleration, and a more massive object requires a larger force to achieve the same acceleration. Force is measured in Newtons (N), where $1 \text{ N} = 1 \text{ kg} \cdot \text{m}/\text{s}^2$.
• Newton's Third Law (Action-Reaction): For every action, there is an equal and opposite reaction. If object A exerts a force on object B, then object B simultaneously exerts an equal and opposite force on object A. These forces always act on different objects.

Types of Forces:

• Weight ($F_g = mg$): The force of gravity on an object, where $g$ is the acceleration due to gravity (approx. $9.8 \text{ m/s}^2$ on Earth, downwards).
• Normal Force ($F_N$): A contact force perpendicular to a surface, pushing outward from the surface. It prevents objects from passing through surfaces.
• Friction Force ($F_f$): A contact force parallel to a surface, opposing relative motion or tendency of motion. It depends on the normal force and the coefficient of friction ($\mu$). $F_f = \mu F_N$.
• Tension Force ($F_T$): A pulling force exerted by a string, rope, or cable.
• Applied Force ($F_A$): A general term for any external pushing or pulling force.

Strategy for Dynamics Problems:

1. Draw a Free-Body Diagram (FBD): This is crucial! Draw the object as a point, and draw all forces acting on that object, starting from the point. Label each force vector.
2. Choose a Coordinate System: Align axes with the expected motion or acceleration if possible.
3. Resolve Forces into Components: Break down any forces not aligned with your axes into x and y components.
4. Apply Newton's Second Law: $\sum F_x = ma_x$ and $\sum F_y = ma_y$.
5. Solve: Use algebra to find the unknowns.

3. Worked Example

Let's say you're pushing a 10 kg box across a rough horizontal floor with a constant force of 50 N. The coefficient of kinetic friction between the box and the floor is 0.3. What's the acceleration of the box?

1. Draw an FBD:

   • Weight ($F_g$) acting downwards ($\downarrow$)
   • Normal Force ($F_N$) acting upwards ($\uparrow$)
   • Applied Force ($F_A$) acting horizontally in the direction you're pushing ($\rightarrow$)
   • Kinetic Friction Force ($F_f$) acting horizontally opposite to the direction of motion ($\leftarrow$)

2. Choose Coordinate System: Let x be horizontal (direction of push) and y be vertical.

3. Resolve Forces: All forces are already aligned or perpendicular to axes.

4. Apply Newton's Second Law:

   • Y-direction (vertical): The box isn't accelerating vertically, so $a_y = 0$.
     $\sum F_y = ma_y$
     $F_N - F_g = m(0)$
     $F_N = F_g$
     We know $F_g = mg = (10 \text{ kg})(9.8 \text{ m/s}^2) = 98 \text{ N}$.
     So, $F_N = 98 \text{ N}$.

   • Calculate Friction: Now that we have $F_N$, we can find the kinetic friction force:
     $F_f = \mu_k F_N = (0.3)(98 \text{ N}) = 29.4 \text{ N}$.

   • X-direction (horizontal): The box is accelerating horizontally, so $a_x = a$.
     $\sum F_x = ma_x$
     $F_A - F_f = ma$
     $50 \text{ N} - 29.4 \text{ N} = (10 \text{ kg})a$
     $20.6 \text{ N} = (10 \text{ kg})a$

5. Solve:
   $a = \frac{20.6 \text{ N}}{10 \text{ kg}} = 2.06 \text{ m/s}^2$.

So, the box will accelerate at $2.06 \text{ m/s}^2$.

4. Key Takeaways

• Kinematics describes how objects move using position, displacement, velocity, and acceleration.
• Dynamics explains why objects move, focusing on forces and Newton's Laws.
• Newton's Second Law, $\vec{F}_{\text{net}} = m\vec{a}$, is the fundamental relationship between force and motion.
• Free-body diagrams are essential tools for visualizing and analyzing all forces acting on an object.
• Forces come in various types (gravity, normal, friction, tension, applied) and always involve an interaction between two objects.
• The kinematic equations are useful for problems with constant acceleration.

Common Mistakes to Avoid:
* Confusing speed with velocity or distance with displacement; remember velocity and displacement include direction.
* Forgetting to draw a Free-Body Diagram; it's the most common reason for errors in dynamics problems.
* Applying Newton's Second Law to an individual force instead of the net force.
* Mixing up action-reaction pairs; remember they act on different objects.
* Assuming normal force always equals weight; it only does on a horizontal surface with no other vertical forces.

5. Now Try It

A car starts from rest and accelerates uniformly at $3.0 \text{ m/s}^2$ for 5.0 seconds. After 5.0 seconds, it stops accelerating and travels at a constant velocity for another 10.0 seconds.

1. Calculate the total distance the car traveled during the entire 15.0 seconds.
2. What was the average speed of the car for the entire trip?

Think about how to break this into two phases. For success, you should be able to clearly calculate the distance for each phase and then combine them for the total distance and average speed.