Notes: Forces, Inertial Frames, and Inclined Planes

Inertial Reference Frames and Motion

• Grounded Idea (Newton's First Law - Law of 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.

  • If an object is at rest or moving with constant velocity in a straight line, it will keep doing so unless acted on by an unbalanced force.

  • Reversal: If something is accelerating (changing its motion), there must be an unbalanced force acting on it, even if you can’t see its source. This directly relates to Newton's Second Law:
    $\sum \vec{F} = m \vec{a}$
    where $\sum \vec{F}$ is the net force, $m$ is mass, and $\vec{a}$ is acceleration.

• Key Concept: Inertial Reference Frames

  • Treat rest and constant velocity as equivalent.

  • You cannot distinguish between being at rest and moving at a constant velocity in an inertial frame.

  • Thought Experiment: Blindfolded in a box, asleep, wake up with no felt acceleration

    • you can’t tell if you’re static or moving at a constant velocity without external reference.

• Types of Reference Frames:

  • Inertial reference frame: A frame that is not accelerating. Here, Newton's Laws of Motion hold true.

  • Non-inertial reference frames (accelerating frames): You feel acceleration, e.g., in a car after pressing the gas pedal or braking.

    • The seat’s force on you (an unbalanced force) accelerates you forward; you feel it in your body until you reach a steady cruising velocity in that frame.

    • Turning, braking, or passing another car introduces a new unbalanced force from the seat, belt, etc.

    • If you’re pulled over (police ahead as you pass the speed trap) and you brake, you feel the seat belt apply a force to change velocity; you’re back in an inertial frame when you stop.

• Practical Note: A boring life happens if there are no unbalanced forces (e.g., a book on a table).

Forces on Objects and Free-Body Diagrams

• Free-Body Diagrams: A visual representation used to analyze forces acting on a single object. All forces are drawn originating from the object's center.

• Normal force: Denoted by N or $F_N$, the contact force exerted perpendicular to a surface.

  • "Normal" refers to perpendicularity to the surface; it is not the same as “usual.”

  • Example: A book on a table.

    • Forces acting: Gravity downward ($F_g$) and normal force upward ($N$).

    • If the book does not accelerate (i.e., it is at rest or moving at a constant velocity vertically), then the sum of forces in the vertical (y) direction is zero:
      $\sum F_y = 0$

    • Therefore, $N - F<em>g = 0 \implies N = F</em>g$. Gravitational force is equal in magnitude and opposite in direction to the normal force.

• Gravity: Force due to mass in a gravitational field.

  • For an object of mass $m$, the gravitational force (also known as weight) is:
    $F_g = m g$
    where $g$ is the acceleration due to gravity.

  • On Earth, $g$ is approximately $9.8 \text{ m/s}^2$ (or $32.2 \text{ ft/s}^2$).
    -