Forces and Free Body Diagrams - Vocabulary
Core ideas and setup
The lecture starts with attempts to parse a mixed algebraic expression that students saw as a force/motion problem, illustrating common notation pitfalls:
Example confusion: interpreting something like "One fifth minus five fifths times six x plus seven"; the placement of parentheses drastically changes the meaning and the result.
Distinguish between expressions such as \frac{1}{5} - \Big(\frac{5}{5}\Big)(6x+7) vs \Big(\frac{1}{5} - \frac{5}{5}\Big)(6x+7); parentheses protect the intended operations by dictating the order of operations (PEMDAS/BODMAS).
The instructor emphasizes being detail-oriented about notation and to start problems early to avoid a messy derivation; even a slight misplacement of parentheses can fundamentally alter the mathematical meaning and physical outcome.
He notes a tendency to misinterpret expressions like "six x plus seven" as either a sum or a product, depending on grouping; parentheses are critical for clarity and avoiding ambiguity.
A practical reminder about signs and direction: when adding forces, you must treat direction as part of the sign convention (e.g., forces acting to the right along a chosen x-axis are positive, to the left are negative).
Physical context referenced:
Gravitational acceleration on Earth is ~ g \approx 9.8\ \mathrm{m/s^2} (often approximated as 10 for quick estimates).
The same ideas apply to rotating or moving reference frames, as discussed later with velocity and acceleration in different frames.
The overarching framework: forces impart accelerations; the net acceleration determines the net force, via Newton’s laws. Forces and accelerations are vector quantities, meaning they have both magnitude and direction.
General statement (to be clarified): each force contributes to the net acceleration; the net force equals the mass times the net acceleration: \mathbf{F}_{\mathrm{net}} = m\mathbf{a} and \mathbf{a} = \frac{\mathbf{F}_{\mathrm{net}}}{m} = \frac{1}{m}\sum_i \mathbf{F}_i
While the transcript says things like "the net accelerations multiplied by the mass gives you the new force," the standard form (Newton's Second Law) is as above: the vector sum of all forces equals mass times the acceleration vector.
Real-world relevance: these ideas underpin everyday decisions about carrying objects, pushing on boxes, levers, and any situation with multiple forces acting on an object, helping us predict motion or equilibrium.
Free-body diagrams (FBD) and force components
Free-body diagram essentials: an FBD is a visual tool to analyze forces affecting a single object.
Isolate the object: Mentally (or physically) separate the object of interest (e.g., a box) from its surroundings.
Represent the object: Draw a simple dot or a small box to represent your isolated object.
Identify and draw all external forces: Draw arrows originating from the object, representing all forces acting ON it. These include:
Gravity (Weight): Always vertically downward, W = mg.
Normal Force (N): Exerted by a surface, always perpendicular to the surface and pushing away from it.
Friction: Acts parallel to the surface, opposing motion or tendency of motion.
Applied Forces: Any pushes or pulls (e.g., a hand pushing a box).
Tension: Force transmitted through a string, rope, cable, etc., always pulling away from the object.
Air Resistance/Drag: Acts opposite to the direction of motion through air or fluid.
Label forces: Label each arrow with its type (e.g., Fg, N, F{app}, F_f) and, if known, its magnitude.
Choose a coordinate system: Define positive x and y directions, often aligning one axis with the direction of expected motion.
Balance of forces and net force:
To find the net force, sum all forces (or their components) in each independent direction (commonly x and y).
This involves resolving any forces not aligned with your chosen axes into their perpendicular components.
If a motion is constrained to a straight line, you may align your x-axis with that line and sum forces along x (and y separately if needed).
Newton’s third law reminder:
Action-reaction pairs act on different bodies; e.g., the Earth pulls downward on the box with weight, and the box pulls upward on the Earth with an equal and opposite force. The Earth’s vast mass (\approx 5.972 \times 10^{24}\ \mathrm{kg}) makes its acceleration due to the box negligible.
Practical tip from the lecture:
If you want the box to be supported by a surface, that surface must exert an upward normal force equal and opposite to the downward forces on the box, ensuring no acceleration perpendicular to the surface.
Example set-up discussed:
A box on a horizontal surface with horizontal forces: the net horizontal force is the vector sum of the horizontal force components, taking signs into account (e.g., F_x = -5\ \mathrm{N} + 2.5\ \mathrm{N}). If forces are collinear (along the same line), you can add them directly with the appropriate signs.
Key equations to memorize
Newton’s second law in vector form:
\mathbf{F}_{\mathrm{net}} = m\mathbf{a}
\mathbf{a} = \frac{\mathbf{F}_{\mathrm{net}}}{m}\n- Component form for a two-dimensional problem, by resolving forces along perpendicular axes:
\sum F_x = m a_x, \quad \sum F_y = m a_y
Weight and normal on a flat surface:
Weight: W = m g
Normal force (when no other perpendicular forces and no motion through the surface): N = W = m g (for a flat horizontal surface without other vertical accelerations or vertical applied forces).
On an inclined plane, the normal force is perpendicular to the plane; the weight has components along and perpendicular to the plane.
Inclined plane decompositions (for the weight vector):
Let the plane be inclined at angle \theta to the horizontal. Then the weight vector \mathbf{W} = m\mathbf{g} can be decomposed into components:
Parallel to plane: Acts down the plane, W_{\parallel} = m g \sin\theta
Perpendicular to plane: Acts into the plane, W_{\perp} = m g \cos\theta
The normal force (N) balances the perpendicular component of weight when there is no acceleration perpendicular to the plane (i.e., the object isn't lifting off or sinking into the surface): N = m g \cos\theta
Directional conventions and axis choice:
If you rotate axes to align the x-axis with the plane's slope and the y-axis perpendicular to it, you must recompute components of all forces in this rotated basis and then apply Newton's Second Law: \sum F_{\parallel} = m a_{\parallel}, \quad \sum F_{\perp} = m a_{\perp}. This often simplifies problems as one of the accelerations (a_{\perp}) is frequently zero.
Worked ideas and examples from the transcript
Hockey puck with two forces:
Setup mentions: a puck with two forces applied; you can compute the instantaneous acceleration by summing forces using vector addition and then applying \mathbf{F}_{\mathrm{net}} = m\mathbf{a}.
The step-by-step approach suggested:
Resolve forces into components: For each force \mathbf{F}i, find its x and y components. If a force \mathbf{F} makes an angle \phi with the x-axis, then Fx = F \cos\phi and F_y = F \sin\phi.
Sum components: Write down F_{1x}, F_{1y}, F_{2x}, F_{2y}, etc. Then sum them to get the net force components: \sum Fx = F_{1x} + F_{2x} + \dots and \sum Fy = F_{1y} + F_{2y} + \dots.
Calculate acceleration: Obtain the acceleration components: ax = (\sum Fx)/m and ay = (\sum Fy)/m.
Find magnitude and direction: If needed, the magnitude of the acceleration is a = \sqrt{ax^2 + ay^2} and its direction \theta = \arctan(ay/ax). (Be mindful of the quadrant for \theta).
Emphasis on early preparation and clean notation:
Start with a clean Free-Body Diagram (FBD) before crunching numbers. This visual representation helps organize thought and prevent omissions.
Be meticulous about parentheses and the meaning of each term; misplacing a parenthesis can dramatically change the result of an algebraic expression.
The instructor notes he will deduct points for sloppy notation to reinforce careful practice, highlighting the importance of clear communication in physics.
Airing out a common scenario on a table or wall:
The board and hand example illustrates action-reaction pairs in a tangible way: the hand exerts a force on the board, the board exerts an equal and opposite force on the hand; crucially, the forces act on different bodies.
When analyzing the system, identify all forces on the box (or object) and separate horizontal and vertical components for net force calculation.
Inclined planes in depth
Normal force concept:
The normal force (N) is always perpendicular (normal) to the surface interface. It is a contact force preventing one object from penetrating another.
On an inclined plane, N is not simply equal to the weight (mg); it specifically balances the component of weight perpendicular to the plane (mg\cos\theta) in the absence of other perpendicular accelerations or forces.
Weight vs. plane reaction:
Weight acts downward toward Earth’s center: its direction is fixed (vertically downward).
The plane exerts a force normal to its surface; if there is motion along the plane, friction (if present) acts parallel to the surface, opposing the motion.
Reorienting axes for incline problems:
It is often convenient to choose the x-axis along the plane (either up or down the incline) and the y-axis perpendicular to the plane. This simplifies Newton's Second Law equations because the acceleration is typically only along the x-axis (a_y = 0), and fewer forces need to be broken into components.
Practical takeaway:
Many introductory problems use the incline to illustrate how the normal force and weight interact, and to show how acceleration along the plane naturally depends on the component of gravity parallel to the plane (mg\sin\theta).
Notes on precision, teaching style, and assessment
The instructor emphasizes:
Starting problems early to avoid messy derivations and allow time for careful thought and error checking.
Being a “stickler for detail” about notation, including careful use of parentheses and consistent sign conventions, as these ensure mathematical accuracy and prevent misinterpretation.
The PowerPoint/Canvas content will cover the material discussed and will be posted for reference, supporting further self-study.
If mistakes occur in real-time (e.g., saying something incorrectly), students should point it out immediately to prevent compounding errors and foster an active learning environment.
Practical implications for study:
Build fluency with: drawing accurate FBDs, choosing convenient axes (especially for inclined planes), decomposing forces into components, and applying \sum F_i = m a_i in each direction independently.
Develop intuition for how different contact surfaces (flat vs. incline) change the relative magnitudes and directions of the normal force and weight, and how these affect motion.
Practice with both static cases (where net force is zero and acceleration is zero) and dynamic cases (where there is a nonzero net force and acceleration).
Quick reference reminders
Gravitational acceleration on Earth: g \approx 9.8\ \mathrm{m/s^2} \; (\text{often rounded to } 10\ \mathrm{m/s^2})
Key vector equation: \mathbf{F}_{\mathrm{net}} = m\mathbf{a}, \quad \mathbf{a} = \frac{\mathbf{F}_{\mathrm{net}}}{m} = \frac{1}{m}\sum_i \mathbf{F}_i
Weight vs. normal:
Weight: W = m g
On a flat horizontal surface (no other vertical forces, no vertical acceleration): N = m g
On a plane: W_{\parallel} = m g \sin\theta, \quad W_{\perp} = m g \cos\theta, \quad N = m g \cos\theta (when no other perpendicular forces and no perpendicular acceleration).
On a horizontal surface with horizontal forces:
Sum components along chosen x and y axes to find net force and total acceleration vector.
Connection to foundational principles
The material connects to Newton’s laws, vector decomposition, and the importance of a proper frame of reference. It is the cornerstone of classical mechanics.
It reinforces how choosing suitable axes (horizontal/vertical vs. along/normal to a surface) can dramatically simplify computations and lead to more elegant solutions.
It ties in real-world intuition (objects on tables, walls, and inclined planes) to the abstract equations, helping bridge conceptual understanding with practical problem-solving skills.
Ethical, philosophical, or practical implications
Accurate notation is not merely pedantic; it is fundamental to scientific integrity. It prevents misinterpretation of physical situations, ensures replicability of results, and avoids critical errors in problem-solving and engineering design.
Clear communication of ideas mirrors scientific practice, where precise language and symbols convey correct meaning across diverse contexts and audiences, fostering collaboration and knowledge transfer.
The discussion of action-reaction pairs highlights the interconnectedness of physical systems and the importance of considering all bodies involved, even when one (e.g., the Earth) is effectively immovable on human timescales, leading to a more complete understanding of interactions