e&m day two

Center of Mass and Gravity Concepts (based on transcript)

• Center of mass (COM): the point where the mass distribution of a body can be treated as if all its mass were concentrated.

• In motion analysis, we often pretend all mass is scrunched into one point (the COM) even though real bodies have distributed mass.

• This is a modeling simplification for applying motion laws.

• Notation: for a system of particles, the COM position is
  $\mathbf{R}<em>{\text{cm}} = \frac{1}{M} \sum</em>i m<em>i \mathbf{r}</em>i$
  where $M = \sum<em>i m</em>i.$

• For a continuous body, the COM is
  $\mathbf{R}_{\text{cm}} = \frac{1}{M} \int \mathbf{r}\,\rho(\mathbf{r})\, dV.$

• The COM concept helps when applying gravitational and inertial equations of motion.

• Point-mass approximation in physics problems:

• We often treat extended bodies as point masses located at their COM to simplify calculations.

• The transcript emphasizes that while this is a common and useful simplification, real bodies are not true point masses.

• Center of mass location in humans (transcript-specific claim):

• The speaker asks, roughly, where a person’s center of mass is.

• The rough answer given: it’s “up here” (around the torso). In practice, COM for an upright human is in the upper torso region and can shift with posture and limb position.

• Earth’s center of mass (transcript-specific claim):

• The transcript asks where Earth’s COM is and states it’s in the core.

• Note: For a perfectly symmetric, uniformly dense sphere, the COM lies at the geometric center. In reality, density variations exist but the COM remains near the center; the transcript cites “in the core” as a simplification.

• The idea of calculating Earth’s COM by summing contributions from chunks is mentioned as a more careful approach (but not carried out in the problem).

• Gravitational force and the inverse-square law:

• Gravitational force between two masses is given by
  $F<em>G = G \frac{m</em>1 m_2}{r^2}.$

• Distance dependence:

• If the distance doubles ($r \to 2r$), then
  $F<em>G' = \frac{F</em>G}{4}.$

• If the distance is halved ($r \to r/2$), then
  $F<em>G' = 4F</em>G.$

• This inverse-square behavior means gravity weakens with distance rapidly.

• Units and dimensional analysis:

• Every physics quantity has units; check consistency with $G$.

• The gravitational constant has units
  $[G] = \mathrm{N\,m^2\,kg^{-2}}.$

• Newton’s second law relation for force is
  $1\ \mathrm{N} = 1\ \mathrm{kg\, m\, s^{-2}}.$

• Using the units of $G$, the units of $F<em>G = G m</em>1 m_2 / r^2$ simplify to Newtons, as expected:
  $\mathrm{N\,m^2\,kg^{-2}} \times (\mathrm{kg})^2 / (\mathrm{m})^2 = \mathrm{N}.$

• In the transcript, $G$ is given as
  $G \approx 6.71 \times 10^{-11}\ \mathrm{N\,m^2\,kg^{-2}}.$

• (Note: CODATA currently lists about $6.67430\times 10^{-11}\ \mathrm{N\,m^2\,kg^{-2}}$, but the transcript uses the value above.)

• Weight vs mass and unit conversions (transcript-focused workflow):

• Weight is the force Earth exerts on you: it’s a force with units of Newtons.

• Weight is related to mass via
  $W = m g,$
  where $g \approx 9.8\ \mathrm{m\,s^{-2}}.$

• Common practical units in the US use pounds (force): $1\ \text{pound-force (lbf)} \approx 4.44822\ \mathrm{N}.$

• The transcript suggests converting your weight in pounds to Newtons, then obtaining mass in kilograms via $m = W/g$.

• Step-by-step idea:

• Given weight in pounds, convert to Newtons: $W<em>{\text{N}} = W</em>{\mathrm{lbf}} \times 4.44822\,\mathrm{N/lbf}.$

• Convert to mass: $m = \frac{W_N}{g}.$

• Now you have your mass in kg to use in other formulas (e.g., $F_G$ with another mass).

• Practical exercise (transcript outline): computing mutual gravity between you and another person

• You will compare your weight (in pounds) to Newtons, then compute your mass in kilograms, and then use an approximate distance to compute gravitational force between two people.

• Required inputs and numbers discussed:

• Gravitational constant: $G \approx 6.71\times 10^{-11}\ \mathrm{N\,m^2\,kg^{-2}}.$

• Distance between the two bodies: $r$ (in meters).

• Masses: $m<em>1$ and $m</em>2$ (from the weight-to-mass conversion).

• Formula to compute the mutual force:
  $F<em>G = G \frac{m</em>1 m_2}{r^2}.$

• Discussion prompts:

• What other numbers do you need to know to compute the force? (Distance $r$, the two masses $m<em>1$ and $m</em>2$.)

• How would changing $r$ affect $F_G$ according to the inverse-square law? (See above.)

• Example calculation (illustrative values, aligned with transcript guidance):

• Suppose your weight is 150 lbf.

• Convert to Newtons: $W_N = 150 \times 4.44822 \approx 667.23\ \mathrm{N}.$

• Mass from weight: $m<em>1 = \frac{W</em>N}{g} \approx \frac{667.23}{9.8} \approx 68.1\ \mathrm{kg}.$

• Suppose another person has weight 160 lbf: $W<em>N' \approx 160 \times 4.44822 \approx 711.7\ \mathrm{N},\quad m</em>2 \approx \frac{711.7}{9.8} \approx 72.6\ \mathrm{kg}.$

• Take distance $r = 2\ \mathrm{m}.$

• Compute mutual gravity:
  $F_G = \frac{(6.71\times 10^{-11}) \times (68.1)(72.6)}{(2)^2} \approx \frac{6.71\times 10^{-11} \times 4943.}{4} \approx 8.3\times 10^{-8}\ \mathrm{N}.$

• Interpretation: The gravitational attraction between two typical humans at human-scale separations is extremely small (on the order of $10^{-7}$ to $10^{-8}\ \mathrm{N}$ for the given example).

• Quick reference: key equations to memorize

• Gravitational force: $F<em>G = G \frac{m</em>1 m_2}{r^2}$

• Weight: $W = m g$

• Mass from weight: $m = \frac{W}{g}$

• Newton unit: $1\ \mathrm{N} = 1\ \mathrm{kg\, m\, s^{-2}}$

• Gravitational constant unit: $[G] = \mathrm{N\,m^2\,kg^{-2}}$

• Convert pounds to Newtons: $1\ \mathrm{lbf} \approx 4.44822\ \mathrm{N}$

• Inverse-square law behavior: doubling $r$ => $F<em>G$ becomes $1/4$ of original; halving $r$ => $F</em>G$ becomes $4$ times original.

• Connections to foundational principles and real-world relevance

• The COM concept links mass distribution to motion and stability in engineering and biomechanics.

• The inverse-square law underpins many phenomena beyond gravity (e.g., electric force, light intensity falloff with distance).

• Unit consistency and dimensional analysis are essential tools for checking physical equations and keeping calculations coherent across unit systems (SI vs. US customary).

• The practical scale of gravitational forces at human separations is minuscule compared to other forces (friction, contact forces), highlighting why gravity is often neglected in everyday interactions but is crucial in astrophysical contexts.

• Ethical, philosophical, and practical implications touched upon in the transcript

• The speaker uses approximations (COM treated as a point, Earth’s COM location) to simplify complex systems; this reflects a broader scientific practice: balance simplicity with accuracy.

• Understanding unit systems and conversions helps prevent errors in interdisciplinary work (engineering, physics, biology).

• End note (summary of takeaway):

• You can treat extended bodies as if their mass is concentrated at their COM for motion analysis, but be aware of the distributed nature of mass in real objects.

• The gravitational force between two objects follows an inverse-square law with distance, governed by the constant $G$ whose units ensure that the resulting force is in Newtons.

• Weight is the gravitational force on a mass; converting pounds to Newtons and then to kilograms via $m = W/g$ lets you use Newtonian gravity formulas consistently.