Chapter 5 Lectures: Weight or Gravitational Force Earth Exerts on a Mass

Mass Versus Weight and Inertial Frameworks

• Inertial Mass:

  • Inertial mass ($m$) is a scalar quantity representing an object's intrinsic resistance to being accelerated.

  • Defined within Newton's physical framework as the ratio of the net total force ($\sum F$) exerted on an object to its resulting acceleration ($a$):     $m = \frac{\sum F}{a}$

  • Intrinsic Property: Mass is an inherent property of matter that remains constant regardless of the environment, location, or gravitational field in which the object is placed.

• Weight:

  • Weight ($W$) is a force vector defined specifically as the gravitational force exerted by a central astronomical body (such as Earth) on an object with mass.

  • In non-SI unit systems (such as the United States system), weight is commonly measured in pounds ($\text{lbs}$), which is a unit of force.

  • General Equation for Weight:     $W = m \cdot g$

  • Key Distinction: Mass ($m$) is independent of location across the universe, whereas weight ($W$) varies dynamically according to the local acceleration due to gravity ($g$).

  • Common Misconception: The equations $F = m \cdot a$ and $W = m \cdot g$ are frequently confused; while structurally similar, $W = m \cdot g$ specifically represents the gravitational force exerted by a celestial body on a given mass.

Fundamental Forces of Nature and Charge Behaviors

• The Four Fundamental Forces:

  • All observed forces derive directly or indirectly from four fundamental interactions in nature:

  1. Gravitational Force: An attractive force existing between any two masses.

  2. Electric (Electromagnetic) Force: An interaction between electric charges that can be either attractive or repulsive. Microscopic contact forces such as the normal force and friction derive directly from electric interactions.

  3. Electroweak Force: Associated with subatomic interactions within atomic nuclei.

  4. Strong Force: Binds protons and neutrons together via quarks inside atomic nuclei.

• Charge Characteristics Across Force Types:

  • Gravitational Charge: Mass serves as the single type of gravitational charge. Gravitational forces are exclusively attractive; masses never repel each other. This unique property allows celestial structures (stars, planets, solar systems) to aggregate and remain stable.

  • Electric Charge: Possesses two charge types: positive and negative. Like charges repel one another, whereas opposite charges attract. A basic force is defined as a push or a pull.

  • Strong Force Charge: Governed by eight distinct charge types (color charges) operating in complex quantum dynamics.

Newton's Universal Law of Gravitation

• Universal Scope:

  • The law is assumed to apply universally across all regions of space throughout the cosmos.

  • Describes the mutual gravitational attraction between any two point masses $m_1$ and $m_2$ separated by a center-to-center distance $r_{12}$.

• Mathematical Formulation:   $F_g = \frac{G \cdot m_1 \cdot m_2}{r_{12}^2}$

  • $F_g$: Magnitude of the gravitational force in Newtons ($\text{N}$).

  • $G$: Universal gravitational constant ($G \approx 10^{-11}\,\text{N} \cdot \text{m}^2/\text{kg}^2$).

  • $m_1$: Mass of object 1 in kilograms ($\text{kg}$).

  • $m_2$: Mass of object 2 in kilograms ($\text{kg}$).

  • $r_{12}$: Distance between the center of mass of $m_1$ and the center of mass of $m_2$ in meters ($\text{m}$).

• Dimensional Analysis of $G$:

  • The SI units of $G$ are Newtons times meters squared per kilogram squared ($\text{N} \cdot \text{m}^2/\text{kg}^2$).

  • Multiplying by the mass product ($\text{kg}^2$) and dividing by distance squared ($\text{m}^2$) isolates force in Newtons ($\text{N}$):     $\frac{\text{N} \cdot \text{m}^2}{\text{kg}^2} \times \frac{\text{kg}^2}{\text{m}^2} = \text{N}$

Derivation of Surface Gravitational Acceleration ($g$)

• Point-Mass Assumption:

  • Gravitational force calculations treat massive spherical bodies (such as Earth) as point objects, assuming all mass ($M_e$) is concentrated precisely at the geometric center.

• Mathematical Derivation:

  • Equating weight $W$ to the universal gravitational force $F_g$ exerted on an object of mass $m$ located on Earth's surface:     $W = F_g = \frac{G \cdot M_e \cdot m}{R_e^2}$

  • Factoring out the object's mass $m$ yields local surface gravitational acceleration ($g$):     $g = \frac{G \cdot M_e}{R_e^2}$

• Quantitative Values for Earth:

  • Mass of Earth ($M_e$): Approximately $6 \times 10^{24}\,\text{kg}$ (or $5.98 \times 10^{24}\,\text{kg}$ to three significant figures).

  • Radius of Earth ($R_e$): Approximately $6,000\,\text{km}$ or $6.39 \times 10^6\,\text{m}$.

  • Calculated Acceleration Value:     $g = 9.8999\,\text{N/kg} \approx 9.80\,\text{m/s}^2$

• Unit Equivalence:

  • Since $1\,\text{N} = 1\,\text{kg} \cdot \text{m/s}^2$:     $\frac{\text{N}}{\text{kg}} = \frac{\text{kg} \cdot \text{m/s}^2}{\text{kg}} = \text{m/s}^2$

Gravitational Acceleration and Weight Across Solar System Bodies

• General Planetary Formula:

  • Local acceleration due to gravity ($g_p$) on any planet or body $p$:     $g_p = \frac{G \cdot M_p}{R_p^2}$

  • Weight on planet $p$:     $W_p = m \cdot g_p$

• Surface Acceleration Across Solar System Bodies:

  • Mercury: $g_{\text{mercury}} = 3.59\,\text{m/s}^2$

  • Venus: Gravitational acceleration is very close to Earth's value, making objects feel slightly lighter on Venus's surface than on Earth.

  • Earth: $g_{\text{earth}} = 9.80\,\text{m/s}^2$

  • Moon:

  • Mass ($M_{\text{moon}}$) $\approx 10^{22}\,\text{kg}$

  • Radius ($R_{\text{moon}}$) $= 1,700,000\,\text{m}$ ($1.7 \times 10^6\,\text{m}$, compared to Earth's radius of $6.4 \times 10^6\,\text{m}$)

  • Calculated Acceleration ($g_{\text{moon}}$) $= 1.62\,\text{m/s}^2$

  • Pluto: $g_{\text{pluto}} = 0.42\,\text{m/s}^2$

Variables Affecting Local Gravitational Acceleration

• Altitude Dependencies:

  • Gravitational acceleration decreases inversely with the square of distance $r$ from the center of mass.

  • Moving to higher elevations (such as the top of a mountain or Mount Everest) increases the effective distance $r_{12}$, resulting in a slight decrease in both $g$ and measured weight.

• Geometrical and Density Non-Uniformities:

  • Earth is not a perfect sphere; it exhibits an elliptical/oblate shape.

  • Material density throughout Earth's interior is non-uniform, shifting the effective center of mass.

  • High-precision measurements (to 4 or 5 significant figures) show local variations in $g$ across geographic coordinates (e.g., Rancho Cucamonga).

  • The standard value $g = 9.80\,\text{m/s}^2$ serves as an accurate global baseline at 3 significant figures.

Questions and Discussion

• Behavior of Opposite Electric Charges:

  • Prompt: What happens when a positive charge and a negative charge are brought close together?

  • Response: They attract each other.

• Behavior of Like Positive Charges:

  • Prompt: What happens when two positive charges are placed near each other?

  • Response: They repel each other.

• Behavior of Like Negative Charges:

  • Prompt: What happens when two negative charges are placed near each other?

  • Response: They repel each other.

Worked Example: Gravitational Weight of a Bowling Ball

• Problem Statement:

  • Given a bowling ball with mass $m = 5.5\,\text{kg}$, calculate its weight on Earth ($g_{\text{earth}} = 9.8\,\text{m/s}^2$) in Newtons and convert to pounds. Calculate its weight on Pluto ($g_{\text{pluto}} = 0.42\,\text{m/s}^2$) in Newtons and pounds.

  • Conversion Factor: $1\,\text{N} = 0.225\,\text{lbs}$

• Calculation for Earth:

  • Weight on Earth in Newtons:     $W_{\text{earth}} = m \cdot g_{\text{earth}} = 5.5\,\text{kg} \times 9.8\,\text{m/s}^2 = 53.9\,\text{N}$

  • Weight Conversion to Pounds:     $53.9\,\text{N} \times \frac{0.225\,\text{lbs}}{1\,\text{N}} = 12.1275\,\text{lbs} \approx 12\,\text{lbs}$

• Calculation for Pluto:

  • Weight on Pluto in Newtons:     $W_{\text{pluto}} = m \cdot g_{\text{pluto}} = 5.5\,\text{kg} \times 0.42\,\text{m/s}^2 = 2.31\,\text{N}$

  • Weight Conversion to Pounds:     $2.31\,\text{N} \times \frac{0.225\,\text{lbs}}{1\,\text{N}} = 0.51975\,\text{lbs} \approx 0.52\,\text{lbs}$