AP Physics 1 - Topic 2.6: Gravitational Force

AP Physics 1 - Topic 2.6: Gravitational Force

Introduction

• Video Series: AP Physics 1 Daily Video

• Speaker: Nefemi Kolayemi

• Location: Boston, Massachusetts

Warm-Up Question

• Scenario: Hiking to the top of Mount Everest

• Question: Would your weight be the same at the top of Mount Everest compared to home?

• Call for viewer reflection: Pause and consider before answering.

Key Concepts of Weight and Gravitational Force

• Gravitational Force:

  • Definition: The force of attraction between the centers of mass of two objects.

  • Example: Earth and an object on its surface.

• Field Force:

  • Nature: No contact is required for gravitational force to act on an object.

• Weight:

  • Definition: The magnitude of gravitational pull that an object experiences from a massive object (e.g., a planet or star).

  • Calculation: Weight is calculated as follows:

  • Formula: $W = mg$

  • Where:

    • $W$ = weight

    • $m$ = mass of the object

    • $g$ = gravitational acceleration (standard value on Earth is approximately $9.81 ext{ m/s}^2$).

New Equation for Gravitational Force

• Universal Law of Gravitation:

  • Formula: $F<em>g = rac{G m</em>1 m_2}{r^2}$

  • Where:

    • $F_g$ = gravitational force between two objects

    • $G$ = universal gravitational constant

    • $m<em>1, m</em>2$ = masses of the two objects

    • $r$ = distance between the centers of mass of the two objects.

  • Important Note:

  • The variable $r$ should not always be interpreted as radius; it is the distance between center of mass of two objects.

Understanding Gravitational Acceleration ($g$)

• Previous Understanding:

  • Defined as the acceleration experienced by an object in free fall near Earth’s surface ($g ext{ approximately } 9.81 ext{ m/s}^2$).

• Distinction:

  • Although $g$ indicates acceleration in free fall, it also represents the strength of the gravitational field.

  • When an object with mass $m$ is placed in a gravitational field with strength $g$:

  • Gravitational force: $F = mg$

Factors Determining the Strength of $g$

• Variations on Different Planets:

  • $g$ varies across different planets due to factors such as mass and radius of the planet.

• Derived Equation for Gravitational Field Strength:

  • Using the gravitational force equation, $g$ can be expressed as:

  • Formula:
    $g = rac{G M}{r^2}$

  • Where:

    • $M$ = mass of the planet

    • $r$ = distance from the center of the planet (radius).

Example: Comparing Gravitational Field Strength on Earth and Mars

• Necessary Information:

  • Radius of each planet

  • Earth: $r_{E}$

  • Mars: $r_{M}$

  • Mass of each planet

  • Earth: $M_{E}$

  • Mars: $M_{M}$

• Ratios for Comparison:

  • $r<em>{M} ext{ is approximately } 0.532$ times $r</em>{E}$ (half of Earth's radius).

  • $M<em>{M} ext{ is approximately } 0.107$ times $M</em>{E}$ (10% of Earth's mass).

• Factor of Change Method:

  • Establish a base of 1 for unchanged variables and plug in corresponding change factors.

  • Calculation Example:

  • $ext{Factor of Change} = rac{0.107}{(0.532)^2}$

  • Calculation yields:

    • Factor of Change: $0.378$

• Interpretation:

  • The gravitational field on Mars is roughly 37.8 ext{%} of the gravitational field on Earth.

  • To find $g$ on Mars:

  • $g<em>{M} = 0.378 imes 9.81 ext{ m/s}^2 ightarrow g</em>{M} ext{ is approximately } 3.71 ext{ m/s}^2$

Practice Question

• Original Question Recap:

  • Considering gravitational force at the top of Mount Everest versus sea level.

• Key Consideration:

  • While mass remains constant, the distance from Earth's center of mass changes when at a higher elevation (e.g., Mount Everest at $9,000$ meters).

• Factor of Change Calculation:

  • Distance at Everest relative to sea level: $1.001$ (distance from Earth's center).

  • Compute Factor of Change:

  • $ext{Factor} = rac{1}{(1.001)^2}$ yielding: $0.998$

• Conclusion:

  • Gravitational field strength at Mount Everest is around 99.8 ext{%} of that at sea level.

  • Resulting value: $g ext{ at Everest } ext{ is approximately } 9.79 ext{ m/s}^2$

• Final Answer:

  • Your weight at Everest is virtually the same as at home.

Key Takeaways

• Gravitational Force:

  • The attraction between the centers of mass of any two objects.

• Weight:

  • Gravitational pull experienced from a large mass such as a planet or star.

• Understanding of g:

  • Beyond being acceleration due to gravity, it also signifies the strength of a gravitational field.

  • Consistent near planetary surfaces, but subject to change with altitude and planetary characteristics.