Newton's Law of Universal Gravitation

Preliminaries

• Greetings.
• Attendance.
• Reminders / Announcements.

Assignment Instructions

• Use the GRESA Method to solve problems.
• Example problem: Fyang (52 kg) experiences a net force of 1678.22 N. Determine Fyang’s acceleration.

Recap Activity: This or That?

• Who developed the 3 laws of motion? Isaac Newton or Galileo Galilei (Answer: Isaac Newton)
• Which law states that acceleration is directly proportional to net force but inversely proportional to mass? 1st law or 2nd law (Answer: 2nd law)
• Which law states that a moving object will continue moving, and an object at rest will remain at rest unless acted upon by an unbalanced force? 1st law or 3rd law (Answer: 1st law)
• Which law states that for every action, there is an equal and opposite reaction? 2nd law or 3rd law (Answer: 3rd law)
• Wearing a seatbelt is an application of which law? 1st law or 2nd law (Answer: 1st law)

Lesson 15: Newton's Law of Universal Gravitation

• Objectives:
  • Explain Newton’s Law of Universal Gravitation.
  • Explain why objects near Earth's surface fall with identical acceleration in the absence of air resistance.
  • Solve word problems related to the universal law of gravitation.

Think and Rank Activity

• Arrange the following from least to greatest:
  • Inertia
  • Weight
  • Force exerted to another object

Force Definition

• Previously defined as the push or pull of objects.
• This definition is vague, as it only defines forces in touch with the object.
• Today's focus: Gravitational Force, a force that doesn't require direct contact.

Force as Interaction

• Force is the interaction between two bodies or a body and its environment.

Types of Forces

• Contact Forces:
  • Involve direct contact between two bodies (push or pull).
  • Examples: Normal Force, Friction, Tension.
• Non-Contact Forces:
  • Act even when bodies are separated by empty space.
  • Examples: Magnetic Force, Gravity.

Identify Contact Force (CF) or Non-contact Force (NF)

• Pushing a cart (CF)
• Moving a rock (CF)
• Falling rock (NF)
• Moon attracting the earth (NF)
• Kicking a ball (CF)

Gravity

• The force by which a planet attracts objects toward its center.
• Keeps planets in orbit around the Sun.
• Measures how fast objects accelerate towards each other.
• Average gravitational acceleration of Earth: $9.8 m/s^2$

Newton and the Apple

• Newton questioned why the apple fell straight down instead of sideways or upward.
• He realized a force pulls objects toward Earth—gravity.

Newtonian Synthesis

• Newton's intuition was a revolutionary break from the Ancient Greeks' notion of separate Terrestrial and Cosmic/Celestial Laws.
• Newtonian Synthesis is the union of these laws based on Newton's observations.

Newton’s Law of Universal Gravitation

• Isaac Newton revised his synthesis based on experiments and published Newton’s Law of Universal Gravitation.
• Every point mass attracts every other point mass in the universe with a force pointing in a straight line between their centers of mass.
  • This force is proportional to the masses of the objects and inversely proportional to the square of their separation.

Formula

• $F = G \frac{mM}{r^2}$
  • F = Force between the masses
  • G = Gravitational constant ($6.673 x 10^{-11} N (m/kg)^2$)
  • m = Mass of one object
  • M = Mass of the other object
  • r = Distance between the centers of the masses

Simplified Equation

• $F<em>g = G \frac{m</em>1 m_2}{r^2}$
  • $F_g$ = Gravitational force (N)
  • G = Gravitational constant ($6.674 x 10^{-11} N m^2/kg^2$)
  • m = Mass (kg)
  • r = Distance between two masses (m)

Problem 1

• Compute the gravitational force between the moon ($7.34 x 10^{22}$ kg) and the earth ($5.97 x 10^{24}$ kg) if their average separation is $3.83 x 10^8$ meters.
• Given:
  • $m_1 = 7.34 x 10^{22} kg$
  • $m_2 = 5.97 x 10^{24} kg$
  • r = $3.83 x 10^8$ m
  • G = $6.674 x 10^{-11} N m^2/kg^2$
• Required: $F_g$
• Equation: $F<em>g = G \frac{m</em>1 m_2}{r^2}$
• Solution:
  • $F_g = (6.674 x 10^{-11}) \frac{(7.34 x 10^{22})(5.97 x 10^{24})}{(3.83 x 10^8)^2}$
• Answer:
  • $F_g = 1.99 x 10^{20} N$

Problem 2

• Determine the gravitational force between the Earth ($m = 5.97 x 10^{24}$ kg) and a 70-kg physics student in an airplane at 40000 feet above Earth's surface (distance of $6.39 x 10^6$ m from Earth's center).
• Given:
  • $m_1 = 5.97 x 10^{24} kg$
  • $m_2 = 70 kg$
  • r = $6.39 x 10^6$ m
  • G = $6.674 x 10^{-11} N m^2/kg^2$
• Required: $F_g$
• Equation: $F<em>g = G \frac{m</em>1 m_2}{r^2}$
• Solution:
  • $F_g = (6.674 x 10^{-11}) \frac{(5.97 x 10^{24})(70)}{(6.39 x 10^6)^2}$
• Answer:
  • $F_g = 683.06 N$

Problem 3

• The mass of one small sphere in a Cavendish balance is 0.0100 kg, the nearest large sphere is 0.500 kg, and the center-to-center distance is 0.0500 m. Find the gravitational force on each sphere.
• Given:
  • $m_1 = 0.0100 kg$
  • $m_2 = 0.500 kg$
  • r = 0.0500 m
  • G = $6.674 x 10^{-11} N m^2/kg^2$
• Required: $F_g$
• Equation: $F<em>g = G \frac{m</em>1 m_2}{r^2}$
• Solution:
  • $F_g = (6.674 x 10^{-11}) \frac{(0.0100)(0.500)}{(0.0500)^2}$
• Answer:
  • $F_g = 1.33 x 10^{-10} N$

Newton’s Law of Universal Gravitation in Real-life

• Artificial satellites are useful in collecting data from Earth.
• Satellites are attracted to Earth and vice versa because of gravitational force.