Deriving the Acceleration due to Gravity on any Planet and specifically Mt. Everest

Summary

This video explores the two key equations for the force of gravity: the planet-specific equation and Newton’s Universal Law of Gravitation. It then applies these equations to find the acceleration due to gravity near the surface of Earth, including the top of Mount Everest. The video highlights how the acceleration changes with altitude and discusses the mass of Earth, the radius of the planet, and altitude in calculating gravity. A fun comparison is made with high jump records, illustrating how gravity variations would affect athletic performance. It provides a deeper understanding of gravity's consistency and fluctuations.

Takeaways

• 😀 The force of gravity can be calculated using two equations: one specific to planets and another based on Newton’s Universal Law of Gravitation.
• 😀 The planet-specific equation for force of gravity is: force = mass × acceleration due to gravity.
• 😀 Newton’s Universal Law of Gravitation states that the force of gravity = (gravitational constant × mass 1 × mass 2) / r², where r is the distance between the objects' centers of mass.
• 😀 By equating both equations, the acceleration due to gravity at any location can be calculated.
• 😀 The mass of the object and the planet are represented as mass 1 and mass 2 in Newton’s Law of Gravitation.
• 😀 The distance r in the law is the sum of the planet's radius and the object's altitude above the surface.
• 😀 The term ‘altitude’ refers to the height of an object above the surface of the planet (or sea level).
• 😀 At the surface of Earth, the acceleration due to gravity is approximately 9.81 m/s², but it varies slightly based on location.
• 😀 The acceleration due to gravity at the top of Mount Everest is 9.77 m/s², slightly lower than at sea level.
• 😀 Even though the acceleration due to gravity is nearly constant on Earth, small variations occur based on factors like altitude.
• 😀 Javier Sotomayor, the high jump world record holder, would clear a higher height (2.46 meters) at the top of Mount Everest due to slightly weaker gravity, but practical challenges like oxygen and temperature would limit this increase.

Q & A

What are the two equations for the force of gravity mentioned in the video?

The two equations for the force of gravity are: 1) The planet-specific equation: force of gravity equals the mass of the object times the acceleration due to gravity. 2) Newton's Universal Law of Gravitation: force of gravity equals the universal gravitational constant times mass 1 times mass 2 all divided by r squared, where r is the distance between the centers of mass of the two objects.

How can we find the acceleration due to gravity at any location?

We can set the two equations for the force of gravity equal to each other to find the acceleration due to gravity at any location.

In Newton’s Universal Law of Gravitation, what do mass 1 and mass 2 represent?

Mass 1 is the mass of the object, and mass 2 is the mass of the planet.

What does 'r' represent in the Universal Law of Gravitation?

'r' represents the distance between the centers of mass of the two objects, which is the radius of the planet plus the altitude of the object above the surface.

What is the definition of altitude in the context of gravity?

Altitude is the height of an object above the surface of the planet or sea level.

What is the acceleration due to gravity on Earth near the surface?

The acceleration due to gravity near the surface of Earth is generally defined as 9.81 meters per second squared.

How does the acceleration due to gravity change at the top of Mount Everest?

The acceleration due to gravity at the top of Mount Everest is slightly lower at 9.77 meters per second squared, compared to the standard 9.81 m/s² on Earth's surface.

Why does the Earth have a varying radius, and how does this affect gravity?

The Earth is an oblate spheroid, meaning its radius is not constant. It has a larger equatorial radius than polar radius due to the planet's rotation and inertia. This affects gravity because the distance from the center of the Earth to an object changes depending on the location.

What is the significance of Javier Sotomayor’s world record in high jump in relation to gravity?

Javier Sotomayor's world record jump of 2.45 meters in Salamanca, Spain, was achieved at an acceleration due to gravity of 9.80 m/s². If he jumped at the top of Mount Everest, with a slightly lower gravity of 9.77 m/s², he would have been able to clear 2.46 meters, which is roughly 1 centimeter higher.

What are some challenges that Javier Sotomayor would face at the top of Mount Everest despite the slight increase in jump height?

Javier Sotomayor would face challenges such as the need for oxygen tanks, low temperatures (never above freezing), snow, and the lack of a suitable space to run and jump.