Newton’s Law and Gravitational field strength

1. Gravity as a universal attractive force:

• Gravity is indeed a fundamental force of nature, shaping the behavior of objects on Earth and in the universe.
• Gravity:
  – Is a universal attractive force between all matter
  – Acts between objects with mass (or energy)
  – Causes objects to fall towards each other
  – Strength depends on mass and distance
  – Gravity is a long-range force, acting over vast distances
  – It’s a weak force compared to other fundamental forces (e.g., electromagnetism, strong and weak nuclear forces)
  – Gravity warps spacetime, causing curvature and bending of light
• Gravity’s effects:
  – Keeps planets in orbit around stars
  – Holds galaxies together
  – Causes tides on Earth
  – Influences time and space (gravitational time dilation and length contraction)
• Theories:
  – Newton’s Law of Universal Gravitation (1687)
  – Einstein’s General Relativity (1915)
• Gravity remains an active area of research, with ongoing studies in:
  – Gravitational waves
  – Dark matter and dark energy
  – Black holes
  – Cosmology
  – Quantum gravity

2. Newton’s Law of Gravity:

• Newton’s Law of Gravity (1687) states:
• “Every point mass attracts every other point mass by a force acting along the line intersecting both points. The force is proportional to the product of the two masses and inversely proportional to the square of the distance between them.”
• Mathematically:

  
  Figure 1 In each of these examples Newton’s law can be used to calculate the gravitational force of attraction that the objects exert on each other.

• [math] F \propto m_1 * m_2 \qquad (1) \\
  F \propto \frac{1}{r^2} \qquad (2)  [/math]
• Comparing equation 1 & 2
• [math] F = G \frac{m_1 * m_2}{r^2} \qquad (3) [/math]
• Where:
  – F is the gravitational force
  – G is the gravitational constant [math] (6.67 * 10^{-11} Nm^2kg^{-2}) [/math]
  –  and  are the masses of the objects
  – r is the distance between the centers of the objects
• Newton’s Law of Gravity:
  – Describes the gravitational force between two objects
  – Predicts the motion of planets, moons, and comets
  – Explains tides and the behavior of objects on Earth
  – Laid the foundation for classical mechanics and celestial mechanics

Figure 2 Gravitational force apply on moon

• Limitations:
  – Doesn’t account for relativity and very high speeds
  – Doesn’t explain gravity’s underlying mechanism
• Newton’s Law of Gravity was a major breakthrough, providing a fundamental understanding of gravity and its effects on the physical world. Later, Einstein’s General Relativity built upon and expanded our understanding of gravity.
• Although Newton law only applies to point masses, it can also be used to calculate the force of attraction between large spherical objects (such as planets and stan) because a sphere behaves as if all the mass were concentrated at its center.
• So, Newton’s law can be correctly used to calculate the force of attraction in each of the cases in Figure (1), the force between two-point mass.
• Newtons law cannot be used to calculate the force between two irregularly shaped objects, unless a complicated summation of the forces is made.

Example

Gravitational force:

• Calculate the gravitational force between the Sun, mass 2 * 10³⁰ kg and Halley’s comet, mass 3*10¹⁴kg, when separated by a distance of 5 * 10⁹

Solution:
Mass [math] m_1 = 2 * 10^{30} kg \\ m_2 = 3 * 10^{14}kg [/math]
distance between these masses [math] = d = 5 * 10^{9}km = 5 * 10^{12}m [/math]
Gravitational constant [math]= 6.67 * Nm^{2}kg^{-2}[/math]
Gravitational force = F=?
Formula:

[math] F = G \frac{(m_1 * m_2)}{r^2} \\
F = 6.67 * 10^{-11} \frac{(2 * 10^{30} * 3 * 10^{14})}{(5 * 10^{12})^2} \\
F = 6.67 * 10^{-11} \frac{(6 * 10^{44})}{25 * 10^{24}} \\
F = 1.6 * 10^9 \, \text{N} [/math]

3. Gravitational field lines:

• Gravitational field lines are a visual representation of the gravitational field around a mass, showing the direction and strength of the gravitational force. They:
  – Emanate from the center of the mass (source)
  – Radiate outward in all directions
  – Are closer together near the source (stronger field)
  – Are farther apart as distance increases (weaker field)
  – Can be used to visualize and predict gravitational forces
• Key properties:
  – Field lines never intersect (no two sources can have the same field line)
  – Field lines are continuous (no gaps or breaks)
  – Field lines can be used to calculate the gravitational force at any point
• Types of field lines:
  – Radial field lines (straight lines from the source)
  – Curved field lines (around spherical or irregular shapes)
• Gravitational field lines help us:
  – Understand the gravitational force and its direction.
  – Visualize the strength and distribution of the field.
  – Predict the motion of objects under gravity.
  – Analyze complex gravitational systems.
• Remember, field lines are a tool for visualization and calculation, not physical entities themselves!

• ⇒ Gravitational Field:

• A gravitational field is a mathematical representation of the gravitational force around a mass or a distribution of mass.
• It’s a vector field that describes the strength and direction of the gravitational force at any point in space.

• ⇒ Gravitational field line:

• A gravitational held is a region in which a massive object experiences a gravitational force.
• Any object with mass produces a gravitational field, but we usually use the term to describe the region of space around large celestial objects such as galaxies, stars, planets and moons.
• The gravitational field strength in a region of space is defined by
• [math] g = \frac{F}{m} \qquad (4) [/math]
• where g is the field strength measured in [math]Nkg^{-11} [/math] and F is the gravitational force in N acting on a mass m in kg.

• ⇒ Uniform field:

• Near the surface of a planet, the gravitational field is very nearly uniform, which means that the field is of the same strength and direction everywhere.
• Figure 3 illustrates a uniform field.

  
  Figure 3 uniform field line

• The field lines show the direction of the gravitational force on an object, and the spacing of the lines gives a measure of the strength of the field.
• You should remember that the spacing of the lines is chosen just for illustrative purposes another person might have represented the field strengths in these diagrams with a different separation of the field lines.

• ⇒ Radial field:

• Figure 4 shows the shape of a gravitational field near to Earth; this is a radial field.
• Here the field lines all point towards the center of the planet.

  
  Figure 4 gravitational radial field

• This is why we can use Newton’s law of gravity to calculate the gravitational forces between two planets.
• The field is exactly the same shape as it would have been if all of the mass of the planet were concentrated at its center C.
• The field lines at a distance of 2r from the center of the planet are further apart than they are at distance r, which is the planet’s surface.
• By Newton’s law of gravity, we know that equation 3
• [math] F = G \frac{(m_1 * m_2)}{r^2} \qquad (5) [/math]
• Also, if [math] m_2 [/math] is the mass of a small object close to the surface of the planet, we know that
• [math] g = \frac{F}{m_2} \\
  F = m_2 g \qquad (6) [/math]
• Comparing equation 5 & 6
• [math] m_2 g = G \frac{m_1 m_2}{r^2} \\
  g = G \frac{m_1}{r^2} [/math]
• Note that we often use a capital M to describe the mass of a large object such as a star or planet.
• So, using M as the mass of a planet, and r as the distance away from the center of the planes.
• we have
• [math]  g = G \frac{M}{r^2} [/math]
• For most planets, treating them as uniform spheres work as a good approximation.
• For most objects with a mass larger than 10¹¹kg, the forces of gravity overcome the massive forces of the rocks turn the planet into a sphere.
• However, smaller moons and minor planets can have irregular shapes, so Newton’s law of gravity cannot be used to simply predict fields near them, but it can be used accurately at large distances.

• ⇒ Examples:

• (1)
• A minor planet has a mass of , and it has a radius of 1200km. Calculate the gravitational field strength its surface.
• Solution:
• Mass of planet [math]= M = 2 * 10^{22}kg [/math]  
  Radius [math] = r = 1200km = 1200 * 1000 m = 1.2 * 10^{6}m [/math]
  Gravitational constant [math] = G =6.67 *10^{-11}Nm^{2}kg^{-2} [/math]
• Formula:
• [math] g = G \frac{M}{r^2} [/math]
• Put values
• [math] g = 6.67 * 10^{-11} \frac{(2 * 10^{22})}{(1.2 * 10^6)^2} \\
  g = \frac{13.34 * 10^{11}}{1.44 * 10^{12}} \\
  g = 9.3 * 10^{-1} \\
  g = 0.9 \, \text{N}  \text{kg}^{-1} \, \text{or} \, 0.9 \, \text{m}  \text{s}^{-2} [/math]
• (2)
• A star has a gravitational field strength at its surface of 300 N. Another star has the same mass but 10 times the radius of the first star. Calculate the gravitational field strength at the surface of the second star.Solution:Gravitational field strength [math] = g = 300 N.kg{-1}[/math]
• Formula:
• [math] g_1 = \frac{GM}{r_1^2} \qquad (7) \\
  g_2 = \frac{GM}{r_2^2} \qquad (8) \\
  \text{Dividing these equations:} \\
  \frac{g_2}{g_1} = \frac{GM}{r_2^2} * \frac{r_1^2}{GM} \\
  \frac{g_2}{g_1} = \left(\frac{r_1}{r_2}\right)^2 \\
  g_2 = g_1 \left(\frac{r_1}{r_2}\right)^2 \\
  g_2 = 300 * \left(\frac{1}{10}\right)^2 = 300 * \frac{1}{100} = 3 \, \text{N}  \text{kg}^{-1} \, \text{or} \, 3 \, \text{m}  \text{s}^{-2} [/math]
• (3)
• Show that the gravitational field strength near to the surface of a planet or star is given by [math] g = \frac{4}{3} \pi \rho Gr [/math], where  is the density of the body and r its radius.
• Solution:
• [math]  \text{Density} = \frac{\text{mass}}{\text{volume}} \\
  \text{Mass of planet} = \text{density} \times \text{volume} \\
  M = \frac{4}{3} \pi \rho r^3 \\
  g = \frac{GM}{r^2} \\
  \text{Substituting the value of } M: \\
  g = \frac{G * \frac{4}{3} \pi \rho r^3}{r^2} \\
  g = G * \frac{4}{3} \pi \rho r [/math]