• One Newton is defined as the force required to accelerate a mass of one kilogram at a rate of one meter per second squared.
  
• In simpler terms, it’s how much push or pull it takes to change the speed of a 1 kg object by 1 meter per second, every second.
  

• This definition leads to the expression: 1 N = 1 kg⋅m/s². This shows how the Newton is derived from the base SI units of kilogram (kg), meter (m), and second (s).
  

• The Newton is named after Sir Isaac Newton, whose laws of motion are fundamental to understanding force.
  
• Specifically, it’s closely tied to Newton’s second law, which states that force (F) equals mass (m) times acceleration (a): F = ma.
  

• To get a sense of scale, the force of gravity on a small apple is roughly equal to 1 Newton.
  
• Also, understanding that gravity on earth, gives a 1 kilogram mass a force of about 9.8 Newtons.
  

In essence, the Newton provides a standardized way to measure and quantify forces, allowing for consistent and accurate calculations in physics and engineering.

• \(\textrm{G}\) is a fundamental physical constant that appears in Newton’s law of universal gravitation. This law describes the attractive force between any two objects with mass.
  
• The formula is: \[F = \textrm{G}\left ( \frac{m_1m_2}{r^2} \right )\]
  
• Where:
  
  • \(F\) is the gravitational force
    
  • \(m_1, m_2\) are the masses of two objects
    
  • \(r\) is the distance between the centers of two objects
    
  • \(\textrm{G}\) is the gravitational constant
    

• To find the units of \(\textrm{G}\) we can rearrange the formula: \[\textrm{G} = \frac{F\cdot r^2}{m_1m_2}\]
  
• Now let’s plug in the SI units:
  
  • \(F\) (force) is in Newtons \((\textrm{N})\)
    
  • \(r\) (distance) is in meters \((\textrm{m})\)
    
  • \(m_1, m_2\) (masses) are in kilograms \((\textrm{kg})\)
    
• Therefore, the units of \(\textrm{G}\) are: \(\textrm{N⋅m}^2/\textrm{kg}^2\) (Newton-meters squared per kilogram squared).
  
• However we can also convert the Newtons into their base units, and therefore find the base SI units of \(\textrm{G}\).
  
• Since Newtons are \( \textrm{kg}\cdot \textrm{m}/\textrm{s}^2\), we can replace the Newtons within the \(\textrm{G} \) unit.
  
• \(\textrm{N = kg⋅m/s^2}\)
  
• Therefore the SI units of \(\textrm{G}\) are also \(\textrm{G}\)= \(\textrm{m}^3/(\textrm{kg}\cdot \textrm{s}^2)\)
  

• Dimensions describe the fundamental nature of a quantity in terms of mass (\(\mathsf{M}\)), length \((\mathsf{L})\), time \((\mathsf{T})\)
  
• To find the dimensions of \(\textrm{G}\),we use the units we just derived:
  
• Units to dimensions:
  
  • \(\textrm{N}\) (force) has dimensions \(\mathsf{MLT^{-2}}\)
    
  • \(\textrm{m}\) (distance) has dimensions \(\mathsf{L}\)
    
  • \(\textrm{kg}\) (mass): \(\mathsf{M}\)
    
• Substituting these into the equation \(\textrm{G} = F\cdot r^2/m_1m_2\) we get:
  
• \(\textrm{G} = (\mathsf{MLT}^{-2})(\mathsf{L}^2)/(\mathsf{M}^2)\)
  
• Simplifying we get: \( \textrm{G} = \mathsf{M}^{-1}\mathsf{L}^3\mathsf{T}^{-2}\)
  
• Therefore, dimensional formula of \(\textrm{G}\) is \(\mathsf{M^{-1}L^3T^{-2}}\)
  

• SI units of \(\textrm{G}\) are Newton-meters squared \( \textrm{N}\cdot \textrm{m}^2/\textrm{kg}^2\) or meters cubed per kilogram second squared \(\textrm{m}^3/(\textrm{kg}\cdot \textrm{s}^2)\)
  
• Dimensional formula of \(\textrm{G}: \mathsf{M^{-1}L^3T^{-2}}\)