Welcome to Gravitational Field Strength!
Hi there! This chapter moves us from simple mechanics (like forces and motion) into the fascinating world of fields. Don't worry if this feels like a big step—we're simply formalizing the idea of gravity that you already understand.
The goal here is to stop thinking about gravity as just two objects pulling on each other, and start thinking about how a massive object (like the Earth) changes the space around it. This 'changed space' is what we call the Gravitational Field.
Understanding field strength is crucial because it connects Newton's Laws with how we measure gravity on Earth (the familiar $g=9.81$ m s\(^{-2}\)) and how satellites behave far out in space.
Review: Gravity as a Universal Force
To define gravitational field strength, we first need to quickly recall the force that creates the field: Newton's Law of Universal Gravitation (Section 3.7.1).
This law states that every particle of mass in the universe attracts every other particle with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between their centres.
$$F = \frac{G M m}{r^2}$$
- \(F\) is the magnitude of the gravitational force (N).
- \(M\) and \(m\) are the masses of the two objects (kg).
- \(r\) is the distance between the centres of the two masses (m).
- \(G\) is the Universal Gravitational Constant (\(6.67 \times 10^{-11} \text{ N m}^2 \text{ kg}^{-2}\)).
The force described here is always attractive.
1. The Concept of a Force Field (Section 3.7.2)
What exactly is a field?
A force field is defined as a region in which a body experiences a force. If you put a mass into a gravitational field, it feels a gravitational force (weight).
Analogy: Imagine someone wearing very strong perfume (the source mass, M). The gravitational field is like the area where you can smell the perfume. The strength of the smell (the field strength, $g$) tells you how much force you would feel if you were a small mass ($m$) standing there.
Key Term: Gravitational Field Strength (\(g\))
The gravitational field strength, $g$, at a specific point is defined as the gravitational force exerted per unit mass on a small test mass placed at that point.
$$g = \frac{F}{m}$$
- \(F\) is the force acting on the test mass (N).
- \(m\) is the small test mass (kg).
The units of $g$ are Newtons per kilogram (N kg\(^{-1}\)).
Did you know? This definition shows why field strength is exactly the same concept as acceleration due to gravity (which has units m s\(^{-2}\)). Since $F=ma$, if $a=g$, then $F/m = g$. So, 1 N kg\(^{-1}\) is equivalent to 1 m s\(^{-2}\)!
Quick Review Box
The field strength $g$ is a vector quantity. Its direction is always the direction in which a mass would be accelerated (i.e., towards the centre of the source mass).
2. Representing the Field: Gravitational Field Lines
We use field lines to visually represent the field strength and direction.
Rules for Drawing Gravitational Field Lines:
- The lines show the direction of the force on a test mass (always pointing inwards towards the source mass).
- The density of the lines (how close they are together) represents the magnitude (strength) of the field. Where lines are close, $g$ is strong; where lines are far apart, $g$ is weak.
a) Radial Fields (Point or Spherical Masses)
For large objects like the Earth, or for point masses, the field lines radiate inwards, meeting at the centre. This is a radial field.
- The lines get further apart as you move away from the centre. This visually shows that $g$ decreases with distance.
- The Earth's field is considered radial outside its atmosphere.
b) Uniform Fields (Near the Surface)
If you zoom in very close to the surface of a planet (like standing on Earth), the field lines appear parallel and equally spaced.
- This represents a uniform field, where the field strength $g$ is constant in magnitude and direction.
- This is why we usually take $g \approx 9.81 \text{ N kg}^{-1}$ when doing mechanics problems on Earth, ignoring small changes in height.
3. Calculating Field Strength in a Radial Field
This is the most important quantitative section. We combine the definition of $g$ with Newton's Law of Gravitation.
We start with the definition: $$g = \frac{F}{m}$$
We substitute the formula for the gravitational force $F$ (where $M$ is the source mass and $m$ is the test mass): $$F = \frac{G M m}{r^2}$$
Substituting $F$ into the expression for $g$:
$$g = \frac{(G M m / r^2)}{m}$$
The test mass \(m\) cancels out! This gives us the equation for gravitational field strength in a radial field:
$$g = \frac{G M}{r^2}$$
STOP! This is incredibly important.
- \(G\) is the gravitational constant.
- \(M\) is the mass of the source object (e.g., the planet or star) creating the field.
- \(r\) is the distance from the centre of the source object to the point where $g$ is being measured.
The value of $g$ at any point depends only on the mass of the object creating the field (\(M\)) and the distance from its centre (\(r\)).
Step-by-Step Application Example: Calculating $g$ on Mars
Imagine you need to find the gravitational field strength ($g$) on the surface of Mars.
- Identify the Source Mass ($M$): This is the mass of Mars.
- Identify the Distance ($r$): Since we are calculating $g$ on the surface, $r$ is equal to the radius of Mars.
- Use the Equation: Plug in the values for $G$, $M_{Mars}$, and $R_{Mars}$ into \(g = G M / r^2\).
If you were asked to find $g$ at a point 1000 km above Mars' surface, $r$ would be (Radius of Mars + 1000 km). Be careful with units!
4. The Inverse Square Law for Field Strength
The formula $g = \frac{G M}{r^2}$ shows that $g$ is proportional to $1/r^2$. This is called the Inverse Square Law.
What does this mean for $g$ as we move away from Earth?
- If you double the distance ($2r$) from the Earth's centre, the field strength becomes $1/(2)^2 = 1/4$ times weaker.
- If you triple the distance ($3r$) from the Earth's centre, the field strength becomes $1/(3)^2 = 1/9$ times weaker.
Common Mistake to Avoid:
When calculating $r$, remember it is the distance from the centre of the mass. If a question gives you the height above the ground ($h$), you must calculate $r$ by adding the radius of the planet ($R$): $r = R + h$.
If you are standing on Earth, you are already one Earth radius ($R$) away from the centre!
Key Takeaway Summary
What you need to know about Gravitational Field Strength ($g$):
- Definition: Force per unit mass, $g = F/m$. (Units: N kg\(^{-1}\)).
- Nature: It is a vector field, always directed towards the source mass.
- Radial Field Formula: $g = \frac{G M}{r^2}$.
- Key Relationship: $g$ obeys the Inverse Square Law ($g \propto 1/r^2$).
- Representation: Field lines show the direction (inwards) and magnitude (density) of $g$.
Great job! You now understand how mass fundamentally alters the space around it to create a gravitational field.