Gravitational Potential Energy and Graphs

Introduction

This section explores how to determine the change in gravitational potential energy (\(\Delta E_g\)) using graphs of force vs. distance and gravitational field strength vs. distance. This is an alternative method to using the formula \(\Delta E_g = mg\Delta h\) which applies specifically to uniform fields.

Gravitational Potential Energy (\(E_g\)) Review

• Definition: The energy an object possesses due to its position in a gravitational field.
• Formula (Uniform Field): \(E_g = mgh\), where \(m\) is mass, \(g\) is gravitational acceleration, and \(h\) is height.
• Reference Point: The zero point for gravitational potential energy is arbitrary; often taken as the Earth’s surface or infinity. Changes in \(E_g\) are physically significant, not absolute values.

KEY TAKEAWAY: Gravitational potential energy depends on the object’s position within a gravitational field.

Force vs. Distance Graphs

Determining \(\Delta E_g\) from Force vs. Distance

• Concept: The area under a force vs. distance graph represents the work done (\(W\)) by the force. In the case of a gravitational force, this work done is equal to the negative change in gravitational potential energy: \(W = -\Delta E_g\).
• Calculation:
  1. Plot the graph of gravitational force (\(F_g\)) against distance (\(r\)).
  2. Calculate the area under the curve between the initial and final positions. This area represents the work done by gravity.
  3. \(\Delta E_g = -W = -\) (Area under the \(F_g\) vs. \(r\) curve).

Example Scenario

Imagine lifting a mass m from a distance \(r_1\) from the center of the Earth to a distance \(r_2\). The area under the force vs. distance curve from \(r_1\) to \(r_2\) will give the work done by gravity. The change in gravitational potential energy is the negative of this work.

EXAM TIP: Always pay attention to the sign. Work done by gravity reduces potential energy, while work done against gravity increases it.

Gravitational Field Strength vs. Distance Graphs

Gravitational Field Strength (\(g\))

• Definition: The gravitational force per unit mass at a given point in space. \(g = \frac{F_g}{m}\)
• Units: N/kg or m/s²
• Formula (General): \(g = \frac{GM}{r^2}\), where \(G\) is the gravitational constant, \(M\) is the mass of the attracting body (e.g., Earth), and \(r\) is the distance from the center of the attracting body.

Determining \(\Delta E_g\) from Field vs. Distance

• Concept: The area under a gravitational field strength vs. distance graph represents the change in gravitational potential energy per unit mass.
• Calculation:
  1. Plot the graph of gravitational field strength (\(g\)) against distance (\(r\)).
  2. Calculate the area under the curve between the initial and final positions. This area represents \(\frac{\Delta E_g}{m}\).
  3. \(\Delta E_g = m \times\) (Area under the \(g\) vs. \(r\) curve).

Mathematical Justification

Since \(g = \frac{F_g}{m}\), then \(F_g = mg\). The area under the F vs distance graph is the work done, \(W = \int F dr\). Substituting in for F, we get \(W = \int mg dr\). Since \(g\) is a function of \(r\), we can rewrite as \(W = m \int g dr\). The integral \(\int g dr\) is the area under the g vs distance graph, so \(W = m \times \text{Area}\). Since \(\Delta E_g = -W\), then \(\Delta E_g = -m \times \text{Area}\).
However, if the area is calculated as the work done against the field, then \(\Delta E_g = m \times \text{Area}\).

Example Scenario

Consider a spacecraft moving from a lower orbit to a higher orbit around a planet. The area under the gravitational field strength vs. distance graph, multiplied by the spacecraft’s mass, gives the change in its gravitational potential energy.

COMMON MISTAKE: Forgetting to multiply the area under the field vs. distance graph by the mass of the object. The area itself represents the change in potential energy per unit mass.

Comparing the Two Methods

STUDY HINT: Practice calculating the area under different types of curves (linear, curved) to prepare for various graph shapes in exam questions.

Non-Uniform Fields

The graphical method is particularly useful when dealing with non-uniform gravitational fields, where the gravitational force and field strength vary with distance. In such cases, simple formulas like \(E_g = mgh\) are not directly applicable.

Applications

• Satellite Motion: Calculating the energy required to change a satellite’s orbit.
• Space Exploration: Determining the energy needed for spacecraft to escape a planet’s gravitational field.
• Astrophysics: Analyzing the motion of celestial objects in varying gravitational fields.

VCAA FOCUS: VCAA often presents problems involving non-uniform fields where the graphical method is essential. Watch for questions that describe a varying gravitational force or field strength.

Examples

Example 1: A 2 kg mass is moved from a point where the gravitational field strength is 5 N/kg to a point where the gravitational field strength is 2 N/kg. The area under the g vs distance graph between these two points is 10 m^2. Calculate the change in gravitational potential energy.

\(\Delta E_g = m \times \text{Area} = 2 kg \times 10 J/kg = 20 J\)

Example 2: The gravitational force on a 1000 kg satellite changes as it moves further from a planet. The area under the force vs distance graph is -5 x 10^8 J. Calculate the change in gravitational potential energy.

\(\Delta E_g = -W = 5 \times 10^8 J\)

REMEMBER: Gravitational potential energy increases as an object moves further away from a gravitational source (e.g., Earth) and decreases as it moves closer.

Summary

• The area under a force vs. distance graph gives the negative change in gravitational potential energy.
• The area under a gravitational field strength vs. distance graph, multiplied by the mass, gives the change in gravitational potential energy.
• These graphical methods are useful for both uniform and non-uniform gravitational fields.

APPLICATION: Understanding gravitational potential energy changes is crucial for analyzing satellite orbits and space travel.