Unit 5.2 – Acceleration Due to Gravity

Key Points About Acceleration Due to Gravity

• Definition:
  • $𝑔$ is the acceleration caused by the gravitational pull of the Earth on an object.
  • It depends on the mass and radius of the Earth.
• Formula for $𝑔$: The acceleration due to gravity can be calculated using the formula:
  $$𝑔=𝐺\cdot \frac{𝑀}{{𝑅}^2}$$
  Where:
  • $𝐺$: Universal gravitational constant ($6.674\times {10}^{-11}{\text{N.m}}^2\mathrm{/}{\text{kg}}^2$)
  • $𝑀$: Mass of the Earth ($5.972\times {10}^{24}\text{kg}$)
  • $𝑅$: Radius of the Earth ($6.371\times {10}^6\text{m}$)
• Dependence on Altitude:
  • As altitude increases, the distance from the Earth's center ($𝑅$) increases, causing $𝑔$ to decrease.
• Direction of $𝑔$:
  • The acceleration due to gravity always acts toward the center of the Earth.
  • g is inversely proportional to the square of the radius.
  • As the distance from Earth’s center increases, gravity decreases rapidly.
  • Therefore, larger radius → smaller value of 𝑔.

Detailed Notes with Bullets

1. Dependence on Mass and Radius of Earth

• Mass of Earth ($𝑀$):
  • A more massive planet would exert a stronger gravitational pull, increasing $𝑔$.
• Radius of Earth ($𝑅$):
  • A larger radius decreases $𝑔$, as the force weakens with distance.

2. Variation of $𝑔$ with Altitude

• At higher altitudes, the value of $𝑔$ decreases because the distance from the Earth's center increases.
• Formula for $𝑔$ at altitude $ℎ$:
  $${𝑔}_{ℎ}=𝐺\cdot \frac{𝑀}{(𝑅+ℎ{)}^2}$$
  Where $ℎ$ is the height above the Earth's surface.

3. Comparison of $𝑔$ on Different Planets

• The value of $𝑔$ varies across planets depending on their mass and radius.
• Example:
  • On the Moon, $𝑔=1.6{\text{m/s}}^2$ (about $\frac{1}{6}$ of Earth's $𝑔$).
  • On Jupiter, $𝑔=24.8{\text{m/s}}^2$ (about 2.5 times Earth's $𝑔$).

4. Applications of $𝑔$

• Free Fall: Objects in free fall accelerate at $9.8{\text{m/s}}^2$ near Earth's surface.
• Weight Calculation: Weight is calculated as:
  $$𝑊=𝑚\cdot 𝑔$$
  Where $𝑚$ is the mass of the object.

Quick Review, Exam Tips, Tricks & Traps

Key Points to Remember

• $𝑔$ is the acceleration caused by Earth's gravity and has a standard value of $9.8{\text{m/s}}^2$ near the surface.
• $𝑔$ decreases with altitude because the distance from Earth's center increases.
• The direction of $𝑔$ is always toward the center of the Earth.

Exam Tips

1. Use the formula $𝑔=𝐺\cdot \frac{𝑀}{{𝑅}^2}$ when calculating $𝑔$ for different planets or celestial bodies.
2. For questions involving altitude, use ${𝑔}_{ℎ}=𝐺\cdot \frac{𝑀}{(𝑅+ℎ{)}^2}$.
3. Remember that $𝑔$ is independent of the mass of the falling object.

Common Traps

1. Students often confuse $𝑔$ with weight. Remember, $𝑔$ is acceleration, while weight depends on both $𝑔$ and mass.
2. Misinterpreting the effect of altitude: $𝑔$ decreases as altitude increases, not the other way around.

Tricks for Competitive Exams

1. Look for keywords like "altitude," "planet," or "free fall" to identify variables affecting $𝑔$.
2. In MCQs, eliminate options where $𝑔$ increases with altitude—it’s impossible unless the planet’s mass changes.
3. Use proportional reasoning:
   • If the radius doubles, $𝑔$ becomes $\frac{1}{4}$.
   • If the mass doubles, $𝑔$ doubles.

Quick Recall Table

Additional Content: Real-Life Examples and Applications

1. Free Fall

• When an object is dropped from a height, it accelerates downward at $9.8{\text{m/s}}^2$.
• Example: A ball dropped from a building gains speed as it falls due to $𝑔$.

2. Weight Differences on Planets

• Your weight on different planets depends on their $𝑔$ values.
• Example: A person weighing $600\text{N}$ on Earth would weigh only $100\text{N}$ on the Moon.

3. Satellites in Orbit

• Satellites experience a reduced $𝑔$ due to their high altitude but remain in orbit because of their forward velocity.

4. Projectile Motion

• The vertical motion of a projectile is influenced by $𝑔$, causing it to accelerate downward.