Unit 2.2: The Gravitational Field in AP Physics 1

Understanding Gravitational Fields and Weight

The gravitational field is a region around a massive object where other objects experience a force due to gravity. On Earth, this field exerts a downward force on all objects, commonly referred to as their weight.

Key Formula:

The gravitational force on an object can be calculated using:

$$F=mg$$

where:

• 
  $F$

   

  is the force due to gravity (weight)
• 
  $m$

   

  is the mass of the object in kilograms (kg)
• 
  $g$

   

  is the acceleration due to gravity (on Earth, $g \approx 9.81 \, \text{m/s}^2$

   

  , but $g = 10 \, \text{m/s}^2$

   

  can be used for simplification on the AP Physics 1 exam)

SI Units:

• Weight: Newtons (N)
• Mass: Kilograms (kg)

Weight vs. Mass

It’s crucial to differentiate between weight and mass:

• Mass is the measure of the amount of matter in an object and remains constant regardless of location.
• Weight is the force exerted by gravity on an object, which varies depending on the gravitational field strength.

Example: An object will weigh less on the Moon than on Earth due to the Moon’s weaker gravitational field.

Key Concepts

• Gravitational Field (
  $g$

   

  ): Region around a massive object where a force of attraction acts on other masses. The strength of this field depends on the mass of the object and its distance from other objects.
• The gravitational constant (
  $G$

   

  ) relates the gravitational force between two objects to their masses and the distance between them:

$$G\approx 6.67\times 10^{-11}\text{N}\cdot {\text{m}}^2\mathrm{/}{\text{kg}}^2$$

Example Problems and Solutions

Example Problem 1: Calculating Gravitational Force

Problem: A 5.00-kg object is placed on a frictionless table. Determine the gravitational force on the object due to Earth’s gravity (

$g = 9.8 \, \text{m/s}^2$

).

Solution:
Using the formula

$F = mg$

:

$$F=(5.00\text{kg})\times (9.8{\text{m/s}}^2)=49\text{N}$$

The gravitational force on the object is 49 N.

Example Problem 2: Gravitational Force on a Different Planet

Problem: A 10.0 g sample of material is placed on Planet X, which has a mass of

$5.00 \times 10^{23} \, \text{kg}$

and a radius of

$6.38 \times 10^6 \, \text{m}$

. The acceleration due to gravity on Planet X is

$8.87 \, \text{m/s}^2$

. Calculate the gravitational force on the sample.

Solution:

1. Convert mass to kilograms:

$$m=\frac{10.0\text{g}}{1000}=0.01\text{kg}$$

2. Calculate the gravitational force using
   $F = mg$

    

   :

$$F=(0.01\text{kg})\times (8.87{\text{m/s}}^2)=0.09\text{N}$$

The gravitational force on the sample is 0.09 N.

Example Problem 3: Weight of an Object

Problem: A textbook has a mass of 2.00 kg. Calculate its weight on Earth (

$g = 9.80 \, \text{m/s}^2$

) and convert the weight to pounds.

Solution:

1. Calculate weight in newtons:

$$F=mg=(2.00\text{kg})\times (9.80{\text{m/s}}^2)=19.6\text{N}$$

2. Convert newtons to pounds (1 N = 0.225 pounds):

$$\text{Weight (pounds)}=19.6\text{N}\times 0.225\text{pounds/N}=4.42\text{pounds}$$

The textbook’s weight is 19.6 N or 4.42 pounds.

Practical Applications of Gravitational Fields

• Understanding weight differences on various planets: Objects weigh less on the Moon than on Earth due to the Moon’s lower gravitational field strength.
• Space travel and celestial mechanics: The concept of gravitational fields is essential for calculating spacecraft trajectories and planetary orbits.