Newton's Laws of Motion and Universal Gravitation

• $F_x = |F| \cos \theta$
• $F_y = |F| \sin \theta$

Exam Tip: When solving problems with multiple forces, always resolve them into their $x$ and $y$ components first. This allows you to sum them up separately to find the total effect in each dimension.

Newton's Second Law of Motion

The relationship between force, mass, and motion is most famously captured in Newton's Second Law. This law states that the acceleration of an object depends on two variables: the net force acting upon the object and the mass of the object.

The mathematical expression is: $F_{\text{net}} = ma$ Where:

• $F_{\text{net}}$ is the sum of all forces acting on the object.
• $m$ is the mass.
• $a$ is the acceleration.

Net Force and Acceleration

The term net force ($F_{\text{net}}$) refers to the vector sum of all forces acting on an object.

1. Equilibrium: If the net force is zero ($F_{\text{net}} = 0$), the object is in equilibrium. This means the object will either remain at rest or continue to move with a constant velocity.
2. Acceleration: If the net force is not zero ($F_{\text{net}} \neq 0$), the object will accelerate. Importantly, the acceleration vector always occurs in the same direction as the net force.

Worked Example: Calculating Acceleration

Suppose a forward force of $2.00 \times 10^3 \text{ N}$ is applied to an object, but there is a resisting force of $1.83 \times 10^3 \text{ N}$. If the mass of the object is $1.00 \times 10^3 \text{ kg}$, what is the acceleration?

First, find the net force: $F_{\text{net}} = (2.00 \times 10^3 \text{ N}) - (1.83 \times 10^3 \text{ N}) = 0.17 \times 10^3 \text{ N}$

Then, use Newton's Second Law: $a = \frac{F_{\text{net}}}{m}$ $a = \frac{170 \text{ N}}{1000 \text{ kg}} = 0.17 \text{ m/s}^2$

The object accelerates at $0.17 \text{ m/s}^2$ in the direction of the forward force.

Newton's Law of Universal Gravitation

While Newton's laws of motion describe how forces affect individual objects, his Law of Universal Gravitation describes the force that exists between all objects in the universe.

Every mass exerts an attractive force on every other mass. This gravitational force ($F_g$) is directly proportional to the product of the two masses and inversely proportional to the square of the distance between their centers: $F_g = G \frac{m_1 m_2}{r^2}$

In this equation:

• $m_1$ and $m_2$ are the masses of the two objects.
• $r$ is the distance between the centers of the masses.
• $G$ is the universal gravitational constant, which is approximately $6.67 \times 10^{-11} \text{ N} \cdot \text{m}^2/\text{kg}^2$.

Because $G$ is such a small number, we do not notice gravitational attraction between everyday objects (like two people standing near each other). However, when dealing with celestial bodies like planets, this force becomes massive.

Weight and Gravity

A common point of confusion is the difference between mass and weight. While mass is an intrinsic property of an object that remains constant everywhere, weight ($W$) is a force. Weight is the specific application of the gravitational force exerted by a celestial body on an object: $W = F_g = mg$

Here, $g$ represents the acceleration due to gravity. On Earth, $g$ is approximately $9.8 \text{ m/s}^2$. However, $g$ is not a universal constant; it depends on the mass and radius of the planet you are standing on.

For example, the acceleration due to gravity on Earth can be calculated using the mass of the Earth ($M_{\text{Earth}}$) and its radius ($R_{\text{Earth}}$): $g = \frac{G M_{\text{Earth}}}{R_{\text{Earth}}^2}$

Critical Distinction: If you travel to the Moon, your mass remains exactly the same because the amount of matter in your body hasn't changed. However, your weight will decrease because the Moon's mass is much smaller than Earth's, resulting in a smaller value for $g$.

Exam Tip: If a question asks how your "mass" changes on Mars, the answer is "it doesn't." If it asks how your "weight" changes, the answer is "it does."