Force laws and laws of motion are a kind of two-sides-of-the-same-coin when it comes to classical physics. Reminds me of a common paradox which I’ll explore below in the form of a narrative.
It’s a wonderful Monday morning at Townsville Middle School. You are sitting next to your best friend Amy. She is telling you about her last family vacation to the Bahamas amid distinct chatter in the class. Suddenly everybody falls quiet and looking up, you see Mr. Thomas, the physics teacher.
Mr Thomas puts his bag on the teacher’s desk at the front of the class and proceeds to remove a mug, three eggs, a chicken doll, and a marker; laying them side by side on the table.
“A bit late for breakfast don’t you think?” whispers Amy.
“I prefer my eggs scrambled, please,” you chuckle back, mimicking the language of a disgruntled restaurant customer.
Mr. Thomas looks up from the teacher’s desk, scans the class, and asks, “What do you think comes first, egg or chicken?”
“The rooster!” comes a voice from the back of the class.
Force laws and laws of motion in classical physics
The egg and chicken dilemma is a great analogy to introduce the relationship between force laws and laws of motion. But before we take a deeper look at force laws and laws of motion, let’s review what the two terms mean and what their differences imply.
Force and motion form a big part of what we call, Classical Physics, an approach to understanding physics that involves forces, motion, and energy. This branch of physics is hugely influenced by theories put forward by English physicist Isaac Newton that it is sometimes referred to as ‘Newtonian Physics’.
The approach to Newtonian physics is quite direct and involves roughly three steps:
- A body with known physical properties (mass, volume, electric charge, etc) is placed at an initial location and moving with an initial velocity.
- We know (or can measure) all the body’s interactions with its environment.
- Can we then predict the subsequent motion of the object? That is, can we find its position and velocity at all future times?
The above diagram demonstrates the physical properties and the initial and final conditions of a body. Note the change in ‘environment’ from windy and rainy in the first instance to cool and sunny in the second instance. The body in question would interact differently with the environment in these two scenarios. For instance, in the first ‘Initial Condition’, the wet and slippery road would interact differently with the tires of the vehicle compared to the dry road of the ‘Final Condition’.
In this context, environment means anything that is outside the body in question. This includes the road, wind, clouds, rain, other vehicles, etc.
Forces and Environment
In classical physics, the interaction between a body and its environment constitutes what we call a ‘Force’. This interaction has the common effect of causing the body to move, slow down, speed up, change direction, and even stop. Force is therefore usually defined as a push or a pull. Forces are best described and represented using vectors. This means all forces have a numerical value attached to them (magnitude) and we must specify a direction in which they act.
Where do forces come from? From other bodies in the environment!
When a force acts on a body, it means there is another body in the environment responsible for that force. For example,
“pushing force on a crate by worker” – means the pushing force on the crate comes from an agent ‘worker’ in its environment.
“frictional force on a crate by floor” – means the frictional force on the crate comes from an agent ‘floor’ in its environment.
“gravitational force on a crate by Earth” – the force of gravity on the crate comes from an agent ‘Earth’ in its environment. And so forth
Forces don’t come isolated, the culprit is always in the vicinity
The Scientist
Force laws and laws of motion joined the chat!
Based on our discussion so far, we know that forces don’t come from within the body themselves, but from other external bodies in the environment. The relationship describing the interaction between the environment and the body constitutes Force Laws.
Force laws describe the interaction between the environment and the body. These laws are important because they tell us the nature of the force and provide a way to calculate the force independent of the laws of motion. Common examples of Force laws include.
Hooke’s law for springs
We are all aware of the principle of this force, you need to exert a force on something to stretch it or compress it. And the simplest possible object here is the spring. The relationship of which is described by Hooke’s law, “the force needed to extend or compress a spring by some distance is proportional to that distance,” or
$$F\ =\ kx$$
Where x is the displacement and k is the spring constant, which depends on the nature of the material.
This law is useful because it provides another way of calculating forces (other than the laws of motion) – which is a useful technique used to calibrate springs.
Gravity: Newton’s law of gravity
Isaac Newton discovered that everybody attracts another body in its vicinity due to the force of gravity. He summarized his findings in the popular law of gravitation which is mathematically represented as:
$$F\ =\ G\frac{m_1m_2}{r^2}$$
This law shows how a body of mass (m1) can experience a force (F) by being in the vicinity (=environment) of another body of mass (m2).
Electrostatics: Coulomb’s law
French physicist Charles-Augustin de Coulomb (1736-1806) discovered a force law that exists between charged bodies:
$$F\ =\ k\frac{q_1q_2}{r^2}$$
Like Newton’s law, this law shows how a body of charge (q1) can experience a force (F) by being in the vicinity (=environment) of another body of charge (q2).
Laws of Motion
The force laws that we just discussed tell a story of the relationship between the environment and the body. They show how force arises from the body’s interaction with other bodies in the environment.
The next set of laws that we are going to discuss, the laws of motion, take us to the other side of the coin. They tell a story of the body’s behavior after interacting with the environment.
The three laws of motion
When studying forces, Isaac Newton deduced three principles that evolved to become the laws of motion as we know them today.
First, he realized that as long as there is no external action, a body’s velocity would remain completely unchanged. This is the first law of motion.
Second, he deduced that if there is indeed a net external force acting on the body, then it will produce a change in motion in the body which is proportional to the force. This is the second law and it is best represented mathematically as:
$$F\ =\ ma$$
Here is a closer look at Newton’s second law of motion.
Thirdly, he noted that “if a body exerts force F1 on a second body, the second body exerts an equal but opposite force F2 on the first body”. This is Newton’s third law of motion.
$$F_1\ =\ F_2$$
Here is a closer look at Newton’s second law of motion
These laws collectively tell us what a force does, but they don’t tell where that force comes from, nor do they care. This is a useful feature because the laws apply to ALL forces.
The centripetal force
Centripetal force is a force acting on a body moving in a circular path and is directed toward the center around which the body is moving. It is an excellent example of a force deduced from the laws of motion.
$$F\ =\ m\frac{v^2}{r}$$
The above formula is very useful in dealing with circular and rotational dynamics because it is independent of the nature of the force.
Whilst the force laws that we just discussed earlier are very useful, they are not generalized i.e. they cannot be applied across all situations. We cannot use Newton’s law of gravity to find the force needed to stretch a spring, or we couldn’t use the law of friction to find the force of gravity between two bodies. And while the laws of motion can be applied across all situations, they don’t tell us anything about the nature of the force which may be of significant importance if we wish to gain some insight into the force in question.
Force laws and laws of motion = classical physics
When force laws and laws of motion are taken together, they make up the laws of classical physics. Force thus appears in both the laws of motion (which tell us what acceleration an object will experience under the action of a given force) and in the force laws (which tell us how to calculate the force on a body in a certain environment).
It is this branch of physics that has successfully enabled us to build great skyscrapers and study the properties of their construction materials; build airplanes that can carry hundreds of people and fly halfway around the world; and send space probes on complex missions to the comets, the planets, and beyond.