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It all started with a man and a falling apple ...
Even if you weren’t learning physics, we’re sure you’d still hear of Newton’s (famous) laws one way or another! But since you’re learning physics, we’ve got to dive deeper into these laws (Netwon’s first law, second law, third law, and Newton’s law of gravitation) so that you can understand what each law means and how they are represented mathematically.
Portrait of Isaac Newton, Wikimedia Commons
Newton’s first law of motion captures the causes of movement and the fundamental equivalence between systems where we (a) aren’t moving and (b) are moving with a constant speed. The official formulation of Newton’s first law of motion is:
A body continues in its state of rest or in uniform motion in a straight line unless acted upon by a force.
The explanation is simple: an object tends to stay in its state if no external force acts on it. If the body is at rest, it will remain that way. If it’s moving at a certain velocity and in a certain direction, it will keep doing so until an external force acts on it.
Any object lying around on a table, the floor, etc. No net force is acting on it since the gravitational force is compensated by the normal force exerted by the horizontal surface. As a result, the object tends to stay at rest.
If we throw a marble so it rolls on the floor, it will eventually stop rolling. However, if no net force acted on it, the marble would continue rolling (forever!) at the speed we threw it. But, in real-life situations, we have the force of friction, and this generates a negative acceleration (deceleration), which eventually stops the marble.
Since the normal force and the gravitational force cancel each other out, they do not play a role when considering movement on flat horizontal surfaces.
If we throw a bowling ball down a very long bowling lane, it will roll until it reaches the end and hits the pins. This happens because the friction is much lower for the bowling ball than for the marble. The bowling ball maintains its speed much longer since its deceleration is lower. An ideal bowling lane (with zero friction) could be arbitrarily long, and the bowling ball would still reach the end since its speed would be constant.
Newton’s second law of motion gives a complete description of the evolution of a system that applies to all systems that do not include quantum or relativistic effects. The official formulation of Newton’s second law of motion is:
A body acted upon by a force moves in such a manner that the rate of change of momentum in time equals the force.
However, you may be more familiar with this:
The total force acting on a body equals its mass times the acceleration the force generates on it.
Both formulations are usually equivalent, although the ‘official’ one is more rigorous. Here is the mathematical expression for Newton’s second law:
F is the force, p is the momentum (the mass multiplied by the velocity), and d/dt indicates derivation with respect to time (the rate of change).
Let’s consider that mass does not change in time. The time derivative of the momentum (i.e. its rate of change) equals the mass times the derivative of the velocity, which we also call acceleration. So, if the mass is constant over time, the above expression is equivalent to:
Take note! The formulation ‘total force equals the mass times the acceleration’ is only true if the mass is constant.
The essence of Newton’s second law of motion is that after considering all the forces and their direction, the total effect captured by the acceleration follows the same direction as the total force, and the proportionality factor is the mass of the object. This mass is called inertial mass.
Billiards is a perfect example of Newton’s laws. The force given to the ball must equal the mass of the ball times its acceleration. Unsplash
Suppose we have four balls and a perfectly horizontal surface. The four balls have masses of 5kg, 10kg, 15 kg, and 15kg. Imagine we exert a force of 150N for 2 seconds for the first ball, second ball, and fourth ball and 4 seconds for the third ball. By applying Newton’s second law of motion, we get the following data:
Mass [kg] | Time [s] | Acceleration [m/s2] a=force/mass | Speed [m/s] speed=a⋅time | Momentum [kg⋅m/s] momentum=mass⋅speed | |
Ball 1 | 5 | 2 | 150/5=30 | 30⋅2=60 | 5⋅60=300 |
Ball 2 | 10 | 2 | 150/10=15 | 15⋅2=30 | 10⋅30=300 |
Ball 3 | 15 | 4 | 150/15=10 | 10⋅4=40 | 15⋅40=600 |
| Ball 4 | 15 | 2 | 150/15=10 | 10·2=20 | 15·20=300 |
Newton’s third law of motion captures the principle of conservation, which is fundamental to nature. It laid the groundwork for all the conservation theorems based on symmetries developed in the twentieth century. The official formulation of Newton’s third law of motion is:
If two bodies exert forces on each other, these forces are equal in magnitude and opposite in direction.
This formulation of Newton’s third law is quite simple, and the best way to understand it is through some examples. However, it is also important to take the momentum into account. Due to Newton’s second law of motion, we know that a force is equivalent to the rate of change of the momentum of a body. If, according to Newton’s third law of motion, reciprocal forces are equal in magnitude and have opposite directions, the same applies to momenta. This is what we know as the conservation of momentum.
Check out our explanation on Momentum.
Imagine sitting on a skateboard on a perfectly horizontal floor while holding a basketball. If you throw the basketball forwards, you will be pushed backwards. This happens because you are exerting a force on the basketball. Thus, according to Newton’s third law of motion, the basketball exerts the same force on you in the opposite direction. And because force equals mass times acceleration (F=m⋅a), the acceleration of the basketball is larger than yours because your mass is greater.
Rockets are perfect examples of Newton’s third law of motion. They cannot rely on friction to move in space, so they need an opposing force to push them forward. They expel gas particles due to combustion, and the sum of their momenta is translated into momentum for the rocket.
In addition to the three laws of motion, Newton formulated the first law of gravitational attraction. Its equation is:
G is the universal constant of gravitation (with an approximate value of 6.67·10-11m3/kg·s2), M is one of the masses being attracted and m is the other one, r is the radial distance separating the masses, and the vector er is the unitary vector joining the masses.
What happens when we connect this idea to Newton’s second and third laws?
We will start with Newton’s third law of motion, which equates the forces that bodies exert on one another, which is an expression of the conservation of momentum. The gravity formulation above fulfils this requirement, as both masses play equivalent roles in the equation. Exchanging masses M and m amounts to the same force since the distance r is the same. However, we must remember that the vector er is the unitary vector pointing from one of the masses to the other. If we exchange their roles, the vector remains the same, except for a net minus sign corresponding to the opposite direction.
Let’s now consider Newton’s second law of motion. If we assume the force exerted on the body of mass m is constant, we get the following equivalence:
We can cancel the mass m on both sides and see that the acceleration of the body depends on the mass of the other body and the distance between them. This is why the acceleration on the surface of the Earth is constant, and all objects fall at the same pace. (The reason this doesn’t happen in reality is because of air friction, which means that we have to take the aerodynamics of the objects into account.)
Dividing by the mass when considering laws of motion like Newton’s seems straightforward. However, the mass measuring the gravitational interaction does not need to be the same as the mass measuring the inertial properties of a body. The principle stating that these quantities are equal is one of the basic axioms of physics, namely the principle of equivalence.
Space technologies give a magnificent example of the law of gravitation during launch. Rockets have to pull a force greater than the pull of the Earth and reach a specific velocity to reach orbit. This force is opposite to the force of gravity. Unsplash
A free-body diagram is a sketch of only the object in question and the forces acting upon it. The object or ‘body’ is usually shown as a box or a dot. The forces are shown as thin arrows pointing away from the centre of the box or dot. The emphasis is on the forces, so they must be drawn accurately and to scale. It is important to label each arrow to show the magnitude of the force it represents. The type of force involved may also be shown.
Here is a very basic example of a free-body diagram. Note that it is not showing all the forces acting on the object (for example, you usually show the normal force and gravitational force acting on the object as well).
Example of a free-body diagram, Wikimedia Commons
Newton’s first law of motion states that a body continues in its state of rest or in uniform motion in a straight line unless acted upon by a force.
Newton’s second law of motion states that a body acted upon by a force moves in such a manner that the rate of change of momentum in time equals the force.
Newton’s third law of motion states that if two bodies exert forces on each other, these forces are equal in magnitude and opposite in direction.
There are three laws of motion. Newton’s laws of motion show the relationship between an object’s motion and the forces that act on it.
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