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# Newton’s Laws

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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

## What is Newton’s first law?

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.

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.

## What is Newton’s second law?

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

• As we can see with balls 1 and 2, if the time is equal, the speed of lighter bodies is greater.
• As we can see with balls 3 and 4, if the force is applied for longer, the final speed will be higher since it has been accelerating for longer.
• When comparing the final momentum of the balls, we find that balls 1, 2, and 4 have the same momentum because the same force was applied for the same amount of time. For ball 3, the momentum is double because the same force was applied for double the time.

## What is Newton’s third law?

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.

## What is Newton’s law of gravitation?

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

## Newton’s laws: free-body diagrams

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

• Free-body diagrams contain quantitative information needed to solve problems.
• System diagrams are only helpful to understand the problem and visualise it.

## Newton’s Laws - Key takeaways

• Newton formulated three laws of motion, which remain fundamental keystones of classical mechanics.
• Newton’s first law of motion states that a body tends to remain in its state of movement or rest unless acted upon by an external force.
• Newton’s second law of motion states that the net force acting on a body equals the rate of change of its momentum. This amounts to the net force acting on a body being equal to its mass times its acceleration when the mass is constant.
• Newton’s third law of motion states that if a force is exerted by a system on another system, the second one exerts a force equal in magnitude and in the opposite direction to the first system.
• Newton formulated the first law of gravitational attraction. It is consistent with his laws of motion.
• A free-body diagram is a sketch of only the object in question and the forces acting upon it.

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.

## Final Newton’s Laws Quiz

Question

Newton’s second law of motion is the law that relates the net force to the rate of change of momentum.

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Question

Newton’s first law of motion implies that moving objects in space move with constant velocity.

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Question

Newton’s law of gravity states that the force of attraction between two bodies is the same for each of them.

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Question

Rockets move because they expel particles with momenta, and, due to Newton’s third law of motion, momentum is generated on the rocket.

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Question

Is the formulation ‘the total force equals the mass times the acceleration’ always true?

No, ‘the total force equals the mass times the acceleration’ is only true if the mass is constant.

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Question

According to Newton’s laws, can we decelerate light by exerting a force?

According to Newton’s laws, we cannot decelerate light by exerting a force because light does not have mass.

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Question

Why do objects lying on a surface remain still (not move) according to Newton’s laws?

Objects lying on a surface do not move because there is no net force acting on them to change their resting state.

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Question

What is the inertial mass?

The inertial mass is the name given to the mass appearing in Newton’s laws.

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Question

Why do objects moving at a certain speed usually stop at some point? Choose the correct answer.

Because there are no ideal conditions for Newton’s first law to be applied. There is always some force, usually friction.

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Question

Explain in terms of Newton’s third law of motion why we can swim? Choose the correct answer.

When we move our arms and legs in the water, we displace particles of water that exert a force back at us, causing us to move.

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Question

If an object is spinning in circles in a no-friction circular motion and the circular motion suddenly stops, what happens to the object? Choose the correct answer.

If an object suddenly stops spinning, it will remain moving in a straight line tangent to the original circle at constant velocity.

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Question

What is the principle of equivalence?

It is the principle that states that the mass measuring the intensity of the gravitation interaction and the inertial mass are the same.

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Question

Do the masses in Newton’s law of gravitation play an equivalent role?

Yes, exchanging them amounts to a global sign in the formula, which accounts for the opposite direction consistent with Newton’s third law.

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Question

Newton’s second law of motion states that the net force acting on an object equals the rate of change of momentum.

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