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Newton’s Second Law

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The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass.

Newton’s Second Law of Motion is one of the fundamental principles of physics. It states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. This law is often written in the following mathematical form:

\Sigma \vec{F}=m \vec{a}

where \Sigma \vec{F} is the net force acting on an object, m is its mass, and \vec{a} is its acceleration. This means that the greater the force acting on an object, the greater its acceleration will be. Conversely, the greater an object’s mass, the less its acceleration will be for a given force.

Sir Isaac Newton’s own words (translated from the original Latin) are:

“The change of motion of an object is proportional to the force impressed; and is made in the direction of the straight line in which the force is impressed.“

Rocket science requires the general form of Newton's Second Law as described below.
Rocket science requires the general form of Newton’s Second Law as described below.

However, this form of Newton’s Second Law is only partially true as it does not account for the situation in which an object’s mass changes. This case is not trivial. In fact, this is one of the primary operating principles of rocket science. The more accurate form of the second law is stated as:

\Sigma \vec{F} = \dfrac{\mathrm{d} \vec{p}}{\mathrm{d}t} = m(t) \dfrac{\mathrm{d} \vec{v}}{\mathrm{d}t} + \dfrac{\mathrm{d} \vec{m}}{\mathrm{d}t} v(t)

where m(t) is the mass of the object as a function of time, v(t) is its velocity as a function of time, and \dfrac{\mathrm{d}}{\mathrm{d}t} is the derivative of each function with respect to time. This longer expression on the right is given by the chain rule when differentiating momentum with respect to time.

This law applies to all objects, regardless of their size, shape, or composition. Whether an object is at rest or in motion, Newton’s Second Law describes how it will behave when acted upon by a net force.

To understand this law in more detail, consider a few examples:

  • If you push a shopping cart with a force of 10 newtons, and the cart has a mass of 20 kilograms, then its acceleration will be 0.5 meters per second squared (m/s^2). This means that its speed will increase by 0.5 m/s every second.
  • If you push the same cart with a force of 20 newtons, then its acceleration will be 1 m/s^2. This means that its speed will now increase by 1 m/s every second.
  • Now imagine pushing the cart with a force of 500 newtons. Its acceleration will be 25 m/s^2. This means that its speed will increase by 25 m/s every second.

In each of these examples, you can see how the acceleration of the object changes as the force acting on it changes.

It’s important to note that Newton’s Second Law is a vector equation, which means that the force and the acceleration are each described by both a magnitude and a direction. This means that the direction of the net force acting on an object determines the direction of its acceleration.

Additionally, when multiple forces are acting on an object, Newton’s Second Law applies to the net force, which is the vector sum of all the individual forces. This means that the acceleration of an object depends on the net force acting on it, not just any one of the individual forces.

Finally, it’s worth noting that Newton’s Second Law is closely related to Newton’s First Law, which states that an object at rest will remain at rest and an object in motion will continue in motion with a constant velocity unless acted upon by an external force. In fact, Newton’s First Law can be easily derived as a special case of the second law when \vec{a} = \vec{0}.

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Comments

2 responses to “Newton’s Second Law”

  1. Newton’s Laws of Motion – Tru Physics

    […] The Second Law is the lifeblood of physics. Any net force will result in the acceleration of an object. You will note that the first law is just a special case of this one. The formula given above is simple and easy to follow. The only caveat is that this formula does not account for the possibility that mass might vary as a function of time. This is not necessary for a Physics I course. However, this is no longer trivial when studying rockets. […]

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  2. Forces – Tru Physics

    […] Normal Force: The force exerted by a surface that supports the weight of an object. always acts perpendicular to the surface. There is not a generic equation to calculate However, in problems involving flat surfaces, we generally deduce that the normal force balances out Thus it will commonly equal When inclined surfaces are involved, is the common treatment (where is the angle of inclination). Generally, is determined by balancing forces via Newton’s Second Law. […]

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