Newton's Second Law of Motion

In Newton's first law of motion we asked what happens when there is no net force on a body (zero net force) but in Newton's second law we ask what happens when there is a nonzero net force on a body.

You'll see this later that if net force is zero in Newton's second law of motion, the motion is described by Newton's first law. It means the statement of Newton's second law is converted back to the Newton's first law of motion (constant velocity or zero acceleration).

We know from many experiments that if a nonzero net force acts on a body, the body must accelerate. It means if a constant nonzero net force acts on a body, the velocity of the body changes continuously in a constant rate. And that rate of change of velocity is called acceleration.

The direction of acceleration is in the same direction of net force. For example, a driver applies brakes of a bus, the directions of force and acceleration are in the opposite direction to the velocity of the bus, that is both force and acceleration are in the same direction, and the bus finally comes to rest.

The statement of Newton's second law based on the relationship between force and acceleration can be stated as

NEWTON'S SECOND LAW: When viewed from inertial frame of reference, a body accelerates in the direction of force if a nonzero net external force acts on it.

The statement of Newton's second law recently stated gives us an idea about the relationship between force and acceleration but not a complete statement of Newton's second law.

Suppose you apply a net force $\vec F$ on a body, the body moves with constant acceleration $\vec a$ in the direction of applied force. We know from experiments that if you double the force ($2\vec F$), the acceleration doubles ($2\vec a$) and if you tipple the force ($3\vec F$), the acceleration triples ($3\vec a$) and so on.

It has been seen from the experiments that the acceleration of a body is directly proportional to net force acting on it. Before stating Newton's second law of motion again we define what mass is.

Mass and Force

Let's start with an example of a tennis ball and a basketball. If you throw both balls with the same force, the tennis ball is accelerated more than the basketball. It means the tennis ball has less resistance to the change in motion than the basketball. Greater the mass of a body, more it "resists" the change in its motion.

We know the property of a body to resist the change in its motion is inertia (tendency of a body to remain in its current state is the same thing as the resistance to the change in motion). The quantitative measure of inertia of a body is the mass of the body.

If the same force is applied to two bodies of masses $m_1$ and $m_2$ and $a_1$ and $a_2$ are the corresponding accelerations, the ratio of $m_1$ to $m_2$ is equal to the ratio of $a_2$ to $a_1$, that is the ratio of masses is equal to the inverse ratio of magnitudes of accelerations.

\[\frac{m_1}{m_2} = \frac{a_2}{a_1}\]

If the mass of one body is known, you can determine the mass of another body using acceleration measurements in above equation. Many observations reveal the fact that the net force on a body is inversely proportional to the mass of the body. We learnt above that the net force on a body is directly proportional to the acceleration of the body.

MASS: The ratio of net force on a body to the acceleration of the body is called mass of the body.

The quantity of matter in a body is determined by the quantitative measure of inertia of the body, that is how much the body resists the change in its motion, that is if we apply a net force on a body the motion of the body changes continuously in a certain rate and the body continuously resists the changes in its motion and the measure of that resistance is mass. If a body has acceleration $\vec a$ due to net force $\sum \vec F$, the mass $m$ is

\[m = \frac{|\sum \vec F|}{a}\]

The above equation gives the quantitative measure of mass of a body. We could have defined the net force before defining mass but either way we reach the same destination. After analyzing the results of many experiments and observations, it turns out that the net force on a body is equal to the product of its mass and acceleration. We know that the acceleration is directly proportional to the the net force and inversely proportional to the mass. Taking the proportionality constant $1$ we obtain the expression of net force.

\[\sum \vec F = m\vec a\]

In the first statement of Newton's second law of motion we just focused on what happens if there is nonzero net force on a body but now a more complete statement (a summarized statement of the observations of nonzero net force on a body) of the Newton's second law is

NEWTON'S SECOND LAW: When viewed from inertial frame of reference, a body accelerates in the direction of force if a nonzero net external force acts on it. The net force is equal to the product of mass and acceleration of the body.

Newton's second law is valid only in inertial frame of reference. It is not valid in an accelerating car, where you move forwards or backwards without any force at all! Newton's first law is the special case of Newton's second law when net force is zero. If the net force is zero, Newton's second law is converted to the Newton's first law, that is acceleration is zero which means constant velocity. The force is always external net force. Internal forces do not cause the change in motion. If they do, you can pull yourself to reach your ceiling!

Now we come to the interesting situation of the above Newton's second law of motion. What if the mass changes not the velocity and there is still force? You'll occasionally encounter such situations in Physics and the above statement of the Newton's second law is invalid in such situations.

Newton realized that the quantity of motion is better described by the momentum which is the product of mass and velocity of a body. The momentum $\vec p$ of a body of mass $m$ and velocity $\vec v$ is

\[\vec p = m\vec v\]

Now we state the final statement of Newton's second law which is equivalent to the Newton's original statement in terms of rate of change of momentum.

NEWTON'S SECOND LAW: When viewed from inertial frame of reference, the net force on a body is equal to the rate of change of momentum of the body.

The above statement of the Newton's second law is expressed in mathematical form as

\[\sum \vec F = \frac{d\vec p}{dt} \tag{1} \label{1}\]

Consider the mass is constant and we know $\vec p = m\vec v$ and solving above equation we get

\[\sum \vec F = m\frac{d\vec v}{dt} = m\vec a \tag{2} \label{2}\]

Both equations represent the mathematical form of Newton's second law of motion. Newton originally stated the second law in terms of the rate of change of momentum expressed by Equation \eqref{1} but we generally use Equation \eqref{2} as the Newton's second law. Remember the Equation \eqref{2} is the special case when the mass is constant!

The product of mass and acceleration, that is $m\vec a$ is not a force! It may be the point of confusion that you may call $m\vec a$ is a force. The acceleration is the result of force and $m\vec a$ is the value of the force not the force itself. The product $m \vec a$ is the result of net force which is the vector sum of all forces acting on a body. You move backwards when your car suddenly accelerates. There is no force on you but if $m\vec a$ is a force, your acceleration mislead you to think there is a net force on you.

As you know now $m\vec a$ is not a force, we never define force as the product of mass and acceleration of a body, that is we don't say force is $m\vec a$ but we say the net force is equal to $m \vec a$. Note that "is equal to" to is used instead of simply "is".

Mechanics
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