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# Newton's first law of motion

Module by: Sunil Kumar Singh. E-mail the author

Summary: The natural tendency of all bodies is to maintain its state of motion.

The first law of motion talks about the motion of a body for a particular situation. There is either "no force" or "zero net force" acting on the body. The first condition of "no force" is not common in our immediate surrounding. All bodies are acted by gravity i.e. force of attraction due to Earth. On the other hand, the second condition of "zero net force" is common in our immediate surrounding, where most bodies are stationary in Earth's reference as net force is zero.

Definition 1: First law of motion
Unless acted upon by a net external force, a body, at rest, will remain at rest and a body, in motion, will remain in motion.

The state of motion of a given body, including the state of rest, is completely defined by its velocity. Stationary state is just one important case of constant velocity or uniform motion. If the object is stationary in a frame of reference, then

v = 0

We can restate the first law of motion more concisely in velocity term as :

Definition 2: First law of motion
If net external force on a body is zero, then its velocity remains constant.

Mathematically equivalent statements of the first law of motion are :

1: If ∑F = 0, then v = a constant.

2: If ∑F = 0, then a = 0.

The substance of first law of motion is expressed in many ways. Here, we sum them all for ready reference (for the condition that the net force on a body is zero) :

• The body may either be at rest or may move with constant velocity.
• The body is not associated with any acceleration.
• If the body is moving, then the body moves along a straight line with a constant speed without any change of direction.
• If the body is moving, then the motion of the body is an uniform linear motion.
• If a body is moving with uniform linear motion, then we can be sure that the net force on the body is zero.

## First law of motion and our experience

Let us now examine the interpretation of the law in a bit more detail as the statement may not be completely in agreement of what we see around.

### A body at rest remains at rest

The part of the statement, which says that the body remains in stationary state, is a comprehensible argument, supported by our daily life experience. We actually experience that a body requires some external force to be moved around form its stationary state. As a matter of fact, this experience instills the notion that the state of rest is the natural state of matter. We, however, know that notion of rest is actually an experience or a perspective in specific reference. A body at rest in Earth's reference is in motion for other heavenly bodies.

Rest does not mean absence of force. The body, in question, may actually experience a system of force whose net force is zero. Consider a book lying on the study table. The book experiences two forces (i) its weight acting downwards and (ii) normal reaction of the table acting upwards. The two forces are equal and opposite and hence their resultant is zero.

In the nutshell, this part of first law of motion provides additional possibility of rest other than when rest may result from “no force” being applied to the body. This part of the law, therefore, characterizes two important aspects of rest as :

• Rest may arise from “no force” being applied on the body.
• Rest may arise from “zero net force” being applied on the body

### A body in motion keeps moving

The second part of the Newton’s first law is not directly supported from daily experience. Our general perception is actually contrary to what this law of motion says. We have seen that all bodies in motion, if left unattended, comes to rest.

We need to look a bit closer at the situation in hand, surrounding us. We live under the force of gravitation and almost always encounter force of friction. The two forces are generally the reason that an object apparently does not follow this part of the law. Since the requirement of “no force” or “zero net force” on a "body in motion" is not fulfilled, we do not find real time example to support this part of the law.

We may get an insight into the basis of the law observing that an object like a base ball travels a longer distance on smoother plane. Lesser the friction longer the distance traveled. This fact is indicative that if there had been no friction, the ball would have kept moving and that would have been possible if the ball did not change its velocity.

In summary, this part of the law characterizes following important aspects of motion :

• The state of uniform linear motion results when “no force” is applied on the body.
• The state of uniform linear motion results, when “zero net force” is applied on the body.
• Uniform linear motion is the natural state of motion, rest being just one important case.

## Newton’s first law and inertia

Newton’s first law of motion can be interpreted to mean that objects do not change its state of motion on its own. This property of an object is known as “inertia” i.e. sluggishness or inactivity.

The property of the object of maintaining state of motion unless being forced externally is characterized by first law. For this reason, Newton’s first law is also referred as “law of inertia”. Incidentally, “inertia” is quantized by Newton’s second law of motion, which measures the “unwillingness” on the part of an object to change. Every object resists change in velocity.

A body moves with acceleration only when it is acted by external net force. The amount of acceleration i.e. rate of change in velocity for a given net force is different for different mass of the bodies.

F = ma

For a given force, F, we have :

a 1 m a 1 m

In words, a smaller mass yields a greater acceleration and a greater mass yields a smaller acceleration. Thus, “mass” of a body is the measure of the inertia of the body in translational motion. Here, we have specified that mass is the measure of inertia in translational motion, because the inertia to rotational motion is measured by a corresponding rotational term called "moment of inertia".

## Newton’s first law and inertial frame of reference

The relation between Newton’s first law and inertial frame of reference is very close. The inertial frame of reference is actually defined in terms of Newton’s first law.

Definition 3: Inertial frame of reference
A frame of reference in which Newton’s laws of motion are valid is called inertial frame of reference.

An inertial frame of reference, where Newton’s first law is valid, moves with constant velocity without acceleration. A frame of reference moving with constant relative velocity with respect to an inertial frame of reference is also inertial frame of reference. The context of inertial frame of reference is important for Newton’s first law. Otherwise, we may encounter situations where a body may be found to be accelerated, even when net force acting on the body is zero. Consider the book lying on the floor of an accelerated lift moving upward.

To an observer in the lift, the books are under a pair of balanced forces : the weight of the books acting downward and an equal normal force acting upward. The net force on the book is zero. An observer on the ground, however, finds that book is accelerated up. To support this observation in Earth's inertial frame, the observations in the lift has to be incorrect.

This apparent paradox is resolved by restraining ourselves to apply Newton's first law in the inertial frame of reference only. The observer on the ground determines that the book is moving with upward acceleration. He concludes that the normal force is actually greater than the weight of the books such that

N - mg = ma

Aletnatively, we may use the technique of psuedo force and convert the accelerated frame into an inertial frame and then apply Newton's first law. We shall discuss this technique subsquently.

Further, we can always convert an accelerated non-inertial frame of reference to an equivalent inertial frame of reference, using the concept of “pseudo force”. This topic will be dealt in detail separately.

We, now, sum up the discussion so far as :

• Inertial frame of reference is one in which Newton’s first law of motion is valid.
• Inertial frame of reference is one which moves at uniform velocity.
• Any reference system, which is moving with uniform velocity with respect to an inertial frame of reference is also an inertial frame of reference.
• Earth’s frame of reference approximates to inertial frame for motion, which is limited in dimension.
• We can convert an accelerated non-inertial frame of reference to an equivalent inertial frame of reference, using the concept of “pseudo force”.

## Exercises

### Exercise 1

In which case(s), the net force is non-zero ?

(a) An air bubble moving up inside soda bottle at a speed 0.1 m/s

(b) A cork floating on water

(c) A car moving with 60 km/hr on a rough horizontal road

(d) None of above

#### Solution

The objects in cases (a) and (c) are moving with constant velocity. Thus, there is no net force in these cases. On the other hand, floatation results when net force is zero.

Hence, option (d) is correct.

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