Law of Inertia: Newton’s first law of motion

Aristotle believed that the natural tendency of objects is to rest, unless there is some external push on them to keep them moving. e.g. if you give a push to a swing, it continues for some time and gradually slows down, swinging less and less, until eventually coming to a halt. Another example is that of a car, it needs the energy from the fuel to keep it moving on the road. If there is no fuel, the car won’t start. If the fuel exhausts when the car is moving, it will suddenly stop. So it does seem reasonable to infer that all objects are likely to maintain a state of rest, and a force is needed to make them move or to maintain their state of motion.

As much as it appeals to us intuitively, this theory is, in fact flawed. In the example of the car, when the fuel exhausts, we have assumed there is no force left to cause the motion of the car.That is true, but there is still the force of friction supplied by the road, trying to stop the car. But if friction weren’t acting to stop it, no force would be required to maintain the motion of the car, even if all fuel ran out.

Galileo Galilei is generally credited with proposing the law of inertia which states that “a body moving on a level surface will continue in the same direction at a constant speed unless disturbed”.

Newton incorporated the law into the three laws of motion, restating the law of inertia as the first law of motion:

Every body continues in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed upon it.

In other words, if no net force acts on it, the object will continue to move in a straight line (if it were moving already) or it will continue to stand still (if it were standing still). A force is needed to change the velocity of the object. Hence the car in question will continue to move with a constant speed along a straight line if there are no obstacles in the way and no friction.

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One can go as far as to say that an object moving in a straight line with uniform velocity and another object standing still, are equivalent. The laws of Physics are valid for both reference frames, one standing still and one moving with uniform velocity. One cannot distinguish between a uniform motion and state of rest, because there is no absolute fixed reference point in space (all stars and planets and galaxies are moving relative to each other). Unless there is acceleration, you can’t tell the difference.

One way to understand this proposition is to do a thought experiment: Imagine one day you wake up in a room with no windows and a locked door that looks like inside of a spaceship. There is a glass of water on a table that is lying still. Since the water is still, you think the spaceship may be stationary, may be parked on the surface of a planet. But you can’t look outside, and there are no jerks. Since you don’t “feel” any movement, can you say for sure that it is actually not moving at all? Or maybe, it is moving in space with a constant velocity? How can you be sure?

This is a fundamental problem which has puzzled scientists for over a thousand years. Maxwell’s equations showed that the speed of light is constant. If Newtonian mechanics is correct, we should be able to measure (i.e. to tell) the absolute velocity of an object using optical (and electromagnetic) experiments, even without a reference point. However, the famous Michelson-Morley experiment (1887) failed to measure the absolute velocity of the Earth. This was a major mystery until Einstein showed that the Newtonian equations of motion need to be modified, to take relativistic effects into account. The application of Special theory of Relativity proved that the absolute velocity cannot be defined. We cannot tell, in principle as well as experimentally, whether something is moving at a constant velocity or staying still without a reference (i.e. without looking outside so as to speak).

Next: Second Law of Motion

Kinematics: Translational Motion

Kinematics is the domain of Mechanics that deals with simple translational motion. In this domain, we simply discuss the relationships among displacement, velocity and acceleration; without considering the cause of the motion.

We know that velocity is the rate of change of displacement:

v = dx/dt

Hence, assuming the velocity stays constant(uniform motion), the net displacement (x) of an object would be the product of its velocity and the time taken to cover that distance. Hence, if a car is moving on a straight road with constant velocity of 100 km/hr, it would have covered 50 kilometres in half an hour. Since it is moving in a straight line, its net displacement is 50 km from its starting point. Therefore, in general we can say that

x’ = x₀ + v × t

where x’ is the final position, x₀ is the initial position and v is the constant velocity. Both x’ and v are vectors. Note that we could have arrived at this formula by integrating the equation (v dt = dx) and setting the constant term to x₀.

Graphically, we can describe it as

1. Displacement vs time plot of a car moving at a constant velocity of 100 km/hr. The slope of the curve denotes velocity. We have assumed x₀ = 0 (initial position)

It is all very straightforward and intuitive. However, in practice you need to start the car first, accelerate all the way from 0 km/hr to 100 km/hr and then only you can use the constant velocity formula above. In order to study this kind of motion, you need to take acceleration into account.

Acceleration is the rate of change of velocity:

a = dv/dt

It is similar in form to the displacement-velocity relation above. Let me take the liberty to simply integrate it rather than go into the full discussion from the point of view of physics, in order to save time (You can learn more about the physics behind this here):

∫a dt = ∫dv

If we assume constant acceleration, a comes out of the integration in the left hand side:

a∫dt = ∫dv

⇒ at = v – v₀

or,

v = v₀ + at

It is easy to see that the constant of integration should be the initial velocity, v₀.

Let’s denote this graphically:

2.Velocity-time plot for uniformly accelerated motion. The slope of the curve indicates acceleration, which is constant. Again, we are assuming initial velocity to be zero, hence the plot starts from the origin.

In order to calculate displacement, we replace v by dx/dt:

⇒  dx/dt = at + v₀

⇒ dx = (at + v₀) dt

Integrating,

∫dx = ∫(at + v₀) dt

or,

x = (1/2)at² + v₀t

This equation enables us to plot the displacement-time plot directly, even with varying velocity:

3. Displacement vs time plot corresponding to the velocity vs time plot(2) above. The slope of the curve denotes velocity, the curvature denotes acceleration.

In the uniformly accelerated motion with acceleration and displacement in the same direction, you get the curve like (3): concave up curve because displacement is constantly increasing function of time.

Let’s see one last example to understand the graphical representation of these concepts. Let’s say a particle starts from rest, has a uniform acceleration of 3 m/s² for the next 8 seconds. Then suddenly the acceleration drops to zero(instantaneously) and stays zero for the next 6 seconds, and finally it decelerates with 2 m/s². We can represent this motion graphically as follows:

4. Displacement vs time plot

5. Velocity vs time plot

It can be seen that the concave up region in plot 4 corresponds to the positive slope line segment in plot 5(from t = 0 to t= 8 seconds), the constant slope region in plot 4 corresponds to the straight horizontal line segment in plot 5(t= 8 to t = 14 seconds) and finally, the concave down region(when the displacement is increasing but more and more slowly) in plot 4 corresponds to the negative slope line segment in plot 5, until finally the velocity reaches zero and particle comes to rest.

Another important relationship can be derived between acceleration and displacement which does not involve time:

we know that

 v = v₀+at

⇒ t = (v-v₀)/a

putting this in equation 2

x = v₀(v-v₀)/a +(1/2) a((v-v₀)/a)²

or,

v² = v₀² +2ax

This is the third equation of Kinematics.

In conclusion, motion of an object can be characterized by its position, velocity and acceleration. If you know the three variables, you can easily describe (and predict) particle trajectory using the three equations of Kinematics.

Next: Forces

Motion and its types

Mechanics is the study of motion and the physical interactions that change it. A moving object is an object that is changing its position with respect to something which is fixed. e.g. a car on a road is said to be moving if it changes its position every second with respect to the road. It is easy to tell if something is moving or not, provided we have a set of coordinate axes and a reference point (the origin); comprising what is called a Reference Frame.

However, depending on the forces acting on a body, it can show a number of patterns of movements. e.g. a wheel rolling on an inclined plane shows two kinds of motions, rotation around its axis and moving down towards the ground.

It gets complicated further if you put an ant on the wheel. that ant, will move all over the wheel, in any direction it pleases, yet will roll down with the wheel. The absolute paths of the ant and the wheel are very complex.

In order to understand this type of complicated motion, you’ll need to separate the different kinds of motions involved, like the rotation of the wheel, its movement down the slope and the path of the ant on it. In order to analyze a problem dealing with motion, you have to analyze the different kinds of motion separately, otherwise it gets very complicated.

The different kinds of motion most commonly encountered in Classical Mechanics are:

1. Rectilinear motion: motion along a straight line, simplest case of translational motion.
2. Translational motion: motion in 3 dimensional space such that the path consists of smaller straight line segments.
3. Simple harmonic motion/oscillation: a particle moves to and fro from its mean position
4. Rotational motion: a solid body rotates around its own axis
5. Uniform Circular Motion: a particle moves in a circle around a point, maintaining a constant distance from it
6. Curvilinear Motion: motion along a curve in space
7. Random motion: No fixed trajectory/path: possibly a combination of some of the above.

We shall begin by examining the simplest motion of all: Translational motion. For that, we first need to understand the concepts of displacement, speed, velocity and acceleration.

Mass and Weight

Weight is something that we are all familiar with, it is how “heavy” something is. It is related to, but not equal to your mass. e.g. the astronauts on the International Space Station have the same mass as they did on Earth, but not the same weight.

Confusing? Try to understand it this way: mass is a measure of how much “stuff” you are made of, and weight is that “stuff” multiplied by the force of gravity.

Weight = Mass x Gravitational Acceleration
= Gravitational Pull

However, the weighing scales (typical bathroom scales) are calibrated to divide the weight by the gravity (9.8 m/s²), hence they show you your mass. But if you take them on the moon, they won’t show the same value, because they’ll still be dividing your weight by 9.8 whereas they should divide it by (1/6)^th of that value. Hence even though your mass is the same, the value on the weighing scale won’t be.

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Another way to define mass is by the concept of Inertia. Inertia, in the literal sense, means to resist or to stay inert.
e.g. if you try to slide a 1 kg block on a road, it does so easily because its mass is less. But if try to slide a block of 100 kgs, it is going to be very difficult because it has more mass. Hence, when a force is pulling on a body, the amount by which that body resists the change in its motion is its mass. The effect of this force is acceleration, the rate of change of that body’s velocity.

Next: Law of Inertia

Matrices (a cheat sheet)

Vector algebra (a cheat sheet)

Scalars and Vectors

Scalar: A quantity that has only one property: magnitude (or value). For example, the temperature only needs to be specified in terms of its value, such as 350 K or 25 ºC, etc.

Vector: A quantity that has two properties; magnitude and direction. For example, 30 km/h to the West is velocity, a vector quantity.

Watch the video below to learn more (credits-TED-Ed):

Actually, both scalars and vectors are special cases of a general class of quantities, known as tensors. You can understand them in terms of multidimensional arrays, if you are familiar with programming. But if you want an intuitive explanation, watch the excellent video by Dan Fleisch:

More on vectors and vector spaces in a future post.

Units and Measurement

Measurements in Science is the process of obtaining the magnitude of a physical quantity and expressing it relative to a certain basic, arbitrarily chosen, internationally accepted reference standard called UNIT.

We need only a limited number of units to express all the physical quantities because these quantities are dependent on each other. We assign a set for fundamental or base quantities only, the units for these are called the Fundamental/Base units.

All remaining quantities can be derived from the base quantities. The units for these derived quantities are called derived units.

A system of units = Fundamental Units + Derived Units

There are different systems of units e.g. CGS (centimeter, gram and second), MKS(meter, gram and second) etc. However, the most commonly used standard is the SI system.

SI System

7 Base quantities

Base Quantity	Name	Symbol
Length	metre	m
Mass	kilogram	kg
Time	second	s
Electric Current	ampere	A
Thermodynamics Temperature	Kelvin	K
Amount of substance	mole	mol
Luminous Intensity	candela	cd

2 additional quantities

Base Quantity	Name	Symbol
Plane Angle	radian	rad
Solid Angle	steradian	sr

Errors

Degree of uncertainty associated with a measurement.

Result = measured value ± absolute error

Accuracy is a measure of how close the measured value is to the true value of a quantity.
Precision tells us to what resolution or limit the quantity is measured.

Reference Frames

Moving objects change their position over time with respect to a reference frame. By reference frame, we mean a set of directions (axes) and a reference point (origin).

We should select a frame of reference in which Newton’s first law is valid i.e., if no force acts on an object, the object continues its state of uniform motion. Such a reference frame is called inertial frame of reference. The ground is most convenient choice for most of everyday problem analysis.

Usually, the set of axes are perpendicular to each other and coincide at the origin. We call the origin O and the set of axes x, y and z. Once you have the set of axes and the origin in place, you can define the position of anything in space by mentioning its distance from the three axes and the origin. We call them coordinates. This system of is called Cartesian system of coordinates and it is the most popular type of coordinates. Other systems include spherical polar coordinates, cylindrical polar coordinates etc.

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The concept of inertial frame of reference is basically an idealization. In Galilean formulation, a fundamental assumption stands that:

A frame of reference can always be chosen in which space is homogeneous and isotropic and time is homogeneous as well; we call it the inertial frame. There are infinitely many inertial frames moving relatively with constant velocity or oriented differently with respect to each other. All of these have same laws of mechanics and same properties of space and time. Therefore, there is no absolute frame of reference.

-Mechanics: Landau and Lifshitz

Any inertial frame is never unique. For example, a frame A moving with constant velocity v with respect to an inertial frame S, is also an inertial frame of reference.

At least 10 linearly independent transformations (getting a new inertial frame S’ by performing operation on S) S-> S’ hold:

1. 3 rotations: r’ = O r; O is a 3×3 orthogonal matrix
2. 3 translations: r’ = r + c; c is a constant vector
3. 3 boosts: r’ = r + ut; u is constant velocity vector
4. 1 time translation: t’ = t+c; c = some real number

(r: position vector in S, r’: position vector in S’)

If motion is uniform in S, it will also be uniform in S’. These transformations make up the Galilean Group under which Newton’s laws are invariant.

Next: Newton’s laws of Motion

Units and Measurement

Measurements in Science is the process of obtaining the magnitude of a physical quantity and expressing it relative to a certain basic, arbitrarily chosen, internationally accepted reference standard called unit.

We need only a limited number of units to express all the physical quantities because these quantities are dependent on each other. We assign a set for fundamental or base quantities only, the units for these are called the Fundamental/Base units.

All remaining quantities can be derived from the base quantities. The units for these derived quantities are called derived units.

A system of units = Fundamental Units + Derived Units

There are different systems of units e.g. CGS (centimeter, gram and second), MKS (meter, gram and second) etc. However, the most commonly used standard is the SI system.

SI System

7 Base quantities

Base Quantity	Name	Symbol
Length	metre	m
Mass	kilogram	kg
Time	second	s
Electric Current	ampere	A
Thermodynamics Temperature	Kelvin	K
Amount of substance	mole	mol
Luminous Intensity	candela	cd

2 additional quantities

Base Quantity	Name	Symbol
Plane Angle	radian	rad
Solid Angle	steradian	sr

Errors

Degree of uncertainty associated with a measurement.

Result = measured value ± absolute error

Accuracy is a measure of how close the measured value is to the true value of a quantity.
Precision tells us to what resolution or limit the quantity is measured.