This chapter explores how we measure the length of objects and understand the movement of things around us. It begins with a scenario involving measuring cloth for a school uniform, highlighting different ways measurements are taken.

How Do We Measure?

Traditional Units

Historically, people used various parts of their body or familiar objects to measure lengths. Examples of these traditional or non-standard units include:

• Angula: The width of a finger.
• Handspan (Balisht): The distance between the tip of the thumb and the tip of the little finger when the hand is fully stretched.
• Length of the arm.
• Length of the foot.
• Stride: The length of a step while walking.

An activity measuring a table using handspans shows that different people get different results (a different number of handspans for the same table) because their handspans are of different sizes.

Need For Standard Units

In any measurement, there are two parts: the number and the unit used for measurement. For example, if a table is measured as 13 handspans, '13' is the number, and 'handspan' is the unit.

However, because traditional units like handspans or angula vary from person to person, measurements of the same length made by different individuals will be different. This variability creates confusion and inconsistency, especially when people need to trade or communicate measurements globally.

This inconsistency highlighted the need for a system of measurement where the unit is fixed and does not change from person to person or place to place. This led to the development and adoption of standard units.

Extra Information: India has a long history of measurement systems. Ancient literature mentions units like `angula`, `dhanusa`, and `yojana`. Scales with markings have been found in archaeological sites of the Harappan Civilisation, suggesting organized measurement practices.

Standard Units

International System Of Units (Si Units)

To ensure consistency and avoid confusion in measurement worldwide, different countries agreed to adopt a common set of standard units. This system is known as the International System of Units, abbreviated as SI units.

Metre, Centimetre, Millimetre

The standard SI unit for measuring length is the metre. Its symbol is m.

A metre scale is a common tool for measuring length. A metre is divided into smaller, equal parts:

• One metre (1 m) is divided into 100 equal divisions. Each division is called a centimetre (symbol cm).
• Each centimetre (1 cm) is further divided into 10 equal parts. Each of these smaller parts is called a millimetre (symbol mm).

The millimetre is the smallest unit of length typically marked on a standard small scale (like a 15 cm ruler).

Kilometre

For measuring very large distances, such as the distance between two cities or the length of a road, a larger unit is used. This unit is the kilometre (symbol km).

Conversion Between Units

The relationships between these standard units are fixed:

• $1 \text{ km} = 1000 \text{ m}$
• $1 \text{ m} = 100 \text{ cm}$
• $1 \text{ cm} = 10 \text{ mm}$

Using standard units like metres, centimetres, or kilometres ensures that a measurement of a specific length will be the same regardless of who makes the measurement.

We choose the unit that is most convenient for the length being measured. For example, measuring the thickness of a book page in kilometres would be impractical. Millimetres or centimetres are more suitable for small lengths, metres for medium lengths (like room height), and kilometres for long distances.

Non-Standard Units In History

Extra Information: Besides the SI units, some older systems of units, such as the inch and foot, were widely used historically and are still used by some people or in certain contexts. For reference, $1 \text{ inch} \approx 2.54 \text{ cm}$.

Note on Writing Units: When writing lengths with standard units, follow these conventions:

• Unit symbols (km, m, cm, mm) are written in lowercase letters.
• Unit symbols are never followed by an 's' to indicate plural (e.g., write 10 m, not 10 ms).
• A full stop is not used after unit symbols unless it's at the end of a sentence.
• Always leave a space between the number and the unit symbol (e.g., 10.4 cm, not 10.4cm).

Correct Way Of Measuring Length

Accurate measurement requires using the measuring tool correctly.

Choosing The Appropriate Scale

Select a measuring instrument suitable for the object and length being measured. For a small object like a pencil, a 15-cm scale is fine. For taller objects or rooms, a metre scale or a long measuring tape is needed. Flexible objects like a tailor's measuring tape are necessary for measuring curved lengths or girths (like around a tree or chest).

Proper Placement Of The Scale

To measure correctly, the scale must be placed in contact with the object and aligned along the length that needs to be measured.

Correct Eye Position

The position of your eye while reading a scale is critical to avoid errors caused by parallax. Your eye should be positioned directly above the mark you are reading. Reading from an angle will give an incorrect measurement.

Measuring With Broken Scales

If the zero marking at the beginning of your scale is damaged or unclear, you can still use the scale to measure. Simply start measuring from any other clear full mark on the scale (e.g., the 1.0 cm mark). Read the measurement at the other end of the object and then subtract the initial reading from the final reading to find the object's length.

For example, if you start from 1.0 cm and the object ends at 10.4 cm, the actual length is $10.4 \text{ cm} - 1.0 \text{ cm} = 9.4 \text{ cm}$.

Extra Information: Scales with raised markings that can be felt by touch are available for visually challenged people to measure lengths.

Measuring The Length Of A Curved Line

Measuring a straight line is simple using a rigid scale. However, measuring a curved line, like the edge of a leaf or a decorated archway, requires different techniques.

Using Flexible Tools

A flexible measuring tape, such as a tailor's tape or a flexible ruler, can be carefully placed along the curved line to measure its length directly.

Method With A Thread

Another method is to use a piece of thread:

1. Place a knot or mark at one end of the thread.
2. Carefully position the thread along the curved line, following its shape closely. You can use your fingers to hold it in place at small intervals.
3. Once the thread has followed the entire curve, mark the point on the thread that corresponds to the end of the curved line.
4. Finally, straighten the thread and measure the length between the initial knot/mark and the final mark using a standard metre scale or ruler. This length of the thread is equal to the length of the curved line.

Describing Position

To locate or describe where something is, we need a point of reference.

Position Relative To A Point

Discussing distances (like whether the garden is closer than the school) highlights that describing a location often involves stating its position or distance relative to another point.

If different people use different starting points (like their individual houses), their descriptions of which place is "closer" might differ, even if the actual locations are fixed.

The Reference Point

To ensure everyone agrees on a position or distance, we use a fixed object or point as a basis. This fixed point is called a reference point.

When we state the position or distance of an object, it is usually understood to be with respect to a particular reference point. For instance, drawing lines for a sports court involves deciding on a starting point, which acts as the reference point for all subsequent measurements.

Position Changes With Time

Consider kilometre stones on a road. A stone marked 'Delhi 70 km' tells you your distance from Delhi. As you travel towards Delhi, the next stone might read 'Delhi 60 km'.

In this example, Delhi is the reference point. Your position is changing relative to Delhi over time (from 70 km away to 60 km away). This change in position with respect to a reference point is key to understanding motion.

Moving Things

Identifying Motion And Rest

We see many things around us – some are still, and some are moving. We need a way to clearly define when an object is moving or not.

Listing objects seen as 'in motion' or 'at rest' and explaining the reasoning helps understand how we make this distinction.

Objects in motion	Justification	Objects at rest	Justification
Cow grazing in the field	Its location on the field is changing over time.	Tree	Its location is not changing over time.
Car driving on the road	Its position relative to buildings or poles along the road is changing.	Building	Its position relative to the ground or nearby objects is not changing.
Bird flying in the sky	Its location in the sky is changing.	Bench in a park	Its position on the ground is not changing.

Definition Of Motion And Rest

The decision of whether an object is in motion or at rest depends on whether its position is changing relative to a specific point or object.

• An object is said to be in motion if its position changes with respect to a reference point over time.
• An object is said to be at rest if its position does not change with respect to a reference point over time.

Importance Of Reference Point

The concept of a reference point is essential for describing motion. For example, passengers sitting inside a moving bus are at rest *relative to* the bus itself (their position inside the bus doesn't change). However, the same passengers are in motion *relative to* objects outside the bus, like buildings or trees (their position relative to these external objects is constantly changing as the bus moves).

Therefore, to accurately state whether something is in motion or at rest, you must specify the reference point being used.

Think it over: If you are inside a ship moving at a constant speed in a straight line on calm water, and there are no windows, it would be difficult to tell if the ship is moving or stationary *without an external reference*. Only by observing something outside the ship or feeling external forces (like acceleration or deceleration) could you determine its motion relative to the water or land.

Types Of Motion

Objects can move in different ways or follow different paths. These different ways of moving are classified into types of motion.

Linear Motion

When an object moves along a straight line, its motion is called linear motion or rectilinear motion.

Examples:

• An object dropped from a height (ignoring air resistance, falls along a straight line).
• Soldiers marching in a parade.
• A car moving on a straight road.
• Pushing a heavy box across a floor in a straight path.

Circular Motion

When an object moves along a circular path, its motion is called circular motion.

In circular motion, the object's distance from a fixed point (the center of the circle) remains constant while it moves.

Examples:

• Whirling an object (like an eraser or potato) tied to a thread around your hand. Your hand acts as the fixed center.
• The motion of a merry-go-round.
• The motion of the hands of a clock.
• The motion of a satellite around the Earth.

Oscillatory Motion

When an object moves to and fro repeatedly about a fixed central position, its motion is called oscillatory motion.

Examples:

• The motion of a swing in a park.
• The motion of a pendulum of a clock.
• The vibrating string of a guitar.
• The motion of a stretched metal strip when released after being slightly pressed.

Periodic Motion

Some motions repeat themselves after a definite or fixed interval of time. This type of motion is called periodic motion.

Both circular motion and oscillatory motion can be periodic if the object repeats its path or to-and-fro movement at regular intervals. For example, the Earth's motion around the Sun (circular and periodic), the motion of a clock pendulum (oscillatory and periodic).

Linear motion is generally not periodic unless it is back and forth along the same straight line at regular intervals (like the piston in an engine cylinder).

Observing activities and objects in a children's park allows for identifying examples of these different types of motion:

Object/Activity	Linear motion	Circular motion	Oscillatory motion	Justification
Swing			Yes	Moves back and forth about a central position.
Merry-go-round		Yes		Children move along a circular path.
Slide	Yes			Movement down the slide is along a straight or nearly straight line.
Child running	Yes (if running in a straight line)	Yes (if running in a circle)		Depends on the path taken; can be linear or circular.

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