Chapter Notes

Measurement of Length and Motion

Deepa, an eleven-year-old girl, needs a new uniform because she has grown taller. Her mother takes her to a cloth shop and asks for a two-metre cloth piece. The shopkeeper measures the cloth using a metal measuring rod. Later, a tailor takes Deepa's measurements with a flexible measuring tape. Her mother asks the tailor to increase the uniform's length by "char angula" (four fingers width). This experience leads Deepa and her friends to discuss different ways of measuring.

How do we Measure?

Historically, people have used various body parts to measure length. For example, a grandmother might measure cloth by the length of her arm, or a farmer might measure a field by counting his strides or using the length of his feet.

Deepa and her friends decide to measure the length of their classroom table using their handspan. A handspan is the distance from the tip of the thumb to the tip of the little finger when the hand is fully extended. Deepa's mother calls this "balisht".

When they measure the table, they find that the number of handspans is different for each student. This occurs because their handspans are of different sizes. For instance:

• Anish: Slightly more than $13$ handspans
• Padma: $13$ handspans
• Tasneem: Slightly less than $13$ handspans
• Deepa: Between $13$ and $14$ handspans
• Hardeep: $14$ handspans

This variation in measurements highlights the problem with using body parts as units: they differ from person to person.

When we express the length of the table as, say, $13$ handspans, "$13$" is the number and "handspan" is the unit selected for the measurement. A unit is a standard quantity used for measurement. Because body parts like handspans, arm lengths, or foot lengths vary between individuals, there is a need for a standard unit that ensures consistent measurements of the same length, regardless of who is measuring.

Standard Units

Over time, different parts of the world developed various systems of units. However, as people began to travel and trade, these different units led to confusion. To address this, countries came together and adopted a universal set of standard units of measurement. This system is known as the International System of Units or SI units.

The SI unit of length is the metre, and its symbol is $\text{m}$. A metre is divided into $100$ equal parts, and each part is called a centimetre ($\text{cm}$). A $15$-$\text{cm}$ scale typically shows markings from $0$ to $15$. Each section of $1 \text{ cm}$ length is further divided into $10$ equal parts. The length of one of these smaller parts is called a millimetre ($\text{mm}$). Therefore:

• $1 \text{ m} = 100 \text{ cm}$
• $1 \text{ cm} = 10 \text{ mm}$
• $1 \text{ mm} = 0.1 \text{ cm}$ (one-tenth of a centimetre)

The millimetre is the smallest value of length that can typically be measured using a standard $15$-$\text{cm}$ scale.

For measuring larger lengths, such as the distance between two cities or the length of a railway track, a larger unit called a kilometre ($\text{km}$) is used.

• $1 \text{ km} = 1000 \text{ m}$

It is important to choose an appropriate unit based on the length being measured. For example, kilometres are used for large distances, metres for moderate lengths (like a room's height), and centimetres or millimetres for smaller lengths (like the thickness of a book page).

When writing lengths, remember these conventions:

• Units of length (kilometre, metre, centimetre, millimetre) begin with a lowercase letter, unless at the start of a sentence.
• Their symbols ($\text{km}$, $\text{m}$, $\text{cm}$, $\text{mm}$) are always written in lowercase letters.
• Symbols are never followed by 's' for the plural (e.g., $5 \text{ m}$, not $5 \text{ ms}$).
• A full stop is not written after the symbol, except if it's at the end of a sentence.
• Always leave a space between the number and the unit (e.g., $10 \text{ cm}$, not $10\text{cm}$).

Correct Way of Measuring Length

Accurate measurement requires selecting the right tool and using it correctly.

• Appropriate Scale: For a pencil, a $15$-$\text{cm}$ scale is suitable. For a room's height, a metre scale or measuring tape is needed. For curved surfaces like the girth of a tree or chest size, a flexible measuring tape (like a tailor's tape) is more appropriate.

When measuring lengths, specific techniques ensure accuracy:

• Correct Placement of the Scale: The scale must be placed directly in contact with the object along its entire length.

• Correct Position of the Eye: To avoid errors caused by parallax, your eye should be positioned directly above the mark you are reading on the scale. If your eye is to the side, the reading will appear different.

• Measuring with Broken Scale Ends: If the zero mark on a scale is worn out or the end is broken, you can still use it. Start measuring from any other clear full mark, for example, $1.0 \text{ cm}$. Then, subtract this initial reading from the final reading at the other end of the object. For instance, if one end aligns with $1.0 \text{ cm}$ and the other with $10.4 \text{ cm}$, the length of the object is $10.4 \text{ cm} - 1.0 \text{ cm} = 9.4 \text{ cm}$.

When recording a length, it is crucial to include both the numerical value and the unit of measurement. For example, if a pencil is $10.5 \text{ cm}$ long, writing just "$10.5$" is incomplete; it must be "$10.5 \text{ cm}$".

Measuring the Length of a Curved Line

Measuring a curved line directly with a rigid scale is difficult. For such measurements, a flexible measuring tape can be used. Alternatively, a simple method involves using a thread:

1. Place a thread along the curved line, carefully tracing its shape from one end to the other.
2. Mark the starting and ending points on the thread.
3. Straighten the thread.
4. Measure the length of the thread between the marked points using a metre scale. This length will be the length of the curved line.

Describing Position

The perceived distance of an object can vary depending on the observer's starting point. For example, if Deepa and her friends discuss the distance to a garden, their individual observations might differ because they are measuring from their respective houses. To make observations consistent, everyone must measure distances from a common, fixed point.

A reference point is a fixed object or point with respect to which a distance or position is stated. For instance, when drawing lines for a Kabaddi court, students must first agree on a reference point on the ground from which all measurements will start. Similarly, kilometre stones on a road, which might read "Delhi $70 \text{ km}$" or "Delhi $60 \text{ km}$", indicate the distance from Delhi. In this case, Delhi serves as the reference point, and the numbers tell you how far you are from it.

Moving Things

An object's state of motion (moving or at rest) is always relative to a chosen reference point and time.

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

The choice of reference point is crucial. For example, if Deepa is on a bus, the passengers seated next to her appear to be at rest if she considers herself or the bus as the reference point. However, if she looks outside and considers a building as the reference point, then the passengers (along with the bus) are in motion because their position is changing relative to the building.

Types of Motion

Objects exhibit various types of motion based on their path.

• Linear Motion: When an object moves along a straight line, its motion is called linear motion. [!example]

  • An eraser dropped from a height moves in a straight line downwards.
  • An orange falling from a tree moves in a straight line.
  • Students marching during a parade move along a straight path.
  • A heavy box pushed across a floor may move in a straight line.

• Circular Motion: When an object moves along a circular path, its motion is called circular motion. [!example]

  • Whirling an eraser tied to a thread in a circle.
  • The motion of children on a merry-go-round.

• Oscillatory Motion: When an object moves to and fro (back and forth) about some fixed position, its motion is called oscillatory motion. This is a repetitive motion around a central point. [!example]

  • The motion of a swing.
  • An eraser hanging by a thread, pulled to one side and released, moves back and forth.
  • A thin metal strip, held at one end and pressed slightly at the other end, moves up and down when released.

• Periodic Motion: If an object repeats its path after a fixed interval of time, its motion is said to be periodic. Both circular motion and oscillatory motion can be periodic in nature if they repeat consistently over time.

• Swing: Oscillatory motion (moving to and fro).
• Slide: Linear motion (sliding down a straight path).
• Merry-go-round: Circular motion.

Keywords

• centimetre
• Circular motion
• Classify
• Distance
• Explore
• Identify
• Investigate
• Justification
• kilometre
• Length
• Linear motion
• Measurement
• metre
• millimetre
• Motion
• Observe
• Oscillatory motion
• Reference point
• SI Unit of Length

Summary

• The International System of Units (SI units) has been adopted globally as standard units of measurement to ensure consistency.
• The SI unit of length is the metre, symbolised as $\text{m}$.
• Key conversions for length units are:
  • $1 \text{ km} = 1000 \text{ m}$
  • $1 \text{ m} = 100 \text{ cm}$
  • $1 \text{ cm} = 10 \text{ mm}$
• A reference point is a fixed object or point used to state the distance or position of another object.
• An object is in motion if its position changes with respect to a reference point over time.
• An object is at rest if its position does not change with respect to a reference point over time.
• Linear motion occurs when an object moves along a straight line.
• Circular motion occurs when an object moves along a circular path.
• Oscillatory motion occurs when an object moves to and fro about a fixed position.