Measurement of Length and Motion — NCERT Solutions, AI Tutor & Practice

Understand: Long ago, people measured lengths using body parts such as the handspan, cubit, or foot.

Analyze: The size of a handspan or foot differs from person to person. So if two people measure the same table using their handspans, they get different numbers of handspans.

Conclude: To avoid confusion and to get the same result everywhere, we use standard units (like the metre). A standard unit has a fixed value that does not change from person to person, so measurements can be understood and compared by everyone.

SI unit of length: the metre (m).

Conversions:

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

Into centimetres: $2.5\text{ m} = 2.5 \times 100 = 250\text{ cm}$

Into millimetres: $250\text{ cm} = 250 \times 10 = 2500\text{ mm}$

So $2.5\text{ m} = 250\text{ cm} = 2500\text{ mm}$.

Precautions while measuring with a scale:

1. Place the scale along the length of the object, with markings close to the object.
2. Start measuring from the zero mark of the scale. If the zero (or the end) is broken/worn, start from another full mark (e.g., $1\text{ cm}$) and subtract that value at the end.
3. Keep your eye directly in front (perpendicular) of the point being read, to avoid the parallax error.
4. Note both the readings — at the beginning and the end of the object.

Reading length = end reading − starting reading.

Given:

• Starting reading = $2\text{ cm}$
• End reading = $11\text{ cm}$

Formula: $\text{Length} = \text{End reading} - \text{Starting reading}$

Substitute: $\text{Length} = 11 - 2 = 9\text{ cm}$

Answer: The length of the pencil is $9\text{ cm}$.

Understand: A straight scale cannot bend along a curved line, so we need a flexible tool.

Method using a thread:

1. Take a thread and place one end at the starting point of the curved line.
2. Carefully lay the thread along the curve, following every bend, until you reach the other end.
3. Mark the point on the thread where the curve ends.
4. Now stretch the thread straight and measure the marked length using a scale.

Conclude: The reading on the scale gives the length of the curved line.

Rest: An object is said to be at rest if its position does not change with time (with respect to its surroundings).

Example: A book lying on a table.

Motion: An object is said to be in motion if its position changes with time (with respect to its surroundings).

Example: A car moving on a road.

Note: Rest and motion are relative — they depend on the reference (surroundings) we choose.

Types of motion:

• Translational motion – object moves from one place to another.
• Rotational motion – object turns about a fixed axis.
• Oscillatory (periodic) motion – object repeats its motion to and fro.

Classification:

| Object | Type of motion | |---|---| | (a) Spinning top | Rotational | | (b) Moving train on straight track | Translational (straight/rectilinear) | | (c) Swinging pendulum | Oscillatory (periodic) | | (d) Rotating fan blades | Rotational |

Rectilinear motion: Motion of an object along a straight line.

Example: A car moving on a straight road; a stone falling straight down.

Circular motion: Motion of an object along a circular path (a closed curve).

Example: The tip of a clock's hand; the Moon moving around the Earth.

Key difference: In rectilinear motion the path is a straight line, while in circular motion the path is a circle.

Conversion: $1\text{ km} = 1000\text{ m}$ $350\text{ km} = 350 \times 1000 = 350000\text{ m}$

Answer: $350\text{ km} = 350000\text{ m}$ (or $3.5 \times 10^{5}\text{ m}$).

Why kilometres? For large distances, the value in metres becomes a very big, awkward number. Using kilometres gives a smaller, convenient number that is easier to write and understand. We choose the unit that suits the size of the quantity being measured.

Understand: When the eye is not placed directly in front of the reading point, the position on the scale appears shifted.

Error caused: This is called the parallax error — the reading appears larger or smaller than the true value depending on the viewing angle.

How to avoid it: Keep the eye exactly in front of and perpendicular to the point being measured, so the line of sight is straight down onto the mark.

Conclude: Correct eye position removes the parallax error and gives an accurate reading.

Increasing order (smallest to largest): $\text{millimetre} < \text{centimetre} < \text{metre} < \text{kilometre}$

Relations between consecutive pairs:

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

Useful chain: $1\text{ km} = 1000\text{ m} = 100000\text{ cm} = 1000000\text{ mm}$

Motion of the blades: The blades turn about a fixed axis (the centre of the fan), so they show rotational motion.

Motion of a point on the tip: A point at the tip of a blade traces a circular path, so it is in circular motion.

Is the fan moving from place to place? No. The fan as a whole stays at the same place — it does not undergo translational motion. Only its blades rotate.

Conclusion: The fan shows rotational motion; each tip point moves in a circle, but the fan's position on the ceiling stays fixed.