What this chapter is about
This chapter explores the mathematics of motion along straight lines and circular paths. The title hints at two kinds of movement: going up and down (linear or vertical motion) and going round and round (circular or rotational motion). You will learn how to describe, measure and calculate quantities related to these motions using coordinate geometry and basic algebraic relationships.
At the Class 9 level, you connect your knowledge of the number line and the Cartesian plane to real-world situations where objects move. This includes understanding how position changes with time, how to measure distances along curved paths like circles, and how angles relate to the arc length travelled. These ideas form the foundation for physics concepts you will meet soon.
After studying this chapter, you should be able to represent motion on a coordinate system, calculate the distance covered in circular motion, relate the circumference of a circle to the radius, and solve problems involving periodic or repeating motion.
Key ideas
- Position on a line: Any point moving up and down along a vertical line can be described by a single coordinate that increases upward and decreases downward.
- The coordinate plane for motion: When an object moves in two dimensions, its position at any moment is given by an ordered pair (x, y), allowing you to track its path.
- Circumference of a circle: The distance around a circle is given by C = 2πr, where r is the radius; this tells you how far an object travels in one complete round.
- Arc length: A portion of the circle's boundary has length proportional to the angle it subtends at the centre; for an angle θ in degrees, arc length = (θ/360) × 2πr.
- Periodic motion: Motion that repeats after a fixed interval (like a point on a rotating wheel) can be described by its period (time for one complete cycle) and frequency (cycles per unit time).
- Relating linear and circular measures: When a wheel rolls without slipping, the distance it covers on the ground equals the arc length traced, linking straight-line distance to rotation.
Formulas and facts to remember
- Circumference of a circle: C = 2πr — the total distance around a circle of radius r.
- Area of a circle: A = πr² — useful when comparing circular regions.
- Arc length: Arc length = (θ/360) × 2πr — the length of an arc that subtends angle θ degrees at the centre.
- Relation between diameter and circumference: C = πd, where d = 2r is the diameter.
- One complete rotation: corresponds to 360° or an arc length equal to the full circumference.
- Value of π: approximately 22/7 or 3.14 for calculations.
Worked examples
Example 1: Distance in one rotation
A bicycle wheel has a radius of 35 cm. How far does the bicycle travel when the wheel makes one complete turn?
Solution: Distance in one turn = circumference = 2πr = 2 × (22/7) × 35 cm = 2 × 22 × 5 cm = 220 cm = 2.2 m
The bicycle moves 2.2 metres forward in one rotation of the wheel.
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Example 2: Arc length for a given angle
A fan blade is 40 cm long. When the fan rotates, the tip of the blade sweeps through an angle of 90°. What is the distance travelled by the tip?
Solution: Here, r = 40 cm and θ = 90°. Arc length = (θ/360) × 2πr = (90/360) × 2 × (22/7) × 40 = (1/4) × 2 × (22/7) × 40 = (1/4) × 1760/7 = 440/7 ≈ 62.86 cm
The tip travels approximately 62.86 cm.
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Example 3: Number of rotations to cover a distance
A circular track has a radius of 21 m. Meera runs around the track and covers a total distance of 660 m. How many complete rounds did she run?
Solution: Circumference of track = 2πr = 2 × (22/7) × 21 = 132 m Number of complete rounds = Total distance ÷ Circumference = 660 ÷ 132 = 5
Meera completed 5 rounds of the track.
Common mistakes
- Confusing radius with diameter when using the circumference formula → remember C = 2πr uses radius, or C = πd uses diameter; do not mix them.
- Forgetting to convert the angle to a fraction of 360° when finding arc length → always divide the given angle by 360 before multiplying by circumference.
- Using π = 3 instead of 22/7 or 3.14, leading to large errors → use the value specified or 22/7 for accuracy.
- Mixing up circumference (a length) with area (a region) → circumference is measured in metres, area in square metres; keep units clear.
- Not converting units before calculation, such as mixing cm and m → convert all measurements to the same unit first.
Quick revision
- Circumference = 2πr; this is the distance for one full rotation.
- Arc length = (angle/360) × circumference.
- When a wheel rolls, distance on ground = number of rotations × circumference.
- π ≈ 22/7 or 3.14; use whichever gives simpler arithmetic.
- One complete turn = 360°; half turn = 180°; quarter turn = 90°.