Calculate speed in circular motion
Scenario
Aapka ek khilona train ek round track par ghoom raha hai. Aapne stop-watch se dekha ki ek round complete karne me usko exact `10 s` lagte hain.
Agar sirf circle ka radius aur time pata ho, toh hum mathematical trick se iski average speed kaise nikal sakte hain?
Mental model
Speed = Distance / Time. Circle me jo distance hota hai, wo uski boundary hoti hai, yani circumference (`2πr`).
Gol raste par ek round pura karne me jo total length aap chalte hain, wahi aapka total distance hota hai. Circle ka perimeter (border) ka formula `2πr` hota hai. Is boundary ki length ko uske lene wale samay (`t`) se divide kar do, you get the speed.
Explanation
Hum pehle padh chuke hain ki speed ka universal formula `Total Distance / Total Time` hota hai. Jab koi object circular path par ek chakkar (revolution) lagata hai, to wo ek perfectly circle ki boundary track par chalta hai. Is mathematical boundary ki total lambaai ko hum Circumference kehte hain.
Geometry se hum jante hain ki radius `r` wale kisi bhi circle ka circumference `2πr` hota hai (yahan `π` ya pi ki value approximate `22/7` ya `3.14` hoti hai). Iska matlab, ek revolution complete karne me object ne bilkul `2πr` ka distance cover kiya hota hai.
Agar is ek chakkar (one revolution) ko poora karne me object ko waqt `T` lagta hai, to uski speed `v` nikalna bohot simple ho jata hai. Formula hai:
Speed = Distance / Time
v = 2πr / T
Ye bohot aasan lagta hai, par board exams isme trick karte hain. Wo raste ka diameter de dete hain, jisse aapko pehle usko half (divide by `2`) karke radius me badalna hota hai. Dusra, wo aapko "5 minutes" time denge, to yaad se usko second (`m/s` speed nikalne ke liye) me convert karna hoga varna answer wrong ho jayega. Calculate karte waqt `π` ko fraction (`22/7`) rakhna zyada aasan hota hai agar `r`, `7` ka koi multiple ho.
Worked example
Problem: An athlete completes one round of a circular track of radius `70 m` in `20 s`. Find his speed.
- Pehle 1 round ka distance (circumference) nikalein. Take `π = 22/7`.
- Calculate it.
- Speed ke liye us distance ko time (`20 s`) se divide karein.
Answer: `22 m/s`
Problem: A communication satellite orbits the Earth at a radius of `42000 km`, taking `24 h` to complete one revolution. What is its speed in `km/h`?
- Radius aur time diya gaya hai.
- Circular orbit ka total distance `2πr` hota hai.
- `π` ki value (`3.14`) rakh kar solve karenge.
Answer: `10995 km/h`
Key points
- Distance in 1 revolution: The distance covered in one full circle is equal to its circumference, which is `2πr`.
- Speed Formula: Speed `v = 2πr / T` (where `r` is radius, `T` is time for one revolution).
- Watch for Diameter: If the problem gives diameter `d`, remember to divide by `2` to get the radius `r` before applying the formula.
Common mistakes
- Using diameter instead of radius: Kabhi kabhi questions directly diameter (`d`) dete hain (jaise `100 m`). Bache galti se formula me `r = 100` daal kar `2π × 100` kar dete hain. Hamesha dhyan do radius diya hai ya diameter.
- Not converting time: Agar time minutes ya hours me diya hai (like satellites taking `24 hours`), aur baccha `2πr` ko seedhe `24` se divide kar deta hai bina usko unit balance kiye, to answers galat aate hain.
Recall questions
- What is the formula to calculate the speed of a body in uniform circular motion?
- What does `2πr` represent in the context of circular motion?
- What is the SI unit of speed?
Questions & answers
An artificial satellite is moving in a circular orbit of radius `42250 km`. Calculate its speed if it takes `24 hours` to revolve around the earth.
`3.07 km/s` (or `11065.4 km/h`)
Approach: `r = 42250 km`. `t = 24 h`. Distance = `2πr` = `2 × (22/7) × 42250` ≈ `265571 km`. Speed = `265571 / 24` ≈ `11065 km/h`. Divide by `3600` to get `km/s`.
A cyclist goes around a circular track once every `2 minutes`. If the radius of the circular track is `105 m`, calculate his speed.
`5.5 m/s`
Approach: `r = 105 m`. `t = 120 s`. `v = (2 × (22/7) × 105) / 120`. `v = (2 × 22 × 15) / 120` = `660 / 120` = `5.5 m/s`.