A vehicle moving on a circular track whose surface is inclined towards the horizon at an angle of π/4. The maximum velocity with which it can move safely is 36 km/hr. Calculate the length of the circular track.
Answers
Solution : Here, angle θ = π/4 = 180/4 = 45⁰,
Maximum velocity of the vehicle, v(max) = 36 km/hr.
Now,
➜ v(max) = 36 km/hr
➜ v(max) = 36 * (5/18)
➜ v(max) = (36 * 5)/18
➜ v(max) = 180/18
➜ v(max) = 10 m/s
Now, by formula of banked road;
➜ tan θ = {v(max)}²/rg
Where, r is radius and g is acceleration due to gravity.
➜ tan 45⁰ = {10}²/(r * 9.8)
➜ 1 = 100/9.8r
➜ r = 100/9.8 m
Now, by formula circumference of the circle;
➜ The length of the circular track = 2πr
➜ The length of the circular track = 2 * (22/7) * (100/9.8)
➜ The length of the circular track = (2 * 22)/7 * (100/9.8)
➜ The length of the circular track = (44/7) * (100/9.8)
➜ The length of the circular track = (44 * 100)/(7 * 9.8)
➜ The length of the circular track = 4400/68.6
➜ The length of the circular track = 64.1399 m
Answer : Hence, the length of the circular track is 64.1399 m.
[Note : 1 km/hr = (1000 m)/(3600 sec) = 5/18 m/s]