Question

A cyclist is moving on a circular track of radius 80 m with a velocity of 72 km/hr. He has to lean from the vertical approximately through an angle

A. tan⁻¹ (1/4)
B. tan⁻¹ (1)
C. tan⁻¹ (1/2)
D. tan⁻¹ (2)

Answer 1

To find the angle through which a cyclist needs to lean on a circular track, use the formula θ = tan^{-1}(v^2/rg) where v is velocity, r is the radius, and g is gravity. Convert velocity to m/s, then calculate the angle, resulting in tan^{-1}(1/4) as the answer. The correct answer is A. tan⁻¹ (1/4).

Step-by-step explanation:

The question asks about the angle a cyclist needs to lean in to maintain balance while moving in a circular track. To calculate this, we can use the fact that the leaning angle (theta) required for balance on a banked curve without friction is given by the equation θ = tan^{-1}(v^2/rg), where v is the velocity, r is the radius of the circular track, and g is the acceleration due to gravity.

To find the leaning angle:

1. First, convert the velocity from km/hr to m/s: 72 km/hr = 20 m/s.
2. Use the given radius of 80 m and the standard value for g, 9.8 m/s^2.
3. Calculate the angle: θ = tan^{-1}(20^2 / (80 * 9.8)) = tan^{-1}(1/4).

Therefore, the correct answer is A. tan^{-1}(1/4).