To determine the angle θ that a bicyclist must tilt their bicycle to make a circular turn with a radius (r) of 33 meters while traveling at a speed (v) of 14 meters per second, we use principles of centripetal force and friction.
The total force exerted on the bicycle by the ground, a combination of normal force and static friction, aligns with the bicycle's direction.
Using the centripetal force equation, ( F_{\text{centripetal}} = \frac{{m \cdot v^2}}{{r}} ), we equate it to the maximum static friction force, ( F_{\text{friction}} = \mu \cdot F_{\text{normal}} ), where ( \mu ) is the coefficient of static friction.
We calculate the normal force as ( F_{\text{normal}} = \frac{{m \cdot v^2}}{{0.5 \cdot r}} ), given ( m = 86.4 ) kg, ( v = 14 ) m/s, and ( \mu = 0.5 ).
With the normal force determined, we find the tangent of the angle θ as the ratio of the frictional force to the normal force: ( \tan(θ) = \frac{{0.5 \cdot F_{\text{normal}}}}{{F_{\text{normal}}}} = 0.5 ). Taking the arctangent of both sides yields ( θ ≈ 26.565^\circ ).
In summary, the bicyclist needs to tilt their bicycle approximately ( 26.565^\circ ) from the vertical to maintain balance while making the circular turn.
This calculation ensures that the frictional force, which aids in turning, remains within the maximum static friction limit, providing the necessary centripetal force for the turn without slipping.
