Savchenko Problem 1.3.29
Physics · mechanics
Problem
To save space, the entrance to one of the highest bridges in Japan is arranged in the form of a spiral line that wraps around a cylinder of radius R. The roadbed forms an angle with the horizontal plane. What is the acceleration of a car moving along it at a constant modulo velocity v?Intuitive explanation
The key insight is that even though the car's speed is constant, it's continuously changing direction as it spirals upward—this means it has acceleration. You need to find the net acceleration by combining two effects: the centripetal acceleration from circular motion (pointing toward the cylinder's axis) and the component of gravitational acceleration along the spiral's path. The tricky part is recognizing that only the horizontal component of velocity contributes to the centripetal term, while gravity acts vertically throughout.
Intuitive explanation
The key insight is that even though the car's speed is constant, it's continuously changing direction as it spirals upward—this means it has acceleration. You need to find the net acceleration by combining two effects: the centripetal acceleration from circular motion (pointing toward the cylinder's axis) and the component of gravitational acceleration along the spiral's path. The tricky part is recognizing that only the horizontal component of velocity contributes to the centripetal term, while gravity acts vertically throughout.