Roller Coaster Safety in Rotation?

📌 CONCEPT: When the angular velocity of a rotating system decreases while the linear velocity remains the same, the force of friction acting on each passenger also decreases.

📐 RULE / FORMULA: The force of friction (F) is directly proportional to the centripetal acceleration (ac) of the passengers, which is given by the formula F ∝ ac = (mv^2)/r, where m is the mass of the passenger, v is the linear velocity, and r is the radius of the circular path.

💡 WORKED EXAMPLE: Suppose the original angular velocity is ω = 5 m/s and the radius of the circular path is r = 10 m. If the linear velocity remains the same and the angular velocity decreases to half its original value, the new angular velocity is ω' = (1/2)ω. Using the formula F ∝ ac, we can see that the force of friction acting on each passenger decreases since ac ∝ (v'^2)/r, where v' = (1/2)v is the new linear velocity.

⚠️ COMMON MISTAKE: Students may incorrectly assume that the force of friction remains constant even if the angular velocity decreases, ignoring the effect of the reduced centripetal acceleration on the frictional force.