Final answer:
The ratio of the centripetal accelerations of two cars moving in circles of different radii, completing the circles in the same time period, is equal to the ratio of the radii of their circular paths.
Step-by-step explanation:
The question deals with the concept of centripetal acceleration in circular motion, a topic covered in high school physics. Given two cars with masses m1 and m2 moving in circles with radii r1 and r2 respectively, and completing their circles in the same time period t, we want to find the ratio of their centripetal accelerations.
The centripetal acceleration is given by the formula ac = v²/r, where v is the tangential speed and r is the radius of the circle. Since both cars complete their circles in the same time, their speeds can be expressed as v1 = 2πr1/t and v2 = 2πr2/t. If we square these speeds and divide each by the corresponding radius, we get the centripetal accelerations a1 and a2.
The ratio of their centripetal accelerations will be:
a1/a2 = (v1²/r1) / (v2²/r2) = ((2πr1/t)²/r1) / ((2πr2/t)²/r2) = (4π²r1/t²) / (4π²r2/t²) = r1 / r2.
Therefore, the ratio of the centripetal accelerations of the two cars is the same as the ratio of the radii of their respective circular paths.