An automobile is accelerating at \(5 \mathrm{~m} / \mathrm{s}^{2}\) on a straight road on a hill where the radius of curvature of the hill is \(200 \mathrm{~m}\). What is the magnitude of the total acceleration when the car's speed is \(30 \mathrm{~m} / \mathrm{s}\) ? a) \(5 \mathrm{~m} / \mathrm{s}^{2}\) c) \(6.04 \mathrm{~m} / \mathrm{s}^{2}\) b) \(5.46 \mathrm{~m} / \mathrm{s}^{2}\) d) \(6.73 \mathrm{~m} / \mathrm{s}^{2}\)

Centripetal acceleration is crucial in the study of mechanics, especially when we observe objects moving in a circular path. It's the acceleration that keeps an object moving along a curved path and is always directed towards the center of the curvature of the path. In our automobile example, the car is moving on a hill with a radius of curvature known as the radius of the circular path it follows.

Using the formula for centripetal acceleration, which is \(a_c = \frac{v^2}{r}\), where \(v\) is the speed of the vehicle and \(r\) is the radius of curvature, we calculated that the centripetal acceleration is \(4.5\,\text{m/s}^2\) for the given speed of \(30\,\text{m/s}\) and a radius of \(200\,\text{m}\). This acceleration component reflects how much the automobile is accelerating towards the center of its circular path at any given instant. Moreover, understanding centripetal acceleration is important for designing safe roads, roller coasters, and for analyzing circular motion in various physical scenarios.

While centripetal acceleration is directed towards the center of a circular path, tangential acceleration pertains to the change in speed along the path of motion itself. If an object is moving in a circular path and speeding up or slowing down, it experiences tangential acceleration in addition to centripetal acceleration.

In our example, the tangential acceleration is given directly as \(5\,\text{m/s}^2\). This represents the rate at which the car's speed is increasing as it moves up the hill. It's important to understand that tangential acceleration contributes to the car's total acceleration as it accounts for changes in the magnitude of velocity, whereas centripetal acceleration is associated with changes in the direction of velocity.

The Pythagorean theorem is a cornerstone not only in mathematics but also in physics, particularly when resolving vector quantities into components. In our context, the total acceleration of an object undergoing circular motion can be found by treating the centripetal and tangential accelerations as perpendicular vector components.

Applying the Pythagorean theorem, \(a_{total} = \sqrt{a_t^2 + a_c^2}\), we combined the tangential acceleration (\(5\,\text{m/s}^2\)) and centripetal acceleration (\(4.5\,\text{m/s}^2\)) to yield the total acceleration of \(6.73\,\text{m/s}^2\). Therefore, utilizing the Pythagorean theorem allows us to effectively calculate the magnitude of total acceleration in scenarios where two perpendicular acceleration components are present. This concept is widely applicable in diverse areas of physical analysis, from simple mechanical systems to complex dynamics of celestial bodies.