Question

A carnival Ferris wheel has a \(15-\mathrm{m}\) radius and completes five turns about its horizontal axis every minute. (a) What is the acceleration, magnitude and direction, of a passenger at the highest point? (b) What is the acceleration at the lowest point? ( \(c\) ) What force (magnitude and direction) must the Ferris wheel exert on a \(75-\mathrm{kg}\) person at the highest point and at the lowest point?

Short answer

The acceleration at the highest point is 0.82 m/s² downward, and at the lowest point is 20.6 m/s² upward. The force the Ferris wheel must exert on a 75 kg person at the highest point is 682 N downward, and at the lowest point is 1548 N upward.

Step-by-step solution

Calculate the frequency and angular speed

First we need to calculate the frequency of the Ferris wheel, which is the number of turns per minute. The wheel completes 5 turns per minute, so the frequency \(f\) is 5/60 revolutions/s. We then convert this frequency into angular speed (in radian/s) using the formula \(\omega = 2\pi f\). Plug in the calculated frequency value to find \(\omega\).

Calculate the acceleration at the highest point

Next, we use the equation for centripetal acceleration \(a = \omega^2 \times r\), where \(r\) is the radius of the ferris wheel, and substitute \(\omega\) and \(r\) into it. Since the passenger is moving in a circular path, however is momentarily at rest at the highest point, the only acceleration acting on them will be the gravity acceleration \(g = 9.8 \, m/s^2 \). Therefore, at this point the total acceleration becomes the vector sum of the gravity and the centripetal acceleration: \(a = sqrt(g^2 + a_c^2)\).

Calculate the acceleration at the lowest point

At the lowest point, the magnitudes of both gravitational and centripetal acceleration add together. Hence the total acceleration at this point is \(a = g + a_c\).

Calculate the force at the highest and lowest point

Finally, we calculate the force the Ferris wheel must exert on a person using the equation \(F = ma\), where \(m\) is the mass of the person. This is done separately for highest and lowest point. At the highest point, the force exerted by the Ferris Wheel is \(F_h = m * g - m * a_{c}\) since ferris wheel force has to compensate for the centripetal force. While at the lowest point, Ferris Wheel has to supply the extra force for centripetal motion in addition to normal force due to gravity, hence, \(F_l = m * g + m * a_{c}\)

Key concepts

Centripetal Acceleration Explained

Imagine you're on a Ferris wheel, clinging to the edges of your seat. As you whirl around, you're pushed into the seat. What's going on here is a classic case of centripetal acceleration. This type of acceleration is what keeps you moving along a circular path instead of shooting off in a straight line. Mathematically, it's defined as \( a_c = \omega^2 \times r \), where \( a_c \) is the centripetal acceleration, \( \omega \) is the angular speed, and \( r \) is the radius of the circle.

In our Ferris wheel example, the radius is given as 15 meters, and you completed this circular path five times in one minute. These numbers let us compute the angular speed—how fast you're turning—and from there, we discover the centripetal acceleration at any point of your ride. Interestingly enough, at the highest point of the Ferris wheel, the force you feel comes from the seat pushing against you, providing the centripetal force needed to keep you moving in a circle, essentially preventing you from 'falling' outwards from the wheel's rotation.

Connecting Centripetal Acceleration and Force

However, what does this mean in terms of force? Using Newton's second law, \( F = ma \), the centripetal force exerted by the Ferris wheel on you can be calculated. If you have a mass of 75 kg, like in our original problem, this force is critical to keep you on that circular path. At the highest and the lowest points, this centripetal force is different due to the direction of your gravitational force, which affects the net force you feel.

Angular Speed Unraveled

Angular speed, often represented by \( \omega \), is a measure of how quickly an object rotates or revolves around a central point. In less jargon-heavy terms: it's the spin rate. For a Ferris wheel, angular speed tells us how fast each seat is moving along the circular path. We derive it from the frequency of rotation. With the given information that the Ferris wheel makes five full turns every minute, we calculate the frequency, \( f \), which is the number of cycles per second. By the equation \( \omega = 2\pi f \), we find our Ferris wheel's angular speed.

From Angular Speed to Real-World Feelings

Ever felt that lurch in your stomach as the wheel starts turning? That's the angular speed having very real consequences. It determines the 'oomph' of the centripetal force you experience. The faster the angular speed, the stronger the push into your Ferris wheel seat. It's vital for solving the physics of Ferris wheels because it links how fast the wheel spins to the forces at play, which affect everything from the excitement of your ride to the structural design of the wheel itself.

The Whirl of Circular Motion

Circular motion is all about objects moving in circles, like cars on racetracks or the Earth orbiting the Sun. When we bring this into the frame of a Ferris wheel, each passenger is undergoing circular motion, constantly changing direction but not leaving the circular path. It's this motion that creates the thrill of the ride.

What's Keeping You in Motion?

While you might suspect it's magic keeping you from flying off into the sky, it is, in fact, the result of physics—specifically, the balance between gravitational forces and the centripetal force provided by the Ferris wheel structure. At the top and the bottom of the wheel, the direction of the acceleration changes. On top, gravity pulls you down while the centripetal force holds you in; at the bottom, both gravity and the centripetal force direct you towards the center of the wheel's rotation.

That's what creates the differing sensations of lightness at the top and heaviness at the bottom. The beauty of circular motion on a Ferris wheel is that, no matter which point you're at, the principles stay the same, but the experienced forces vary, creating a memorable ride powered by the fundamental rules of physics.