Problem 27 - University Physics with Modern Physics

Question

A Ferris wheel with radius 14.0 m is turning about a horizontal axis through its center (\(\textbf{Fig. E3.27}\)). The linear speed of a passenger on the rim is constant and equal to 6.00 m/s. What are the magnitude and direction of the passenger's acceleration as she passes through (a) the lowest point in her circular motion and (b) the highest point in her circular motion? (c) How much time does it take the Ferris wheel to make one revolution?

Step-by-Step Solution

Answer

The acceleration is 2.57 m/s² upwards at the lowest point, downwards at the highest point; one revolution takes about 14.66 s.

Step 1: Understand Circular Motion Dynamics

To solve this problem, we need to understand that a passenger on a Ferris wheel moves in a circular path, hence they are under a constant centripetal acceleration directed towards the center of the circle. Two key formulas are relevant: the centripetal acceleration formula, \( a = \frac{v^2}{r} \), where \( v \) is the linear speed and \( r \) is the radius of the circle; and the time for one complete revolution, \( T = \frac{2\pi r}{v} \).

Step 1: Calculate the Centripetal Acceleration

Using the formula for centripetal acceleration \( a = \frac{v^2}{r} \), substitute \( v = 6.00 \, \text{m/s} \) and \( r = 14.0 \, \text{m} \). Calculate the acceleration: \[ a = \frac{(6.00)^2}{14.0} = \frac{36.0}{14.0} \approx 2.57 \, \text{m/s}^2. \]

Step 2: Determine the Acceleration Direction at the Lowest Point

At the lowest point of the circular motion, the centripetal acceleration is directed upwards towards the center of the wheel.

Step 3: Determine the Acceleration Direction at the Highest Point

At the highest point of the circular motion, the centripetal acceleration is directed downwards towards the center of the wheel.

Step 4: Calculate the Time for One Revolution

Using the formula \( T = \frac{2\pi r}{v} \), substitute \( r = 14.0 \, \text{m} \) and \( v = 6.00 \, \text{m/s} \). Calculate the time:\[ T = \frac{2\pi \times 14.0}{6.00} \approx \frac{87.96}{6.00} \approx 14.66 \, \text{s}. \]

Key Concepts

Centripetal Acceleration

Centripetal acceleration is essential in understanding circular motion. When an object follows a curved path, it experiences an acceleration directed towards the center of the circle. This is known as centripetal acceleration and ensures that the object keeps its circular path. The formula to calculate this acceleration is given by: \( a = \frac{v^2}{r} \). Here, \( v \) represents the linear speed, while \( r \) is the radius of the circle.
For a Ferris wheel, as described in the exercise, the centripetal acceleration ensures that the passenger experiences a smooth circular ride. - At the lowest point of the wheel, this acceleration is pointed upwards, aligning towards the center of the wheel. - Conversely, at the highest point, the centripetal acceleration directs itself downwards, still aiming towards the circle's center.These directional accelerations play a crucial role in balancing out the gravitational forces on the passenger, keeping them safely in their seats throughout the ride.

Linear Speed

Linear speed refers to how fast a point on the outer edge of the circular path travels. It's the distance covered by the passenger along the Ferris wheel's circumference per unit of time. In this context, it is constant and set at 6.00 m/s. Understanding linear speed is vital as it helps to determine the centripetal force needed to keep passengers moving in a circle. With the constant linear speed, passengers maintain a uniform circular motion.
Despite the speed not changing, the direction of velocity does. This continuous change in direction is why centripetal acceleration exists. It pulls the passenger towards the center of the Ferris wheel to sustain the circular motion. - Providing a consistent experience, linear speed combined with the wheel's radius enables you to compute other important motion characteristics, such as velocity and the time for one complete revolution.

Time of Revolution

Time of revolution is the time it takes for the Ferris wheel to complete one full circle, bringing the passenger back to their starting point. The formula used for discovering the time of revolution is: \( T = \frac{2\pi r}{v} \). It calculates how long one circuit of the wheel takes by dividing the full circular distance traveled by the speed. Using the given values: radius \( r = 14.0 \, \text{m} \), and speed \( v = 6.00 \, \text{m/s} \), the time for one complete revolution on the Ferris wheel becomes approximately 14.66 seconds.
- This duration reflects how swiftly the wheel spins, impacting the ride experience. - A shorter time makes for a quicker ride, while a longer time allows for a gentler experience.