Problem 50 - University Physics with Modern Physics

Question

The Cosmoclock 21 Ferris wheel in Yokohama City, Japan, has a diameter of 100 \(\mathrm{m}\) . Its name comes from its 60 arms, each of which can function as a second hand (so that it makes one revolution every 60.0 s). (a) Find the speed of the passengers when the Ferris wheel is rotating at this rate. (b) A passenger weighs 882 \(\mathrm{N}\) at the weight-guessing booth on the ground. What is his apparent weight at the highest and at the lowest point on the Ferris wheel? (c) What would be the time for one revolution if thepassenger's apparent weight at the highest point were zero? (d) What then would be the passenger's apparent weight at the lowest point?

Step-by-Step Solution

Answer

(a) 5.24 m/s, (b) 832.66 N at highest, 931.34 N at lowest, (c) 14.18 s for zero weight, (d) 1764 N at lowest.

Step 1: Find the Radius of the Ferris Wheel

To find the speed of the passengers, first determine the radius of the Ferris wheel. The diameter is given as 100 m, so the radius \( r \) is half of that: \( r = \frac{100}{2} = 50 \text{ m} \).

Step 2: Calculate the Speed of Passengers

The speed \( v \) of the passengers is given by the formula \( v = \frac{2\pi r}{T} \), where \( T \) is the period of revolution (60.0 seconds). Thus, \( v = \frac{2\pi \times 50}{60} = \frac{100\pi}{60} \approx 5.24 \text{ m/s} \).

Step 3: Calculate Apparent Weight at the Highest Point

At the highest point, the apparent weight \( W_{h} \) is the actual weight minus the centripetal force: \( W_h = mg - F_c \). Here, \( F_c = \frac{mv^2}{r} \). Given weight \( W = 882 \text{ N} = mg \), calculate \( F_c \). First, find \( m = \frac{882}{9.8} \approx 90 \text{ kg} \). Then, \( F_{c} = \frac{90 \times (5.24)^2}{50} \approx 49.34 \text{ N} \). Hence, \( W_h = 882 - 49.34 = 832.66 \text{ N} \).

Step 4: Calculate Apparent Weight at the Lowest Point

At the lowest point, the apparent weight \( W_l \) is the actual weight plus the centripetal force: \( W_l = mg + F_c \). So, \( W_l = 882 + 49.34 = 931.34 \text{ N} \).

Step 5: Time for One Revolution When Apparent Weight is Zero at the Highest Point

If the apparent weight is zero at the highest point, \( mg = F_c \). Hence, \( mg = \frac{mv^2}{r} \to g = \frac{v^2}{r} \to v = \sqrt{gr} \). This speed corresponds to \( v = \frac{2\pi r}{T} \). Solving for \( T \), \( T = \frac{2\pi \times 50}{\sqrt{9.8 \times 50}} = \frac{100\pi}{22.14} \approx 14.18 \text{ s} \).

Step 6: Apparent Weight at the Lowest Point for New Revolution Time

At the higher speed making the apparent weight zero at the top, the speed \( v = \sqrt{gr} \approx 22.14 \text{ m/s} \). The new centripetal force at the lowest point \( F_{c,new} = \frac{mv^2}{r} = \frac{90 \times (22.14)^2}{50} = 882 \text{ N} \). Therefore, the apparent weight \( W_{l,new} = mg + F_{c,new} = 882 + 882 = 1764 \text{ N} \).

Key Concepts

Apparent Weight

Apparent weight is a fascinating concept in physics, particularly when discussing scenarios involving motion like circular motions on a Ferris wheel. Apparent weight is what you feel when you are subjected to acceleration, unlike actual weight which is simply your mass multiplied by the gravitational force ( mg ). When on the Ferris wheel, passengers experience a change in apparent weight at different points because they are in motion along a curved path.

• At the highest point, passengers feel lighter because the centripetal force, which acts towards the center, is subtracted from their actual weight. They experience less than their normal gravitational pull.
• Conversely, at the lowest point, the apparent weight increases because the centripetal force adds to the force of gravity.

Thus, the apparent weight varies depending on where you are along the circular path of the Ferris wheel.

Centripetal Force

Centripetal force is the key to keeping you moving in a circle without flying off into the straight line path that inertia would normally take you on. In the case of a Ferris wheel, the force that maintains the circular motion is known as the centripetal force, which acts towards the center of the circle.

• This force is necessary to change the direction of the passenger's velocity without altering their speed.
• For an object with mass (\( m \)) moving at speed (\( v \)) on a circular path with radius (\( r \)), the centripetal force (\( F_c \)) can be given by the equation: \[ F_c = \frac{mv^2}{r} \]

When riding a Ferris wheel, centripetal force affects how heavy or light a passenger feels due to changes in speed and direction.

Uniform Circular Motion

Uniform circular motion refers to motion along a circular path at a constant speed. Despite the speed being constant, the direction is always changing. This necessitates a net force acting towards the center of the circle (centripetal force) to sustain the motion.

• A key characteristic is that the net force required points inward, keeping the object on its circular trajectory.
• For a Ferris wheel that maintains uniform rotation, passengers would experience continuous motion with varying apparent weight but unchanged speed.
• This consistent speed allows precise calculations for aspects like the period of rotation and velocity of the passengers, as demonstrated in Ferris wheel problems.

Understanding uniform circular motion is critical for analyzing machinery like Ferris wheels, where stability and precision are paramount.

Ferris Wheel

The Ferris wheel is a prime example of uniform circular motion and offers passengers a continuous, circular ride while rotating at a set speed. This attraction is designed to balance the thrilling feel of weightlessness at the top with the reassuring sense of weight at the bottom.

• The Ferris wheel moves in a circle at a constant speed, giving a delightful sensation of soaring when at the peak and grounding at the base.
• Passenger perceived weight varies due to the changing centripetal force as they move along the Ferris wheel's circular path.
• Specifically, in our exercise, the Cosmoclock 21 Ferris wheel has a diameter of 100 meters and completes one full rotation every 60 seconds, influencing the apparent weight felt by riders across their journey.

This remarkable invention showcases principles of physics in a real-world setting, offering not only a ride but also a lesson in the dynamics of circular motion.