Question

Amusement Park Ride A centrifugal force ride, similar to the Gravitron, spins at a rate of 22 revolutions per minute. If the diameter of the ride is 13 meters, what is the linear speed of the passengers in kilometers per hour?

Short answer

Approximately 53.91 kilometers per hour.

Step-by-step solution

- Determine the radius of the ride

The diameter of the ride is given as 13 meters. The radius is half of the diameter. Therefore, the radius is \(\frac{13}{2} = 6.5 \) meters.

- Calculate the circumference of the ride

The circumference of a circle is calculated using the formula \(\text{Circumference} = 2 \pi r\). Substituting the radius we found in Step 1, \(\text{Circumference} = 2 \pi \times 6.5 \approx 40.84 \) meters.

- Convert revolutions per minute to revolutions per hour

Since the ride spins at 22 revolutions per minute, to find revolutions per hour we multiply by 60 (since there are 60 minutes in an hour): \(\text{Revolutions per hour} = 22 \times 60 = 1320 \) revolutions per hour.

- Calculate linear speed in meters per hour

The linear speed in meters per hour is found by multiplying the number of revolutions per hour by the circumference of the ride: \(\text{Linear speed} = 1320 \times 40.84 \approx 53908.8 \) meters per hour.

- Convert linear speed to kilometers per hour

To convert the speed from meters per hour to kilometers per hour, divide by 1000 (since there are 1000 meters in a kilometer): \(\text{Linear speed} = \frac{53908.8}{1000} \approx 53.91 \) kilometers per hour.

Key concepts

Circumference of a Circle

To understand the linear speed of the Gravitron, you need to know about the circumference of a circle. The circumference is the distance around the edge of a circle.
To find the circumference, use the formula: \[ \text{Circumference} = 2 \times \text{π} \times \text{radius} \]
Here, \( \text{π} \) (pi) is a constant, approximately 3.14159.
First, let's find the radius of the ride. Given the diameter is 13 meters, the radius is half of that: 6.5 meters.
Plug the radius into the formula: \[ \text{Circumference} = 2 \times 3.14159 \times 6.5 \]
This comes out to approximately 40.84 meters.
This means the ride’s edge covers a distance of 40.84 meters in one full revolution.

Angle Conversion

Understanding angle conversion is important when dealing with circular motion. In our problem, the Gravitron spins at 22 revolutions per minute (RPM).
To get the linear speed, we need to convert RPM to a more useful measurement: revolutions per hour (RPH).
There are 60 minutes in an hour, so we multiply 22 revolutions per minute by 60 to convert: \[ \text{RPH} = 22 \times 60 = 1320 \]
This simple multiplication shows how many revolutions the ride makes in one hour.
Longer timings help to understand speed over practical units like kilometers per hour, making it easier to grasp the ride’s speed.

Linear Speed

Linear speed is the distance traveled per unit of time in a straight line. For circular motion, linear speed can be found by multiplying the circumference of the circle by the number of revolutions in a given time.
After converting the revolutions per minute to revolutions per hour earlier, which was 1320 RPH, the next step is to find the total distance traveled in meters per hour.
Using our previous circumference calculation: \[ \text{Linear speed in meters per hour} = 1320 \times 40.84 = 53908.8 \text{ meters per hour} \]
Finally, to convert meters per hour to kilometers per hour, we divide by 1000: \[ \text{Linear speed} = \frac{53908.8}{1000} \text{ km/h} \approx 53.91 \]
Therefore, the linear speed of the passengers on the Gravitron is about 53.91 kilometers per hour, illustrating how fast they are moving.