Problem 172 - Calculus Volume 3

Question

Find the maximum speed of a point on the circumference of an automobile tire of radius \(1 \mathrm{ft}\) when the automobile is traveling at \(55 \mathrm{mph}\).

Step-by-Step Solution

Answer

The maximum speed of a point on the tire's circumference is 80.67 ft/s.

Step 1: Convert Speed to Feet Per Second

First, we need to convert the automobile's speed from miles per hour to feet per second. Since there are 5280 feet in a mile and 3600 seconds in an hour, we use the conversion factor:\[55 \text{ mph} \times \frac{5280 \text{ ft}}{1 \text{ mile}} \times \frac{1 \text{ hour}}{3600 \text{ s}} = \frac{55 \times 5280}{3600} \approx 80.67 \text{ ft/s}\]

Step 2: Determine Angular Velocity

The angular velocity \( \omega \) of the tire in radians per second is determined by the formula:\[\omega = \frac{v}{r}\]Where \(v\) is the linear speed (80.67 ft/s) and \(r\) is the radius of the tire (1 ft):\[\omega = \frac{80.67 \text{ ft/s}}{1 \text{ ft}} = 80.67 \text{ rad/s}\]

Step 3: Calculate Maximum Linear Speed at the Tire's Edge

The maximum speed \(V_{max}\) reached by a point on the edge of the tire's circumference during its rotation can be calculated as:\[V_{max} = \omega \cdot r = 80.67 \text{ rad/s} \times 1 \text{ ft} = 80.67 \text{ ft/s}\]Thus, the maximum speed is the same as the linear speed of the car because of the tire's rotation.

Key Concepts

Linear Speed Conversion

When dealing with speed measurements, converting between units can often appear daunting but is quite straightforward with the right approach. Linear speed conversion is the process of changing a speed value from one unit of measurement to another. In the exercise, the automobile's speed is initially provided in miles per hour (mph). To find the speed in a more usable unit for calculations, like feet per second (ft/s), we follow these steps:

• Understand that there are 5280 feet in a mile.
• Recognize there are 3600 seconds in an hour.
• Use these values to convert miles per hour to feet per second.We multiply 55 mph by 5280 to convert miles to feet and then divide by 3600 to convert hours to seconds:\[55 \text{ mph} \times \frac{5280 \text{ ft}}{1 \text{ mile}} \times \frac{1 \text{ hour}}{3600 \text{ s}} = 80.67 \text{ ft/s}\]

This conversion allows us to work with the speed in a form that aligns naturally with the physical properties of the tire’s rotation.

Maximum Linear Speed

The maximum linear speed of a point on the circumference of a rotating object, like a tire, refers to the fastest speed that the point reaches during rotation. For this problem, once we know the angular velocity, we find the maximum linear speed by multiplying it by the radius of the circle. In essence:

• The linear speed of the car is equivalent to the speed at the tire's edge when it's moving without slipping.
• Given the radius of the tire is one foot and the calculated angular velocity is 80.67 rad/s, the maximum speed of a point on the tire's edge is: \[ V_{max} = \omega \cdot r = 80.67 \text{ rad/s} \times 1 \text{ ft} = 80.67 \text{ ft/s} \]

Hence, the tire's rim point travels as fast as the car's speed converted to feet per second, underlining that linear speed along the tire's circumference is synchronized with the vehicle's forward motion.

Radial Velocity

Radial velocity generally describes how fast something moves along a path that extends radially outward or inward. However, in the context of rotational motion like a spinning tire, radial velocity becomes less about distance along a line and more about angular dynamics.

• Angular velocity (\(\omega\)) is closely tied to radial velocity as it describes how quickly the tire rotates in radians per second.
• In this exercise, radial velocity is essentially the same as angular velocity since every point on the tire moves in a circle around the tire's center.
• The tire's angular velocity calculated using \( \omega = \frac{v}{r} \) gives us:\[ \omega = \frac{80.67 \text{ ft/s}}{1 \text{ ft}} = 80.67 \text{ rad/s} \]

This value signifies how fast each point on the tire rotates around its axis, providing a critical understanding of its rotational movement rather than linear trajectory.