Problem 28 - College Physics

Question

The driver of a car sets the cruise control and ties the steering wheel so that the car travels at a uniform speed of \(15 \mathrm{~m} / \mathrm{s}\) in a circle with a diameter of \(120 \mathrm{~m}\). (a) Through what angular distance does the car move in \(4.00 \mathrm{~min} ?\) (b) What arc length does it travel in this time?

Step-by-Step Solution

Answer

(a) 60 radians; (b) 3600 meters.

Step 1: Find the circumference of the circle

The circumference of a circle is given by the formula \(C = \pi \cdot d\), where \(d\) is the diameter of the circle. Given that the diameter is \(120 \text{ m}\), the circumference is \(C = \pi \times 120 = 120\pi \text{ m}\).

Step 2: Convert time to seconds

Convert the time from minutes to seconds to match the units of speed (m/s). Given \(4.00\) minutes, the time in seconds is \(4.00 \times 60 = 240 \text{ s}\).

Step 3: Calculate the distance traveled in 4 minutes

Using the formula \(d = vt\), where \(v = 15 \text{ m/s}\) and \(t = 240 \text{ s}\), calculate the distance: \(d = 15 \times 240 = 3600 \text{ m}\).

Step 4: Calculate arc length (same as distance traveled)

The arc length traveled by the car is the same as the linear distance calculated in the previous step, thus the arc length is \(3600 \text{ m}\).

Step 5: Convert linear distance to angular distance

Since one complete circle is \(2\pi\) radians, calculate the portion of the circle traveled by the formula: \(\theta = \frac{d}{C} \times 2\pi\). Substitute \(d = 3600\) and \(C = 120\pi\), \(\theta = \frac{3600}{120\pi} \times 2\pi = 60\) radians.

Key Concepts

Angular Distance

When a car moves in a circular path, it not only travels a certain distance, but it also sweeps out an angle. This angle is known as the angular distance. It is usually measured in radians. A complete circle has an angular distance of \(2\pi\) radians.
The angular distance tells us how far around the circle the object has traveled. If a car moves around the entire circle, it covers \(2\pi\) radians; if it covers half the circle, it sweeps out \(\pi\) radians. This unit is particularly useful for understanding rotational motion.
To determine the angular distance over a partial circle, you calculate it based on how much of the circle has been covered. You use the formula:

• \(\theta = \frac{d}{C} \times 2\pi\)

where \(\theta\) represents the angular distance, \(d\) is the linear distance traveled, and \(C\) is the circumference of the circle.

Arc Length

Arc length is the distance measured along the curved path of a circle segment, or 'arc'. In circular motion, especially at a constant speed, the arc length is simply the linear distance covered along the circular path during a specific time.
It is essential because it represents the actual path your object has traveled. In this case, for a car moving in a circular path, the arc length is the same as the linear distance it has covered.
The formula to find the arc length is:

• Arc Length \( = r \times \theta \)

However, if you know the speed and time, you can calculate arc length using:

• \( ext{Arc Length} = v \times t \)

where \(v\) is the uniform speed and \(t\) is the time traveled in seconds.

Uniform Speed

Uniform speed refers to the consistent speed maintained by an object throughout its motion. It implies that the car or object covers equal distances in equal intervals of time without acceleration or deceleration.
In calculations involving uniform speed, we employ the basic formula of motion:

• Distance = Speed \(\times\) Time

This feature is crucial in circular motion problems to simplify findings of time, distance, and arc length for accurately predicting an object's path. When the speed is uniform, as in our exercise, it leads to straightforward calculations.

Diameter of a Circle

The diameter is a critical measure in a circle, defined as the straight line passing from one side of the circle to the other, through the center. It is twice the length of the radius.
Understanding the diameter is crucial as it directly helps determine the circle's circumference using the formula:

• \(C = \pi \times d\)

where \(C\) is the circumference, and \(d\) is the diameter.
For circular motion problems, knowing the diameter means you can easily calculate the circle's circumference, which is essential in finding other values such as arc length or angular distance.