Problem 2 - Algebra and Trigonometry

Question

Approximate the angular speed of the second hand on a clock in rad/sec. (Round to three decimal places.)

Step-by-Step Solution

Answer

The angular speed of the second hand is approximately 0.105 radians per second.

Step 1 - Determine the total rotation in one revolution

Recall that a full rotation, or one complete revolution, is equivalent to an angle of 360 degrees. In radians, one complete revolution is equivalent to \( 2\pi \text{ radians} \).

Step 2 - Time for one complete revolution

The second hand of a clock completes one full revolution every 60 seconds. Therefore, the period of the second hand is 60 seconds.

Step 3 - Calculate the angular speed

Angular speed is given by the formula \( \omega = \frac{\theta}{t} \), where \( \omega \) is the angular speed, \( \theta \) is the angle of rotation in radians, and \( t \) is the time period. Substituting the known values: \( \omega = \frac{2\pi}{60} \).

Step 4 - Simplify the expression

Simplify the calculated angular speed: \( \omega = \frac{2\pi}{60} = \frac{\pi}{30} \approx 0.105 \text{ radians/second} \)

Key Concepts

Radian Measure

Understanding radian measure is crucial for calculating angular speed. A radian is a way of measuring angles based on the radius of a circle. It's different from degrees.
In simpler terms, one complete rotation around a circle is 360 degrees. However, this can also be measured as 2π radians. Here’s why radians are useful:

• They relate directly to the radius of a circle.
• They simplify many mathematical formulas.

For example, if you walk around a circle with a radius of 1 unit and complete a full circle, you've walked 2π radians. This makes radians a natural choice for measuring angles in various calculations, especially when dealing with periodic functions like those of a clock.

Angular Speed Formula

Angular speed refers to how quickly an object rotates or revolves around a point. It can be expressed as \( \omega = \frac{\theta}{t} \), where:\( \omega \) is the angular speed in radians per second.\( \math{theta} \) is the angle rotated (in radians).\( t \) is the time taken.To calculate the angular speed of the second hand on a clock, we use this formula.The angle of one complete rotation is \( 2\pi \) radians.The time period for one full rotation, the period, is 60 seconds. By substituting these values in, we get \( \omega = \frac{\2\pi}{60} = \frac{\pi}{30} \).This equation shows how radians and time directly affect the speed calculation.

Period of the Second Hand

The period refers to the duration it takes for one complete cycle or rotation. In the context of the second hand on a clock, the period is the time it takes to complete one revolution. This is 60 seconds.Using the period in calculations is essential, especially when dealing with periodic motion like that of a clock.Key points to remember about periods:

• It’s the time for one full rotation or cycle.
• Important in determining angular speed.
• Directly relates to frequency, where frequency is the inverse of the period.

In this exercise, knowing that the period is 60 seconds helps us find the angular speed by letting us understand how long it takes for one full revolution. This period of 60 seconds is a critical piece of information that feeds into our calculation using the angular speed formula.

Unit Conversion

Unit conversion is critical in many physics and engineering problems. Here, it allows us to switch between degrees and radians, and to simplify our results for angular speed.Important conversions to remember include:

• Degrees to radians: 1 degree = \( \frac{\theta \pi}{180} \) radians
• Complete revolution: 360 degrees = 2π radians

In our problem, the second hand makes a full revolution of 360 degrees, but it’s easier to use radians.Using the conversion that 360 degrees = 2π radians simplifies the math and helps us directly plug values into formulas involving π, such as the angular speed formula mentioned. This makes unit conversion a powerful tool in our problem-solving toolkit.