Question

The angular speed of the hour hand of a clock (in radians per second) is a) \(\frac{\pi}{7200}\) b) \(\frac{\pi}{3600}\) c) \(\frac{\pi}{1800}\) d) \(\frac{\pi}{60}\) e) \(\frac{\pi}{30}\) f) The correct value is not shown.

Short answer

Answer: f) The correct value is not shown (The actual angular speed is π/21600 radians/second).

Step-by-step solution

Convert 12 hours to seconds

First, we need to convert 12 hours to seconds. We can do this using the conversion factors 1 hour = 60 minutes and 1 minute = 60 seconds. 12 hours × (60 minutes/hour) × (60 seconds/minute) = 43200 seconds

Calculate the total angle covered by the hour hand

We need to determine the total angle covered by the hour hand in one full rotation. As there are 360° in a full rotation, it corresponds to 2π radians: Total angle = 2π radians

Calculate the angular speed

To find the angular speed, we divide the total angle (2π radians) by the total time (43200 seconds): Angular speed = (2π radians)/(43200 seconds) = π/21600 radians/second Now let's compare the value found to the answer options: a) \(\frac{\pi}{7200}\) b) \(\frac{\pi}{3600}\) c) \(\frac{\pi}{1800}\) d) \(\frac{\pi}{60}\) e) \(\frac{\pi}{30}\) f) The correct value is not shown The value found, π/21600 radians/second, is not listed among the options. Therefore, the answer is: f) The correct value is not shown.

Key concepts

Radians per Second

Radians per second is a unit of angular speed or rotational speed. It tells us how many radians an object rotates or revolves in one second. In the context of circular motion, one complete revolution corresponds to an angle of \(2\pi\) radians. To understand this concept more intuitively, picture a pizza sliced into equal parts. If one slice represents a certain angle in radians, then 'radians per second' tells us how many pizza slices the object passes by each second as it moves along its circular path.
When working with angular speed, it's helpful to remember that radians are a way to measure angles based on the radius of a circle. There are about 6.28 radians (which is \(2\pi\) exactly) in a full circle, rather than the 360 degrees most people are familiar with. This is why understanding radians is critical when calculating angular speed in physics and engineering problems.

Angular Velocity

Angular velocity, often denoted by the Greek letter omega (\(\omega\)), is a vector quantity that represents the rate of change of an object's angular position. In simpler terms, it's how fast something rotates or spins. While the magnitude of angular velocity tells us the angular speed (how fast), its direction usually points along the axis of rotation according to the right-hand rule.
For example, a merry-go-round spinning at a constant rate, or the Earth rotating on its axis, both have angular velocities. In solving problems related to angular velocity, such as the hour hand on a clock, it’s important to keep the distinction between linear velocity (measured in units like meters per second) and angular velocity (measured in radians per second) clear. Understanding both concepts helps in grasping the nature of rotational movements.

Hour Hand Clock Problem

The hour hand clock problem is a classic example used to illustrate angular speed and angular velocity. This type of problem involves calculating how fast the hour hand of a clock moves in terms of radians per second. Considering a clock face as a circle, the hour hand completes one full rotation (or \(2\pi\) radians) in 12 hours. To solve this, one typically divides the total angle the hand moves by the total time taken to move that angle.

Common Misconceptions

Students sometimes confuse the hour hand's angular speed with the minute hand's, which moves twelve times faster. Another source of confusion can be the conversion from angular position or displacement (measured in radians or degrees) to angular speed or velocity (measured in radians per second or degrees per second), which requires careful attention to units of time.

Converting Time Units

Converting time units is a fundamental skill needed for solving problems related to angular velocity and speed. It involves changing one measure of time, like hours or minutes, into another, like seconds. For instance, there are 60 minutes in an hour and 60 seconds in a minute, thus there are \(60 \times 60 = 3,600\) seconds in an hour. This conversion is crucial when calculating angular speed because angular velocity is often expressed in radians per second.
Converting time correctly ensures that the units in our calculations match, providing accurate and meaningful results. In the context of homework problems involving clocks or rotating objects, always double-check your time unit conversions to avoid simple mistakes that could lead to incorrect answers.