Question

The hour hand on a certain clock is \(8.2 \mathrm{~cm}\) long. Find the tangential speed of the tip of this hand during normal operation.

Short answer

The tangential speed is approximately \(1.19 \times 10^{-3}\text{ m/s}\).

Step-by-step solution

Understanding Circular Motion

The tip of the hour hand moves in a circle. Knowing that the length of the hour hand is the radius of this circular path, we identify that it is a circular motion problem, where the radius, \(r\), is \(8.2 \text{ cm}\).

Identify the Time Period

The hour hand completes a full circle (360 degrees) in 12 hours. Thus, the angular velocity can be calculated based on this period.

Calculate Angular Velocity

The angular velocity, \(\omega\), is given by the formula \(\omega = \frac{2\pi}{T}\), where \(T\) is the time period. Here, \(T = 12 \times 3600 = 43200 \text{ seconds}\). So, \(\omega = \frac{2\pi}{43200}\) radians per second.

Calculate Tangential Speed

The tangential speed, \(v_t\), is related to the angular velocity by the equation \(v_t = r \times \omega\). Substituting \(r = 8.2 \text{ cm} = 0.082 \text{ m} \) and \(\omega\) from Step 3, we find \(v_t = 0.082 \times \frac{2\pi}{43200}\approx 1.19 \times 10^{-3}\text{ m/s}\).

Conclusion

The tangential speed of the tip of the hour hand is approximately \(1.19 \times 10^{-3}\text{ m/s}\).

Key concepts

Circular Motion

When an object moves along a circular path, it is involved in circular motion. Circular motion is characterized by movement around a fixed central point or axis. In our exercise, this motion is exemplified by the tip of the hour hand on a clock moving in a circle.

• The radius of the circle is the length of the hour hand, which is given as 8.2 cm.
• This motion means that every point on the hour hand rotates around the clock's center.
• In a clock, each part of an hour hand moves through the same angle in the same time, maintaining a constant speed.

The concept that describes how fast the object moves around the circle in terms of angular displacement and time is called angular velocity. In summary, circular motion is a fundamental concept that leads us to explore other properties like angular velocity and tangential speed.

Angular Velocity

Angular velocity is a measure of how quickly an object traverses a circular path. It tells us the rate at which an angle changes as the object moves along the circular path.
In this case:

• For a complete revolution, the hour hand moves through an angle of 360 degrees.
• This 360-degree rotation is equivalent to moving through a complete angle of \(2\pi\) radians.
• To find the angular velocity \( \omega \), we use the formula \( \omega = \frac{2\pi}{T} \), where \( T \) is the time period for one revolution.

For the hour hand, which completes a full circle in 12 hours, \( T \) is 43200 seconds, so we calculate \( \omega = \frac{2\pi}{43200} \). This tells us the rate of rotation in radians per second. Understanding angular velocity helps us know how fast the hour hand is moving through its circular path.

Time Period

The time period in the context of circular motion refers to the time it takes to complete one full cycle of motion along the circular path. It is an essential parameter because it helps calculate other important values like angular velocity.
Consider the clock's hour hand:

• It takes precisely 12 hours to complete one full revolution.
• Thus, the time period \( T \) for one complete turn of the hour hand is equivalent to \( 43200 \) seconds (12 hours multiplied by 3600 seconds/hour).

By knowing the time period, we can find out how fast the hour hand is moving around its circular path, reflecting on how quickly it traces the circle. This time measure is crucial for understanding the dynamics of circular motion.

Radius

The radius is a crucial factor in studying circular motion, as it represents the distance from the center of the circle to its outer edge. In the clock-related activity, the radius refers to the length of the hour hand.
Key points about the radius in this context include:

• In the given exercise, the radius is 8.2 cm, which is about 0.082 meters, converting centimeters to meters.
• The radius affects the tangential speed since this speed depends directly on both the radius and the angular velocity, given by \( v_t = r \times \omega \).
• Larger radii result in quicker tangential speeds because a longer path is traced in the same amount of time.

Understanding the concept of the radius allows us to appreciate how changes in distance from the circle's center impact the speed of movement along that path.