Problem 35 - College Physics

Question

\(\bullet\) A wall clock has a second hand 15.0 \(\mathrm{cm}\) long. What is the radial acceleration of the tip of this hand?

Step-by-Step Solution

Answer

The radial acceleration of the second hand's tip is approximately 0.00328 m/s².

Step 1: Understand the problem

We need to find the radial (centripetal) acceleration of the tip of the second hand of a wall clock. We are given that the length of the second hand is 15.0 cm, and we need to consider that the second hand completes a full rotation in 60 seconds.

Step 2: Use the formula for radial acceleration

The radial acceleration is given by the formula \(a_r = \frac{v^2}{r}\), where \(v\) is the linear velocity and \(r\) is the radius of rotation. Here, \(r = 15.0\) cm, which we need to convert to meters (\(0.15\) m).

Step 3: Calculate the linear velocity

The linear velocity \(v\) can be calculated using \(v = \omega \times r\), where \(\omega\) is the angular velocity. For one full revolution, \(\omega = \frac{2\pi}{T}\), where \(T = 60\) seconds.

Step 4: Calculate angular velocity

Substitute \(T = 60\) seconds into the angular velocity formula: \(\omega = \frac{2\pi}{60} = \frac{\pi}{30}\) radians per second.

Step 5: Calculate linear velocity

Substitute \(\omega = \frac{\pi}{30}\) and \(r = 0.15\) m into the formula \(v = \omega \times r\): \(v = \left(\frac{\pi}{30}\right) \times 0.15\).

Step 6: Solve for linear velocity

Calculate \(v: v = \frac{\pi}{30} \times 0.15 = \frac{0.15\pi}{30} = \frac{\pi}{200}\) m/s.

Step 7: Calculate radial acceleration

Substitute \(v = \frac{\pi}{200}\) m/s and \(r = 0.15\) m into the radial acceleration formula: \(a_r = \frac{\left(\frac{\pi}{200}\right)^2}{0.15}\).

Step 8: Solve for radial acceleration

Calculate \(a_r: a_r = \frac{\pi^2}{40000 \times 0.15} = \frac{\pi^2}{6000} \approx 0.00328\) m/s².

Key Concepts

Angular Velocity

Understanding angular velocity is key to solving problems involving rotational motion. Angular velocity, denoted as \( \omega \), is a measure of how fast something rotates or revolves around a particular axis. It's typically expressed in radians per second.

For our problem with a clock’s second hand, which completes one full revolution in 60 seconds, we use the formula for angular velocity:

• \( \omega = \frac{2\pi}{T} \)

Where \( T \) is the period, the time for one complete cycle.

Substituting \( T = 60 \) seconds, we find that \( \omega = \frac{2\pi}{60} = \frac{\pi}{30} \) radians per second.

This calculation lets you comprehend how fast the second hand moves in rotational terms rather than in straightforward, linear movement.

Linear Velocity

Linear velocity comes into play when you want to understand the speed along a path. While angular velocity tells us how fast something spins, linear velocity gives the rate at which an object covering a segment of that spin moves along its circular path.

Calculated through the formula:

• \( v = \omega \times r \)

where \( \omega \) is angular velocity and \( r \) is the radius. Using our previous calculation, with \( \omega = \frac{\pi}{30} \) radians per second and \( r = 0.15 \) meters, the linear velocity \( v \) becomes:

• \( v = \left( \frac{\pi}{30} \right) \times 0.15 = \frac{0.15\pi}{30} = \frac{\pi}{200} \)

meters per second.

This helps you quantify how fast a point (like the tip of the second hand) is moving along its circular path.

Centripetal Acceleration

Centripetal acceleration, also known as radial acceleration in some contexts, is an essential concept in circular motion. This acceleration keeps an object moving in a curved path and points toward the center of the circle.

The formula for centripetal acceleration \( a_r \) is:

• \( a_r = \frac{v^2}{r} \)

where \( v \) is linear velocity, and \( r \) is the radius.

Plugging in our values, \( v = \frac{\pi}{200} \) m/s and \( r = 0.15 \) m, we find:

• \( a_r = \frac{(\frac{\pi}{200})^2}{0.15} = \frac{\pi^2}{6000} \)

which equals approximately 0.00328 m/s².

This calculation highlights how the centripetal force acts to continually change the direction of the velocity, maintaining circular motion.

Conversion of Units

Conversion of units is a necessary math skill needed to solve physics problems accurately. Often, measurements are not given in the necessary units for calculation, so converting them ensures consistent analysis.

In this problem, the second hand length is provided as 15.0 cm. It’s crucial to convert this to meters as the standard unit in physics (SI units), especially when calculating with metrics like acceleration or velocity.

To convert centimeters to meters:

• 1 cm = 0.01 meters

Thus, 15.0 cm becomes:

• 15.0 cm \( = 15.0 \times 0.01 = 0.15 \) meters

Learning to check and convert units keeps calculations precise and prevents errors.