Problem 27 - College Physics

Question

\(\bullet\) Electric drill. According to the shop manual, when drilling a 12.7 -mm-diameter hole in wood, plastic, or aluminum, a drill should have a speed of 1250 rev/min. For a 12.7 - -mm-diameter drill bit turning at a constant 1250 \(\mathrm{rev} / \mathrm{min}\) , find (a) the maximum linear speed of any part of the bit and (b) the maximum radial acceleration of any part of the bit.

Step-by-Step Solution

Answer

The maximum linear speed is 0.831 m/s and the maximum radial acceleration is 108.7 m/s².

Step 1: Determine the Linear Speed Formula

The linear speed \( v \) of any part of a rotating object can be calculated using the formula \( v = r \cdot \omega \), where \( r \) is the radius and \( \omega \) is the angular speed in radians per second. First, we need to convert the angular speed from revolutions per minute to radians per second.

Step 2: Convert Angular Speed

Given that \( \omega = 1250 \) rev/min, convert this to radians per second:\[\omega = 1250 \frac{\text{rev}}{\text{min}} \times \frac{2\pi \text{ rad}}{1 \text{ rev}} \times \frac{1 \text{ min}}{60 \text{ s}} \]After calculating, \( \omega \approx 130.9 \text{ rad/s} \).

Step 3: Calculate Maximum Linear Speed

The maximum linear speed occurs at the outer edge of the drill bit, which is half its diameter. Given the diameter is 12.7 mm, the radius \( r = \frac{12.7 \text{ mm}}{2} = 6.35 \text{ mm} = 0.00635 \text{ m} \). Now, use the formula:\[v = r \cdot \omega = 0.00635 \times 130.9 \approx 0.831 \text{ m/s}\]This is the maximum linear speed of the drill bit.

Step 4: Determine the Radial Acceleration Formula

Radial or centripetal acceleration \( a_r \) can be calculated using the formula \( a_r = r \cdot \omega^2 \).

Step 5: Calculate Maximum Radial Acceleration

Using the maximum radius (6.35 mm or 0.00635 m) and the angular speed (\( \omega = 130.9 \text{ rad/s} \)), calculate radial acceleration:\[a_r = r \cdot \omega^2 = 0.00635 \times (130.9)^2 \approx 108.7 \text{ m/s}^2\]This is the maximum radial acceleration of the drill bit.

Key Concepts

Angular Velocity

Angular velocity is a crucial concept when dealing with rotational motion, such as the spinning of a drill bit. It tells us how fast an object is rotating and is usually measured in radians per second (rad/s).

In our exercise, the angular velocity, denoted by the symbol \( \omega \), is given initially in revolutions per minute (rev/min). We converted this measure into radians per second to better align with standard equations for rotational motion.

To perform this conversion:

• First, we multiply the number of revolutions by \(2\pi\) because one full revolution is \(2\pi\) radians.
• Then, we divide by 60 to change the time unit from minutes to seconds.

By following these steps, we found that the angular velocity of the drill bit is approximately \(130.9\) rad/s. This angular velocity helps us calculate other important aspects of the drill's motion, like linear speed and radial acceleration.

Linear Speed

Linear speed relates to how fast a point on the edge of a rotating object is traveling along a path. This speed changes with the radius of the circle created by rotation.

In the context of our drill bit, the linear speed is highest at the outermost edge. To find this speed, we use the formula \(v = r \cdot \omega\), where \(v\) is the linear speed, \(r\) is the radius, and \(\omega\) is the angular velocity.

For our drill bit:

• The radius is half of the diameter, which means \(r = 6.35\) mm or \(0.00635\) meters when converted into meters for consistency in the units.
• Using the angular velocity \(\omega = 130.9\) rad/s, the calculation yields a linear speed of approximately \(0.831\) m/s.

This value indicates how quickly a section on the outer edge of the drill bit is moving as the drill spins.

Radial Acceleration

Radial acceleration, also known as centripetal acceleration, is the acceleration directed toward the center of the circular path in which an object is moving. This acceleration keeps the rotating object in its circular path.

The formula for radial acceleration \(a_r\) is \(a_r = r \cdot \omega^2\). It highlights that this type of acceleration is dependent on both the radius of rotation and the square of the angular velocity.

In our example:

• The radius of the drill bit is \(0.00635\) meters.
• The angular velocity is \(130.9\) rad/s.

We then calculate the radial acceleration to be approximately \(108.7\, \text{m/s}^2\). This measures the rate of change of direction for a point at the drill bit's edge, which is essential for maintaining its rotational motion. Understanding radial acceleration is key to analyzing forces in rotational systems.