Problem 27
Question
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 rev/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
Verified Answer
Maximum linear speed is 0.832 m/s; maximum radial acceleration is 108.65 m/s².
1Step 1: Understand the Problem
We need to find two things: (a) the maximum linear speed of the drill bit, and (b) the maximum radial acceleration. We know the diameter of the drill bit is 12.7 mm and the rotational speed is 1250 revolutions per minute (rev/min).
2Step 2: Convert Diameter to Radius
To find the linear speed, we first convert the diameter of the drill bit to radius: Divide the diameter by 2 to get the radius.Given: Diameter = 12.7 mm.Radius: \( r = \frac{12.7}{2} = 6.35 \) mm.
3Step 3: Convert Radius to Meters
Convert the radius from millimeters to meters because SI units are preferred for physics calculations.Radius in meters: \( r = 6.35 \text{ mm} = 0.00635 \text{ m} \).
4Step 4: Calculate Linear Speed
The linear speed \( v \) of the edge of the drill bit is given by the formula:\[ v = 2 \pi r f \]where \( r \) is the radius in meters, and \( f \) is the frequency in revolutions per second.First, convert 1250 rev/min to rev/s: \( f = \frac{1250}{60} \approx 20.83 \text{ rev/s} \).Now calculate \( v \):\[ v = 2 \times \pi \times 0.00635 \times 20.83 \approx 0.832 \text{ m/s} \].
5Step 5: Calculate Radial Acceleration
Radial acceleration \( a_r \) is given by the formula:\[ a_r = r \omega^2 \]where \( \omega \) is the angular velocity in radians per second and \( r \) is the radius.First, we need to convert \( f \) to \( \omega \):\[ \omega = 2 \pi \times f = 2 \pi \times 20.83 \approx 130.88 \text{ rad/s} \].Now calculate \( a_r \):\[ a_r = 0.00635 \times (130.88)^2 \approx 108.65 \text{ m/s}^2 \].
Key Concepts
Linear Speed CalculationRadial Acceleration CalculationUnit Conversion in Physics
Linear Speed Calculation
To determine the linear speed of an object moving in a circular motion, we rely on a formula that links the object's radius and frequency. The linear speed, often denoted by the letter \( v \), measures how fast a point on the edge of the rotating object is moving in a straight line.
To calculate it, we use the formula:
\[ v = 2\pi r f \]
Here, \( r \) represents the radius of the circle, and \( f \) is the frequency of rotation, or how many revolutions occur each second. Frequency is particularly important because it defines the speed at which the rotation occurs.
For example, if you have a drill bit with a radius calculated from a 12.7 mm diameter (0.00635 m in meters) and a rotation speed of 1250 revolutions per minute, you first convert the speed to revolutions per second:
\[ f = \frac{1250}{60} \approx 20.83 \text{ rev/s} \]
This conversion allows us to accurately find the linear speed as:
\[ v = 2 \times \pi \times 0.00635 \times 20.83 \approx 0.832 \text{ m/s} \]
Moving forward with calculations, ensuring units are consistent is crucial for accuracy.
To calculate it, we use the formula:
\[ v = 2\pi r f \]
Here, \( r \) represents the radius of the circle, and \( f \) is the frequency of rotation, or how many revolutions occur each second. Frequency is particularly important because it defines the speed at which the rotation occurs.
For example, if you have a drill bit with a radius calculated from a 12.7 mm diameter (0.00635 m in meters) and a rotation speed of 1250 revolutions per minute, you first convert the speed to revolutions per second:
\[ f = \frac{1250}{60} \approx 20.83 \text{ rev/s} \]
This conversion allows us to accurately find the linear speed as:
\[ v = 2 \times \pi \times 0.00635 \times 20.83 \approx 0.832 \text{ m/s} \]
Moving forward with calculations, ensuring units are consistent is crucial for accuracy.
Radial Acceleration Calculation
Radial acceleration, or centripetal acceleration, is the acceleration experienced by an object moving in a circular path. This type of acceleration is directed towards the center of the circle, and it is crucial for understanding how forces keep an object moving in a curved path.
The formula for radial acceleration \( a_r \) is:
\[ a_r = r \omega^2 \]
Where \( r \) is the radius and \( \omega \) is the angular velocity. Angular velocity \( \omega \) (expressed in radians per second) is connected to frequency \( f \) by the formula:
\[ \omega = 2 \pi f \]
In our example, \( f \) is approximately 20.83 rev/s, leading to:
\[ \omega = 2 \times \pi \times 20.83 \approx 130.88 \text{ rad/s} \]
Plugging the values into the radial acceleration formula, we compute:
\[ a_r = 0.00635 \times (130.88)^2 \approx 108.65 \text{ m/s}^2 \]
This shows how significant the radial force is, keeping the drill bit's edge on its circular path instead of flying off tangent.
The formula for radial acceleration \( a_r \) is:
\[ a_r = r \omega^2 \]
Where \( r \) is the radius and \( \omega \) is the angular velocity. Angular velocity \( \omega \) (expressed in radians per second) is connected to frequency \( f \) by the formula:
\[ \omega = 2 \pi f \]
In our example, \( f \) is approximately 20.83 rev/s, leading to:
\[ \omega = 2 \times \pi \times 20.83 \approx 130.88 \text{ rad/s} \]
Plugging the values into the radial acceleration formula, we compute:
\[ a_r = 0.00635 \times (130.88)^2 \approx 108.65 \text{ m/s}^2 \]
This shows how significant the radial force is, keeping the drill bit's edge on its circular path instead of flying off tangent.
Unit Conversion in Physics
When dealing with physics problems, unit conversion is essential to maintain consistency and accuracy. Different regions and industries might use various units, but physics calculations typically rely on the International System of Units (SI units).
In many situations, calculations start with a variety of measurements, such as lengths in millimeters or speeds in revolutions per minute, which need to be converted into meters and seconds respectively.
Key steps in unit conversion:
In many situations, calculations start with a variety of measurements, such as lengths in millimeters or speeds in revolutions per minute, which need to be converted into meters and seconds respectively.
Key steps in unit conversion:
- Convert length units: If needing to work in meters instead of millimeters, divide the original measurement by 1000. For the drill bit example, 12.7 mm becomes 0.00635 m.
- Convert time units: Rotational speed often given in revolutions per minute should be converted to revolutions per second by dividing by 60.
Other exercises in this chapter
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