Acceleration is a fundamental concept in physics that describes the rate of change of an object’s velocity with respect to time. It is commonly expressed in various units, depending on the scale of the problem being considered. Some of the most frequently used units of acceleration are meters per square second (m/s²), nanometers per square second (nm/s²), and decimeters per square second (dm/s²). Converting between these units is a necessary skill, especially when dealing with problems that require precise measurements and a consistent unit system. This article will explain how to convert acceleration from nanometers per square second (nm/s²) to decimeters per square second (dm/s²), step by step.
Step 1: Understanding the Units
Before we proceed with the conversion, it is important to understand the relationship between the base units involved.
- Nanometer (nm): A nanometer is a unit of length equal to one-billionth of a meter (1 nm = 10−910^{-9}10−9 meters).
- Decimeter (dm): A decimeter is a unit of length equal to one-tenth of a meter (1 dm = 10−110^{-1}10−1 meters).
In terms of acceleration, the units of time are typically seconds (s), so the unit of acceleration is in meters per square second (m/s²). However, we are tasked with converting from nanometers per square second (nm/s²) to decimeters per square second (dm/s²).
Step 2: Conversion Factor for Length Units
To convert between nanometers (nm) and decimeters (dm), we first need to convert the length units. We know:1 nm=10−9 meters1 \, \text{nm} = 10^{-9} \, \text{meters}1nm=10−9meters1 dm=10−1 meters1 \, \text{dm} = 10^{-1} \, \text{meters}1dm=10−1meters
Therefore, the conversion factor between nanometers and decimeters is:1 nm=10−9 mand1 dm=10−1 m1 \, \text{nm} = 10^{-9} \, \text{m} \quad \text{and} \quad 1 \, \text{dm} = 10^{-1} \, \text{m}1nm=10−9mand1dm=10−1m
To convert nanometers to decimeters, divide the length in nanometers by 10810^{8}108 because:10−9 m10−1 m=10−8\frac{10^{-9} \, \text{m}}{10^{-1} \, \text{m}} = 10^{-8}10−1m10−9m=10−8
Thus, the conversion factor from nanometers to decimeters is 10−810^{-8}10−8.
Step 3: Converting the Acceleration Units
Since acceleration involves squared units of length, we must apply the square of the conversion factor when changing from nm/s² to dm/s².
The conversion factor for acceleration is:(1 nm1 dm)2=(10−8)2=10−16\left( \frac{1 \, \text{nm}}{1 \, \text{dm}} \right)^2 = (10^{-8})^2 = 10^{-16}(1dm1nm)2=(10−8)2=10−16
Thus, to convert from nanometers per square second (nm/s²) to decimeters per square second (dm/s²), we multiply by 10−1610^{-16}10−16.
Step 4: Applying the Conversion
Given an acceleration value in nanometers per square second (nm/s²), to convert it to decimeters per square second (dm/s²), simply multiply the value by 10−1610^{-16}10−16.
For example:
- If an object is accelerating at 5×106 nm/s²5 \times 10^6 \, \text{nm/s²}5×106nm/s², to convert to decimeters per square second, multiply by 10−1610^{-16}10−16:5×106 nm/s²×10−16=5×10−10 dm/s²5 \times 10^6 \, \text{nm/s²} \times 10^{-16} = 5 \times 10^{-10} \, \text{dm/s²}5×106nm/s²×10−16=5×10−10dm/s²
Thus, 5×106 nm/s²5 \times 10^6 \, \text{nm/s²}5×106nm/s² is equivalent to 5×10−10 dm/s²5 \times 10^{-10} \, \text{dm/s²}5×10−10dm/s².
Conclusion
The conversion from nanometers per square second (nm/s²) to decimeters per square second (dm/s²) involves multiplying the given acceleration value by 10−1610^{-16}10−16, as derived from the conversion of nanometers to decimeters and considering the square of the conversion factor due to the units of acceleration. This simple but crucial conversion allows for consistent scaling of acceleration measurements in different contexts, ensuring accuracy and precision in scientific calculations.