Problem 41 - College Physics

Question

Express the length \(50500 \mu \mathrm{m}\) (micrometers) in centimeters, decimeters, and meters, to three significant figures.

Step-by-Step Solution

Answer

50500 μm = 5.05 cm or 0.505 dm or 0.0505 m.

Step 1: Convert Micrometers to Centimeters

To convert micrometers to centimeters, we use the conversion factor: 1 cm = 10,000 μm. Divide the given value in micrometers by 10,000: \[ 50500 imes 10^{-6} = 5.05 ext{ cm} \] Thus, 50,500 μm is equivalent to 5.05 cm.

Step 2: Convert Micrometers to Decimeters

To convert micrometers to decimeters, we use the conversion factor: 1 dm = 100,000 μm. Divide the given value in micrometers by 100,000: \[ 50500 imes 10^{-5} = 0.505 ext{ dm} \] Hence, 50,500 μm is equal to 0.505 dm.

Step 3: Convert Micrometers to Meters

To convert micrometers to meters, we use the conversion factor: 1 m = 1,000,000 μm. Divide the given length by 1,000,000: \[ 50500 imes 10^{-6} = 0.0505 ext{ m} \] Therefore, 50,500 μm is equivalent to 0.0505 m.

Key Concepts

micrometers to centimeters

Understanding the conversion from micrometers to centimeters is quite simple once you grasp the basic idea of unit conversion. Micrometers (\( \mu \text{m} \)) are a smaller unit compared to centimeters (cm). Therefore, you will find more micrometers in one centimeter. For this conversion, it is important to remember the key conversion factor:

• 1 centimeter = 10,000 micrometers

This means that you will divide the number of micrometers by 10,000 to get the equivalent in centimeters. With this factor, you can easily convert any given length from micrometers to centimeters. For instance, if you have 50,500 micrometers and want to express it in centimeters, you simply calculate:\[50500 \div 10000 = 5.05 \text{ cm}\]As shown, 50,500 micrometers equals 5.05 centimeters. This result is rounded to three significant figures, which is a common practice in science to maintain precision without implying unwarranted accuracy.

micrometers to decimeters

Converting micrometers to decimeters involves a similar strategy, just with a different conversion factor. Here, decimeters (dm) are larger than centimeters, so even more micrometers fit into a decimeter than a centimeter. The conversion factor for micrometers to decimeters is:

• 1 decimeter = 100,000 micrometers

To express micrometers in decimeters, you divide by 100,000.Let's apply it to the same example: converting 50,500 micrometers into decimeters.\[50500 \div 100000 = 0.505 \text{ dm}\]The conversion shows that 50,500 micrometers is equal to 0.505 decimeters, once rounded to three significant digits. This process demonstrates the utility of conversion factors in transitioning between units succinctly.

micrometers to meters

Finally, converting micrometers to meters requires using a larger conversion factor because a meter is much bigger than both centimeters and decimeters. The conversion factor is critical here:

• 1 meter = 1,000,000 micrometers

This means any given number in micrometers must be divided by 1,000,000 to convert it to meters.Let's look into converting 50,500 micrometers to meters:\[50500 \div 1000000 = 0.0505 \text{ m}\]Thus, 50,500 micrometers convert to 0.0505 meters, again keeping with three significant figures. By comprehending each conversion factor, students can move seamlessly through unit conversions, allowing for better scientific literacy and understanding of measurement systems.