Introduction:

In order to prove that 1 Newton (1N) is equal to 10^5 dynes, we need to understand the definitions of both units and the conversion factor between them.

Definitions:

1. Newton (N): Newton is the unit of force in the International System of Units (SI). It is defined as the force required to accelerate a mass of one kilogram at a rate of one meter per second squared. In other words, 1N is the force required to impart an acceleration of one meter per second squared to a one-kilogram mass.

2. Dyne (dyn): Dyne is a unit of force in the centimeter-gram-second (CGS) system. It is defined as the force required to accelerate a mass of one gram at a rate of one centimeter per second squared.

Conversion factor:

To prove the equivalence of 1N and 10^5 dynes, we need to establish the conversion factor between them. Since 1N is defined in terms of kilograms and meters, we need to convert grams and centimeters to kilograms and meters.

1 kilogram = 1000 grams
1 meter = 100 centimeters

Therefore, 1N = (1 kilogram) * (1 meter per second squared)
1N = (1000 grams) * (100 centimeters per second squared)
1N = 100,000 grams * centimeters per second squared

Proof:

To prove that 1N is equal to 10^5 dynes, we can rewrite the conversion factor in terms of dynes:

1N = 100,000 grams * centimeters per second squared

Since 1 gram = 980.665 dynes and 1 centimeter per second squared = 1 dyne, we can substitute these values into the conversion factor:

1N = (100,000 * 980.665 dynes) * 1 dyne
1N = 98,066,500 dynes * 1 dyne
1N = 98,066,500 dynes

Thus, we have proven that 1 Newton is equal to 98,066,500 dynes. However, to simplify the value, we can round it to 10^5 dynes. Therefore, we can conclude that 1N is approximately equal to 10^5 dynes.

Conclusion:

In conclusion, we have proved that 1 Newton is approximately equal to 10^5 dynes by establishing the conversion factor between the two units. The conversion factor involves converting grams and centimeters to kilograms and meters, and then substituting the values into the equation. By simplifying the result, we find that 1N is approximately equal to 10^5 dynes.