Unit conversion in mechanics: In classical mechanics and SI–CGS conversions, one Newton (N) is equal to how many dynes?

Introduction / Context:
Force units are frequently converted between the SI system (Newton, N) and the CGS system (dyne). Being comfortable with these conversions is essential in physics, chemical engineering calculations, and instrumentation where legacy CGS data may appear alongside modern SI data.

Given Data / Assumptions:

• SI base: 1 N = 1 kg * m / s^2.
• CGS base: 1 dyne = 1 g * cm / s^2.
• Exact metric relations: 1 kg = 10^3 g; 1 m = 10^2 cm.

Concept / Approach:
Write both units in terms of base dimensions and convert systematically using the kilogram–gram and meter–centimeter relations. Avoid memorizing isolated numbers; derive once from first principles to reduce mistakes.

Step-by-Step Solution:
Start with 1 N = 1 kg * m / s^2.Convert kg to g: 1 kg = 10^3 g → 1 N = 10^3 g * m / s^2.Convert m to cm: 1 m = 10^2 cm → 1 N = 10^3 * 10^2 g * cm / s^2.Multiply powers of 10: 10^3 * 10^2 = 10^5.Therefore 1 N = 10^5 g * cm / s^2 = 10^5 dynes.

Verification / Alternative check:
The dyne is defined so that 1 dyne accelerates 1 g at 1 cm/s^2. Using dimensional analysis consistently always yields 10^5 as the factor between Newtons and dynes.

Why Other Options Are Wrong:
10^2, 10^3, and 10^4 dynes underestimate the conversion because they miss one or both metric scale factors between kg↔g and m↔cm.

Common Pitfalls:

• Converting only mass or only length but not both.
• Confusing dyne with pound-force; use consistent metric bases.

Final Answer:
10^5 dynes