Understanding Force Units: Dyne and Newton

The question asks us to find the equivalent value of 1 dyne in terms of kg m/s². Dyne is the unit of force in the CGS (Centimetre-Gram-Second) system, while the unit kg m/s² is the unit of force in the SI (Système International) system, which is also known as the Newton (N).

Force is defined by Newton's second law as mass times acceleration (\(F = ma\)). The units of force are derived from the units of mass, length, and time in a specific system of measurement.

• In the CGS system:
  • Mass is measured in grams (g).
  • Length is measured in centimeters (cm).
  • Time is measured in seconds (s).
  • The unit of force, the dyne, is \( 1 \text{ g} \times \text{ cm/s}^2 \), so \( 1 \text{ dyne} = 1 \text{ g cm/s}^2 \).
• In the SI system:
  • Mass is measured in kilograms (kg).
  • Length is measured in meters (m).
  • Time is measured in seconds (s).
  • The unit of force, the Newton (N), is \( 1 \text{ kg} \times \text{ m/s}^2 \), so \( 1 \text{ Newton} = 1 \text{ kg m/s}^2 \).

Converting Dyne (CGS) to kg m/s\(^2\) (SI)

To convert dyne to kg m/s², we need to convert the CGS base units (gram and centimeter) into their corresponding SI base units (kilogram and meter).

We know the following conversions:

• \( 1 \text{ gram (g)} = 10^{-3} \text{ kilograms (kg)} \)
• \( 1 \text{ centimeter (cm)} = 10^{-2} \text{ meters (m)} \)
• \( 1 \text{ second (s)} = 1 \text{ second (s)} \) (The unit of time is the same in both systems)

Now, let's substitute these conversions into the definition of 1 dyne:

\( 1 \text{ dyne} = 1 \text{ g cm/s}^2 \)

Substitute the values in terms of kg and m:

\( 1 \text{ dyne} = (10^{-3} \text{ kg}) \times (10^{-2} \text{ m}) / (\text{s}^2) \)

\( 1 \text{ dyne} = (10^{-3} \times 10^{-2}) \text{ kg m/s}^2 \)

Using the rule for exponents \( a^m \times a^n = a^{m+n} \):

\( 10^{-3} \times 10^{-2} = 10^{(-3) + (-2)} = 10^{-5} \)

So, the conversion becomes:

\( 1 \text{ dyne} = 10^{-5} \text{ kg m/s}^2 \)

Since \( 1 \text{ kg m/s}^2 \) is equal to \( 1 \text{ Newton} \), we can also say that \( 1 \text{ dyne} = 10^{-5} \text{ N} \).

Comparing with the Options

We found that \( 1 \text{ dyne} = 10^{-5} \text{ kg m/s}^2 \). Let's look at the given options:

• Option 1: \( 10^{3} \text{ g cm/s}^2 \). This is just the definition of 1000 dynes in CGS units, not in kg m/s².
• Option 2: \( 10^{-3} \text{ g cm/s}^2 \). This is \( 10^{-3} \) dynes in CGS units, not in kg m/s².
• Option 3: \( 10^{5} \text{ kg m/s}^2 \). This is \( 10^{5} \) Newtons. Since \( 1 \text{ N} = 10^5 \text{ dynes} \), this option represents \( 10^{5} \times 10^5 \text{ dynes} = 10^{10} \text{ dynes} \).
• Option 4: \( 10^{-5} \text{ kg m/s}^2 \). This matches our calculated value for 1 dyne. This is equivalent to \( 10^{-5} \text{ N} \).

Thus, 1 dyne equals \( 10^{-5} \text{ kg m/s}^2 \).

Revision Table: Key Force Units and Conversions

Here is a summary of the relationship between the CGS and SI units of force:

Additional Information on Force and Units

Understanding unit conversions is crucial in physics. Different systems of units evolved historically, and the SI system is now the internationally accepted standard for most scientific and engineering purposes. The CGS system is still used in some specific fields, particularly in theoretical physics.

• Dimensional Analysis: The process we used to convert dyne to Newton is an example of dimensional analysis. It involves converting the base units of a physical quantity from one system to another using known conversion factors.
• Base and Derived Units: Mass, length, and time are considered base quantities with base units (kg, m, s in SI). Force is a derived quantity, meaning its unit (\( \text{kg m/s}^2 \)) is derived from the base units according to a physical law (\( F=ma \)).
• Other Units of Force: Besides dyne and Newton, other units of force exist, such as the pound-force (lbf) in the imperial system, although Newton is the standard in scientific contexts.

Being able to convert between different units ensures that calculations are accurate regardless of the system used for initial measurements.