Problem 13 - Master Resource Book in JEE Main Physics

Question

The surface tension of mercury is 32 dyne \(\mathrm{cm}^{-1}\). Its value in SI units is (a) \(0.032\) (b) \(0.32\) (c) 3200 (d) 32000

Step-by-Step Solution

Answer

The surface tension in SI units is 0.032 N/m (option a).

Step 1: Understand surface tension units

Surface tension in the CGS unit system is measured in dynes per centimeter ( dyne cm^{-1} ). We need to convert this into SI units, which is measured in Newtons per meter (N/m).

Step 2: Conversion factor

Recall that 1 dyne is equal to 0.00001 Newtons (1 dyne = 10^{-5} N), and 1 cm is equal to 0.01 meters (1 cm = 10^{-2} m).

Step 3: Apply conversion

We start with a surface tension of 32 dyne/cm and convert it using the conversion factors. Substitute the units: \[ 32 ext{ dyne/cm} = 32 imes 10^{-5} ext{ N/0.01 m} \].

Step 4: Simplify the conversion

Simplify the conversion equation: \[ 32 imes 10^{-5} ext{ N/0.01 m} = 32 imes 10^{-3} ext{ N/m} \].

Step 5: Final calculation

Perform the multiplication: \[ 32 imes 10^{-3} = 0.032 \]. Thus, the surface tension in SI units is 0.032 N/m.

Key Concepts

CGS Unit System

The CGS unit system is one of the systems used to measure physical quantities. It stands for Centimeter-Gram-Second, based on the metric system. In the world of science and engineering, CGS was among the most widely used unit systems before the International System of Units (SI) was established. The CGS system employs centimeters for length, grams for mass, and seconds for time. For forces and related measurements, like surface tension, the proper CGS unit is the dyne. Here, one dyne is defined as the force necessary to accelerate a 1 gram mass by 1 centimeter per second squared. Surface tension, in particular, is measured in dynes per centimeter (dyne/cm) in CGS. This reflects the force acting along the surface of a liquid, pressing into another medium. For those familiar with surface tension, understanding its CGS measurement provides a good base for converting into other unit systems, especially the SI system.

SI Unit Conversion

Converting units from CGS to SI can initially seem complex due to different base unit sizes, but with the use of conversion factors, it becomes straightforward. The SI unit system, or International System of Units, is more prevalent today, known for utilizing meters, kilograms, and seconds as its fundamental units. When converting force in the CGS system (dyne) to the SI system (Newton), remember:

• 1 dyne is equivalent to 10\(^{-5}\) Newtons.

For example, if you have a measurement in dynes and need to convert it to Newtons, simply multiply by 10\(^{-5}\). Likewise, when working with distances, where the measurement in the CGS system is in centimeters, you'll need to convert to meters for SI units:

• 1 centimeter equals 0.01 meters.

This means converting dyne/cm to N/m involves multiplying by 10\(^{-5}\) for the force and dividing by 0.01 for the distance. The overall conversion factor becomes 10\(^{-3}\), resulting in easier computational steps. The approach allows transformations where precision and accuracy are crucial for scientific study or engineering tasks.

Surface Tension Units

Surface tension refers to the force per unit length, acting at the surface of a liquid. It's what allows insects to walk on water or water droplets to form.In the CGS unit system, surface tension is denoted by dyne per centimeter (dyne/cm). This represents how much force is required to stretch or break the liquid surface. On the other hand, in the SI unit system, surface tension is measured in Newtons per meter (N/m).When converting surface tension units from CGS to SI, attention must be given to both force and length units, as explained previously:

• Conversion of force involves transforming dynes to Newtons: 32 dyne/cm becomes 32 \(\times\) 10\(^{-5}\) N.
• The length conversion from centimeters to meters results in dividing by 0.01, giving you 32 \(\times\) 10\(^{-5}\) / 0.01 = 0.032 N/m.

Understanding these conversions helps in comparing substance surface tensions accurately, whether in scientific research, industrial applications, or educational environments.