Question

When a pressure of 100 atmosphere is applied on a spherical ball then its volume reduces to \(0.01 \%\). What is the bulk modulus of the material of the rubber in \(\left(\right.\) dyne \(\left./ \mathrm{cm}^{2}\right)\) (A) \(10 \times 10^{12}\) (B) \(1 \times 10^{12}\) (C) \(100 \times 10^{12}\) (D) \(20 \times 10^{12}\)

Short answer

The Bulk Modulus (B) of the rubber is approximately \(1 \times 10^{12}\) dyne/cm^2. (Option B)

Step-by-step solution

Determine the given values from the problem statement

The given values in the problem are the applied pressure (100 atm) and the reduced volume (0.01%). We need to calculate the Bulk Modulus (B) in dyne/cm^2.

Convert the pressure to dyne/cm^2

We are given the pressure in atmospheres, but we need it in dyne/cm^2. To convert from atmospheres to dyne/cm^2, use the conversion factor: \(1 atm = 1.01325 \times 10^6 dyne/cm^2\). Therefore, the pressure in dyne/cm^2 is: \(100 atm \times \frac{1.01325 \times 10^6 dyne/cm^2}{1 atm} = 1.01325 \times 10^8 dyne/cm^2 \)

Calculate the relative volume change

The given volume reduction is 0.01% or 0.0001 in decimal form. Since the volume reduces, we must include a negative sign, so the relative volume change is -0.0001.

Apply the Bulk Modulus formula

The formula for Bulk Modulus (B) is: \(B = - \frac{P}{\Delta V / V}\) Where P is the applied pressure and \(\Delta V / V\) is the relative volume change. Now plug in the values we found in steps 2 and 3: \(B = \frac{1.01325 \times 10^8 dyne/cm^2}{0.0001} = 1.01325 \times 10^{12} dyne/cm^2 \)

Identify the correct answer in the multiple-choice options

Comparing our calculated Bulk Modulus to the given options, we see that our result is closest to the option (B). Hence, the correct answer is: (B) \(1 \times 10^{12}\) dyne/cm^2

Key concepts

Volume Reduction

Volume reduction is a critical concept when discussing aspects of elasticity and compressibility of materials. In simple terms, volume reduction happens when a material is compressed under an applied force or pressure, leading to a decrease in its original volume.

In the context of the problem, a spherical ball experiences a compression that reduces its volume by a very small percentage. Specifically, the volume reduction is given as 0.01%, which, when expressed as a decimal, is 0.0001.

Understanding the concept of volume reduction helps us in calculating the Bulk Modulus of a material, which is the measure of a substance's resistance to uniform compression. The negative sign associated with volume reduction represents the decrease, reinforcing the concept that the volume has decreased from its original state.

Pressure Conversion

Pressure conversion is essential for problems involving different units of measurement. Pressure is the force per unit area applied on an object. It is often measured in various units like atmospheres, pascals, or dyne per square centimeter.

In this exercise, converting the pressure from atmospheres to dyne per cm squared is necessary because the Bulk Modulus is to be calculated in dyne/cm² units. The conversion factor applied here is crucial:

• 1 atmosphere = 1.01325 x 10⁶ dyne/cm²

By multiplying 100 atmospheres by this conversion factor, we change the pressure to 1.01325 x 10⁸ dyne/cm². Understanding this conversion ensures accuracy in calculations and helps in aligning problem values with the desired units for solutions.

Spherical Object

Spherical objects have unique properties distinguishing them from other shapes in physics. A sphere is perfectly symmetrical, which influences how it responds to external forces like pressure.

In problems involving spheres, pressure applied uniformly will affect the entire surface equally, leading to a uniform volume reduction. This uniformity simplifies calculations, such as those involving the Bulk Modulus, as assumptions made about changes in dimensions (volume) are consistent throughout the object.

Spherical geometry contributes significantly to understanding how materials behave under stress, as it provides a simplified and idealized context. This, combined with the fundamental principles of physics, helps predict outcomes such as the degree of compression or material deformation.

Dyne Per Cm Squared

Dyne per cm squared ( ext{dyne/cm}^2) is a unit of pressure in the centimeter-gram-second (CGS) system. It is widely used in physics to express force applied per unit area. Understanding units such as dyne/cm² is vital when engaging in precise scientific calculations.

In this specific exercise, the requirement to solve for Bulk Modulus in dyne/cm² means first ensuring all applied pressures are expressed in this unit. The conversion from atmospheres to dyne/cm² becomes essential because it aligns the problem parameters with the desired unit for a coherent solution.

Emphasizing the importance of consistent units across calculations ensures accurate and credible outcomes, thereby providing insights into the material characteristics like compressibility or resistance to deformation.