Question

A uniform cube is subjected to volume compression. If each side is decreased by \(1 \%\) Then what is bulk strain ? (A) \(0.01\) (B) \(0.06\) (C) \(0.02\) (D) \(0.03\)

Short answer

The bulk strain is approximately -0.03, so the correct answer is (D) \(0.03\).

Step-by-step solution

Calculate the initial volume

Let's denote the initial side length of the cube as s. The initial volume of the cube V₁ = s³

Find the new side length

As each side is subjected to a volume compression of 1%, the new side length, s', will be 99% of its original length: s' = 0.99s

Calculate the new volume

The new volume of the cube, V₂, can be found using the new side length: V₂ = (s')³ = (0.99s)³

Determine the change in volume

Now, we can find the change in volume, ΔV, by subtracting the initial volume from the new volume: ΔV = V₂ - V₁ = (0.99s)³ - s³

Calculate the bulk strain

The bulk strain is defined as the ratio of the change in volume to the initial volume: Bulk Strain = \( \frac{ΔV}{V₁} \) Substituting the expressions for ΔV and V₁ we found earlier, we get: Bulk Strain = \( \frac{(0.99s)^3 - s^3}{s^3} \)

Simplify the expression

Now, let's simplify the expression for the Bulk Strain: Bulk Strain = \( \frac{(0.99^3)s^3 - s^3}{s^3} \) Since the denominator and the numerators all have the same s³, we can cancel them out: Bulk Strain = \( 0.99^3 - 1 \)

Calculate the numerical value

Now, we will compute the numerical value of the Bulk Strain: Bulk Strain = \( 0.99^3 - 1 ≈ -0.0297 \)

Select the correct answer

Since the calculated Bulk Strain ≈ -0.0297 ≈ -0.03, the correct answer is: (D) \(0.03\)

Key concepts

Volume Compression

Volume compression refers to the decrease in volume of a three-dimensional object due to external forces. In our exam question, we are analyzing how a change in the side length of a cube affects its overall volume. When each side of the cube is compressed by a certain percentage, the whole structure reduces in size.

For instance, if each side of our cube reduces by 1%, it affects the total volume significantly because volume is proportional to the cube of the side length. This means a small change in side length results in a larger change in volume. This kind of multiplication effect is at the heart of understanding volume compression and helps us calculate bulk strain - which is a measure of deformation due to compression.

Cube Geometry

Understanding cube geometry is crucial when studying how changes to a cube's side length affect its volume. A cube is a simple shape where each edge is equal in length, and its volume is calculated as the cube of its side length, expressed mathematically as:

• Volume, \( V = s^3 \)

Therefore, if any side length changes, you must cube this change to find the new volume. This principle is especially helpful in exercises involving geometric transformations like volume compression. In our question, when the side length was reduced by 1%, the new side length is 99% of the original length. Hence, the new volume is calculated using

• New volume, \( V_2 = (0.99s)^3 \)

The cube's symmetry and uniformity make calculations straightforward, but also highlight how sensitive volume is to changes in edge length.

Percentage Change

Percentage change is a way to express the degree of change over an original amount in percentage terms. In the context of the exercise, a 1% decrease in the side length translates into significant changes in volume.

To calculate the percentage change in length:

• New Length = Old Length - (Percentage Change × Old Length)

In this case, for a 1% decrease:

• New Length, \( s' = 0.99s \)

When compressed, the new metric doesn’t just alter the side length but causes a cascading effect affecting the cube's volume. This is why understanding percentage changes is crucial for predicting behaviors in physical systems like our cube.

Change in Volume

Change in volume is the key parameter we need to determine the bulk strain of the cube. This is computed as the difference between the initial and new volumes, which acts as a comparative measure of deformation. In our exercise, it is defined as:

• Change in Volume, \( \Delta V = V_2 - V_1 \)

With the initial volume specified as \( V_1 = s^3 \) and the new volume as \( V_2 = (0.99s)^3 \), the change is calculated using:

• \( \Delta V = (0.99s)^3 - s^3 \)

This simplifies to:

• \( \Delta V = s^3 (0.99^3 - 1) \)

This loss in volume directly translates to our understanding and quantification of bulk strain, yielding numerical insights into the cube's deformation due to compression.