Problem 42 - College Physics

Question

\(\cdot\) A cube of metal with sides of length \(a\) sits at rest in the laboratory with one edge parallel to the \(x\) axis. Therefore, in the laboratory frame, its volume is \(a^{3} .\) A rocket ship flies past the laboratory parallel to the \(x\) axis with a velocity \(v .\) To an observer in the rocket, what is the volume of the metal cube?

Step-by-Step Solution

Answer

The volume of the metal cube as observed from the rocket is \( a^3 \sqrt{1 - \frac{v^2}{c^2}} \).

Step 1: Understanding Length Contraction

In special relativity, an object moving at velocity \( v \) along an axis exhibits length contraction in the direction of motion. The contraction is given by \( L = L_0 \sqrt{1 - \frac{v^2}{c^2}} \), where \( L_0 \) is the rest length, and \( c \) is the speed of light.

Step 2: Applying Length Contraction to the Cube

Since the cube moves parallel to the \( x \) axis, only the side of length \( a \) parallel to the \( x \) axis contracts. The new length of this side as observed from the rocket is \( a' = a \sqrt{1 - \frac{v^2}{c^2}} \).

Step 3: Calculation of the New Volume

The cube's other two dimensions (which are perpendicular to the direction of motion) remain unchanged at length \( a \). Thus, the new volume \( V' \) as observed from the rocket is given by \( V' = a' \times a \times a = (a \sqrt{1 - \frac{v^2}{c^2}}) \times a^2 \).

Step 4: Simplifying the Expression for the Volume

Substitute \( a' \) into the expression for the volume, resulting in \( V' = a^3 \sqrt{1 - \frac{v^2}{c^2}} \). This is the volume of the cube as observed from the rocket.

Key Concepts

Length Contraction

Let's dive into the concept of length contraction, a fascinating phenomenon of special relativity. When an object moves at a significant fraction of the speed of light, it experiences changes in its dimensions, specifically along the direction of motion. This effect is not something we encounter in our daily experiences, as it only becomes significant at relativistic speeds. This is the speed at which relativity theories become crucial.
Length contraction can be mathematically described by the formula \( L = L_0 \sqrt{1 - \frac{v^2}{c^2}} \). Here, \( L_0 \) is the original or rest length of the object, \( v \) is the velocity of the moving object, and \( c \) is the speed of light, which is approximately 299,792,458 meters per second.

• The contraction only affects the dimension parallel to the direction of motion.
• The closer the object’s speed is to the speed of light, the more noticeable the contraction.

In our specific problem, a cube's side parallel to the direction of motion contracts to a shorter length \( a' = a \sqrt{1 - \frac{v^2}{c^2}} \), while the other dimensions remain the same.

Volume Calculation

Once we grasp the concept of length contraction, calculating the new volume of an object in motion becomes straightforward. In the context of our cube, we need to consider how its volume changes as observed by someone moving with respect to it.
The original volume of a cube is \( a^3 \), because all its sides are equal, having a length of \( a \).
However, due to length contraction, the side of the cube that is aligned with the velocity vector, typically the \( x \)-axis in our exercise, contracts to \( a' = a \sqrt{1 - \frac{v^2}{c^2}} \). The other two sides, which are perpendicular to the direction of motion, remain unaffected with the length \( a \). Therefore, the volume \( V' \) as observed from the moving rocket is \[ V' = a' \times a \times a = (a \sqrt{1 - \frac{v^2}{c^2}}) \times a^2 = a^3 \sqrt{1 - \frac{v^2}{c^2}} \].

• Notice how the contraction affects only one dimension of the cube.
• The expression for the new volume involves the original volume multiplied by the contraction factor \( \sqrt{1 - \frac{v^2}{c^2}} \).

Speed of Light

The speed of light, often symbolized as \( c \), stands as a cornerstone of modern physics, particularly in the realm of special relativity. It defines the ultimate speed limit in our universe, which holds profound implications for objects traveling at high speeds. The constant value of \( c \) is approximately 299,792,458 meters per second, an incredibly fast velocity that everyday objects never achieve.
In special relativity, the speed of light serves as a critical component in many formulae, including the one for length contraction. Here, it acts as a baseline to measure how much an object's length contracts at various speeds. The formula \( L = L_0 \sqrt{1 - \frac{v^2}{c^2}} \) shows how this speed is used to calculate the factor that determines contraction.

• No object with mass can reach the speed of light, as it would require infinite energy.
• The speed of light is consistent in all frames of reference, meaning it does not change regardless of the observer's motion.

Thus, understanding the role of the speed of light allows us not just to apply the mathematical equations correctly but also to appreciate the behavior of moving objects when viewed from different inertial frames.