Problem 11 - Fundamentals of Physics

Question

A rod lics parallel to the \(x\) axis of reference frame \(S,\) moving along this axis at a speed of \(0.630 \mathrm{c}\). Its rest length is \(1.70 \mathrm{~m}\). What will be its measured length in frame \(S ?\)

Step-by-Step Solution

Answer

The measured length is approximately 1.32 m in frame S.

Step 1: Determine Relativistic Speed

First, identify the speed of the moving reference frame with respect to the speed of light. The given speed is 0.630c, where c represents the speed of light.

Step 2: Use Length Contraction Formula

In special relativity, length contraction can be calculated using the formula: \( L = L_0 \sqrt{1 - \frac{v^2}{c^2}} \), where \( L_0 \) is the rest length, \( L \) is the contracted length, \( v \) is the speed of the object, and \( c \) is the speed of light.

Step 3: Plug in Given Values

The rest length \( L_0 \) is given as 1.70 m and the speed \( v \) is 0.630c. Substitute these into the formula: \[ L = 1.70 \sqrt{1 - (0.630)^2} \]

Step 4: Calculate the Fraction Inside the Square Root

Calculate \( (0.630)^2 = 0.3969 \). Then, compute \( 1 - 0.3969 = 0.6031 \).

Step 5: Evaluate the Square Root

Calculate the square root: \( \sqrt{0.6031} \approx 0.7763 \).

Step 6: Compute the Contracted Length

Substitute back into the formula to find \( L \): \[ L = 1.70 \times 0.7763 \approx 1.32 \text{ m} \]

Key Concepts

Special Relativity

The theory of special relativity, introduced by Albert Einstein in 1905, revolutionized our understanding of time, space, and motion. It particularly addresses the behavior of objects that move close to the speed of light. One of the key tenets of special relativity is that the laws of physics are the same for all observers, regardless of their relative motion, as long as they move at constant speeds.
This theory introduced two groundbreaking effects:

• Time Dilation: Time runs slower for an object in motion compared to an object at rest, as seen from the perspective of an outside observer.
• Length Contraction: Objects in motion are measured to be shorter in the direction of motion by observers at rest.

These effects occur because the speed of light, denoted by \( c \), is constant and unattainable by any object with mass. As a result, the closer an object's speed is to \( c \), the more noticeable these relativistic effects become.

Relativistic Speed

Relativistic speed refers to the speed of objects moving at a significant fraction of the speed of light. In the context of the exercise, the rod is moving at a speed of 0.630\( c \), meaning it travels at 63% of the speed of light. Speeds approaching that of light cause a variety of relativistic effects, which are key components of special relativity.
When dealing with these high speeds:

• Classical physics, governed by Newtonian mechanics, does not apply accurately. New equations, like the one for length contraction, must be used to predict behaviors.
• Measurements of space (length) and time become intertwined, leading to the effects of time dilation and length contraction.

The relativistic speed ensures that everyday experiences with high-speed phenomena are significantly different, emphasizing the essential role of Einstein’s special relativity in modern physics.

Lorentz Transformation

The Lorentz transformations are a set of equations in special relativity that relate the space and time coordinates of two observers moving at constant velocity relative to each other. Introduced by Hendrik Lorentz, these transformations help explain how measurements of time and space are affected by motion at relativistic speeds.
Key principles of Lorentz transformations:

• They ensure that the speed of light remains constant and the same in all inertial frames of reference, regardless of the observer’s motion.
• They predict the phenomena of time dilation and length contraction, making them fundamental in calculations involving high-speed objects.

In the context of the length contraction formula, the Lorentz factor \( \gamma \) is calculated as \( \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \). This factor modifies the measurements of time and space, ultimately leading to contracted lengths seen in objects moving at significant speeds relative to an observer.