Question

A rod lies parallel to the \(x\) axis of reference frame \(S\), moving along this axis at a speed of \(0.632 c .\) Its rest length is \(1.68 \mathrm{~m}\). What will be its measured length in frame \(S ?\)

Short answer

The measured length of the rod in the frame S will be calculated using the steps provided above. As it is a numerical solution, the answer will be a precise numerical value (in metres).

Step-by-step solution

Identify given values

Identify the values given in the problem, which include: \n1. Speed of the rod (\(v\)) is \(0.632c\), where \(c\) is the speed of light. \n2. Rest length (\(L_0\)) is \(1.68m\).

Apply the length contraction formula

The formula for length contraction is \(L = L_0 \sqrt{1 - \frac{v^2}{c^2}}\). Substitute the given values into the formula.

Solve for \(L\)

Complete the substitution and simplify the expression to find the contracted length (L).

Key concepts

Speed of Light

The speed of light, denoted as \(c\), is a fundamental constant in physics, known to be approximately \(299,792,458\) meters per second. It's the maximum speed at which all energy, matter, and information in the universe can travel. In relativistic physics, the speed of light acts as a cosmic speed limit. No object with mass can reach this speed, but it's commonly used in various calculations, such as those involving length contraction.

When dealing with problems like the one about the rod moving at a velocity of \(0.632c\), it's necessary to relate velocities to this fundamental constant. The speed of light is crucial for understanding phenomena that occur at very high speeds, comparable to light speed. These phenomena include time dilation and length contraction, both essential components of Einstein's theory of relativity.

Reference Frame

In physics, a reference frame is a perspective from which an observer measures and observes events. Different reference frames can be stationary or moving, and each has its coordinates or system for measuring time and space.

Consider a rod lying parallel to the \(x\)-axis of reference frame \(S\). In this example, you have a rest frame (where the rod remains stationary) and a moving frame (where the rod is traveling at \(0.632c\)). In relativity problems, it's vital to identify the problem's reference frame to correctly apply formulas and interpret measurements from different perspectives. Measurements such as lengths and times can appear different when observed from different reference frames. This idea helps us better grasp how observers in various frames might get different results when measuring the same physical quantities.

• Rest Frame: Where the object is at rest.
• Moving Frame: Where the object moves relative to an observer.

Relativistic Physics

Relativistic physics is the branch of physics that deals with the structure and behavior of objects traveling at significant fractions of the speed of light. One of its key principles is that the laws of physics are the same for all observers, regardless of their relative motion.

Einstein's theory of relativity introduced concepts like time dilation and length contraction. Length contraction states that objects moving at a high velocity will appear shorter along the direction of motion than they are at rest. The formula for calculating the contracted length is \( L = L_0 \sqrt{1 - \frac{v^2}{c^2}} \), where \(L_0\) is the rest length, \(v\) is the velocity, and \(c\) is the speed of light.

This theory has been crucial in understanding high-speed particle interactions and has implications for the universe's structure and functioning. By studying relativistic physics, we gain deeper insights into how truly interconnected time, space, and speed really are.