Problem 14 - Fundamentals of Physics

Question

A rod is to move at constant speed \(v\) along the \(x\) axis of reference frame \(S\), with the rod's length parallel to that axis. An observer in frame \(S\) is to measure the length \(L\) of the rod. Figure \(37-23\) gives length \(L\) versus speed parameter \(\beta\) for a range of values for \(\beta\). The vertical axis scale is set by \(L_{a}=1.00 \mathrm{~m}\). What is \(L\) if \(v=0.95 c\) ?

Step-by-Step Solution

Answer

The length \(L\) is approximately 0.312 meters.

Step 1: Understand the Relationship

The length of a rod measured in a different frame moving at a speed relative to the rod's rest frame is governed by the formula for length contraction. It states that the length in the observer's frame, denoted as \(L\), is given by \(L = L_0 \sqrt{1-\beta^2}\) where \(L_0\) is the proper length of the rod (length in its rest frame), and \(\beta\) is the speed parameter defined as \(\beta = \frac{v}{c}\).

Step 2: Identify Known Values

We are given that \(v = 0.95c\). Thus, \(\beta = \frac{v}{c} = 0.95\). The vertical axis scale, \(L_a\), corresponds to \(L_0\) which is 1.00 m (the proper length of the rod in its rest frame).

Step 3: Apply the Length Contraction Formula

Using the length contraction formula: \[ L = L_0 \sqrt{1-\beta^2} \]Substitute in the known values to calculate the contracted length \(L\).

Step 4: Calculate \(1-\beta^2\)

Since \(\beta = 0.95\), calculate \(1-\beta^2\): \[ 1-\beta^2 = 1-(0.95)^2 = 1-0.9025 = 0.0975 \]

Step 5: Compute the Contracted Length \(L\)

Now plug the value from Step 4 into the length contraction formula:\[ L = 1.00 \sqrt{0.0975} \] Calculate the square root of 0.0975.

Step 6: Final Calculation

Compute \(\sqrt{0.0975} \approx 0.3122\). Multiply this by the proper length to find \(L\):\[ L = 1.00 \times 0.3122 \approx 0.3122 \text{ m}\]

Step 7: Conclusion

The measured length \(L\) when \(v = 0.95c\) is approximately 0.3122 meters.

Key Concepts

Length Contraction

Length contraction is a concept from Einstein's theory of relativity that describes how the length of an object changes when it is moving at a significant fraction of the speed of light. This effect becomes noticeable when dealing with speeds approaching the speed of light, known as relativistic speeds.
For length contraction to occur, the object must be moving relative to an observer's frame of reference. The length you measure in this moving frame is not the same as the length measured in the object's rest frame. Instead, it is shorter.
The formula for length contraction is:

• \[ L = L_0 \sqrt{1-\beta^2} \], where:
  • \( L \) is the contracted length (the length an observer measures)
  • \( L_0 \) is the proper length (the length of the object in its rest frame)
  • \( \beta \) is the speed parameter

The effect of length contraction is imperceptible at everyday speeds, and it only becomes significant at velocities close to the speed of light.

Speed Parameter

The speed parameter, often represented by \( \beta \), is a crucial concept in relativity. It helps us understand the effects of traveling at high speeds, such as length contraction and time dilation.
Speed parameter \( \beta \) is defined as:

• \( \beta = \frac{v}{c} \)

Here:

• \( v \) is the velocity of the object
• \( c \) is the speed of light (approximately \( 3 \times 10^8 \text{ m/s} \))

So, \( \beta \) represents the ratio of the object's speed to the speed of light. It’s always a value between 0 and 1. When \( \beta \) approaches 1, the equations of relativity start to have significant effects, as evident from length contraction, where high \( \beta \) leads to significantly contracted lengths.

Lorentz Factor

The Lorentz factor, also known as the Lorentz transformation factor, plays a key role in relativity, particularly when exploring high-speed physics.
Defined as:

• \[ \gamma = \frac{1}{\sqrt{1-\beta^2}} \]

It shows how much time, length, and relativistic mass appear to change for an object moving at a velocity \( v \). The Lorentz factor comes into play in scenarios of high-speed travel, and affects calculations that involve time dilation and length contraction.
When you apply the Lorentz factor:

• As \( \beta \) becomes closer to 1 (meaning speed \( v \) is close to \( c \)), \( \gamma \) increases significantly, highlighting strong relativistic effects.
• It is crucial for ensuring the consistency of physical laws across different reference frames, providing a bridge for transforming measurements between stationary and moving observers.

Reference Frame

Understanding reference frames is fundamental when discussing relativity. A reference frame is essentially a viewpoint or coordinate system used to measure and observe physical phenomena.
In physics, an observer's measurements of space and time can vary depending on their reference frame, which could be stationary or moving relative to other frames. Reference frames are crucial for understanding how different observers perceive motion, time, and length differently when they aren’t experiencing the same inertial conditions.
Key points to remember:

• Each observer in a different reference frame may measure different values for quantities like velocity, time intervals, and distance.
• The laws of physics are the same in all inertial reference frames, a principle known as the relativity principle.
• Transformations like the Lorentz transformation help convert measurements from one frame to another.

Relativity famously tells us that there is no absolute reference frame; all motion is relative to some observer or framework.