Ch 36: Special Relativity

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 36: Special Relativity

Problem 48

Chapter 36, Problem 48

The Stanford Linear Accelerator (SLAC) accelerates electrons to v = 0.99999997c in a 3.2-km-long tube. If they travel the length of the tube at full speed (they don’t, because they are accelerating), how long is the tube in the electrons’ reference frame?

Verified step by step guidance

Identify the concept involved: This problem deals with length contraction, a phenomenon in special relativity where the length of an object moving relative to an observer is shorter than its proper length (the length measured in the object's rest frame). The formula for length contraction is:

L = L o_{1} \sqrt{1 - v^{2} / c^{2}}

, where

L

is the contracted length,

L o_{1}

is the proper length,

v

is the velocity, and

c

is the speed of light.

Write down the given values: The proper length of the tube in the lab frame is

L o_{1} = 3.2 km

. The velocity of the electrons is

v = 0.99999997 c

. The speed of light is

c = 3 \times 10 ¹² m/s

Substitute the given values into the length contraction formula:

L = 3.2 km \sqrt{1 - (^{0.99999997) 2}}

. Note that the velocity is already expressed as a fraction of the speed of light, so

v / c = 0.99999997

Simplify the term inside the square root: Calculate

(^{0.99999997) 2}

, subtract it from 1, and then take the square root of the result. This will give the factor by which the proper length is contracted.

Multiply the proper length by the contraction factor: Once the square root is calculated, multiply it by

3.2 km

to find the contracted length

L

, which is the length of the tube in the electrons’ reference frame.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Relativity of Length

According to the theory of relativity, the length of an object as measured in a frame of reference moving relative to the object is contracted. This phenomenon, known as length contraction, occurs at speeds approaching the speed of light (c). The formula for length contraction is L = L0√(1 - v²/c²), where L0 is the proper length, v is the velocity of the moving object, and L is the contracted length observed in the moving frame.

Speed of Light

The speed of light in a vacuum, denoted as c, is a fundamental constant in physics, approximately equal to 299,792,458 meters per second. It represents the maximum speed at which information or matter can travel. In the context of relativity, as an object approaches this speed, its relativistic effects, such as time dilation and length contraction, become significant, altering the perception of time and space.

Reference Frames

A reference frame is a perspective from which measurements are made, including the position, velocity, and time of objects. In physics, different reference frames can yield different observations of the same event, particularly in relativistic contexts. For example, an observer at rest will measure different lengths and times compared to an observer moving at a significant fraction of the speed of light, highlighting the relativity of simultaneity and spatial measurements.

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