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Length contraction is the phenomenon in which the length of a moving object is measured to be shorter than its actual length, which is the length measured in the rest frame of the object.
- It is also known as Lorentz contraction or Lorentz-FitzGerald contraction (after Hendrik Lorentz and George Francis FitzGerald).
- It can be only visible at a significant fraction of the speed of light.
- Length contraction occurs only in the direction that the body travels.
- For standard objects, the effect is negligible at normal speeds and can be ignored for all practical purposes
- It becomes significant only when the object approaches the speed of light relative to the observer.
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Key Terms: Length contraction, Length contraction formula, Speed of light, Relative velocity, Speed, Distance, Length, Lorentz factor, Relative velocity
What is Length Contraction?
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Length contraction is the length measured in the rest frame of an object.
- It is also known as Lorentz-FitzGerald contraction or Lorentz contraction.
- It can be noticeable at a significant fraction of the speed of light.
- The length contraction phenomenon is only considered in the direction where an object is moving.
- Length contraction is only essential when the object moving compared to the speed of light with respect to an observer.
- At everyday speeds, this effect is negligible for standard objects and can be ignored for all practical applications.
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Length Contraction Formula
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The length contraction formula is given by
\(L = L_o\sqrt{1-\frac{v^2}{c^2}}\)
Where
- Lo is the length of the object at rest
- L is the length of an object within relativistic speed
- v is the velocity of the object
- c is the speed of light
Length Contraction Derivation
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Suppose a cosmic ray collides with a nucleus in the upper atmosphere of Earth, producing a muon; then, to the Earth-bound observer, the velocity relative to the muon is given as
v = Lo/ΔT
As the object moves relative to the observer, the time (Δt) is related to the Earth-bound observer. The velocity relative to a moving observer is given by
v = L/ΔTo
The moving observer travels with the muon at the proper time Δto, which leads to equal velocities. Therefore
Lo/ΔT = L/ΔTo
We know ΔT = γΔTo
On substituting this expression, we get
Lo/γΔTo = L/ΔTo
⇒ L = Lo/γ ...(i)
Where γ is known as the Lorentz factor and it is given by
\(\gamma = \frac {1}{\sqrt {1- \frac {v^2}{c^2}}}\)
On substituting the value of γ, we get
\(L = \frac {L_o}{\sqrt {1- \frac {v^2}{c^2}}}\)
Where
- v is the relative velocity between the moving object and the observer
- c is the speed of light
Solved Examples
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Ques. Calculate the contracted length of an object whose initial length is 20 m and travels with a velocity of 0.5 c, where c is the speed of light.
Ans. Given
- The original length of the object, Lo = 20 m
- The velocity of the object, v = 0.5 c
According to the length contraction formula, the contracted length of the object is given by
\(L = \frac {L_o}{\sqrt {1- \frac {v^2}{c^2}}}\)
On substituting the value, we get
\(L = \frac {20}{\sqrt {1- \frac {(0.5c)^2}{c^2}}}\)
\(\Rightarrow L = \frac {20}{\sqrt {0.75}}\)
\(\Rightarrow L = 22.9 \: m\)
Ques. Calculate the contracted length of an object whose initial length is 15 m and travels with a velocity of 0.8 c, where c is the speed of light.
Ans. Given
- The original length of the object, Lo = 15 m
- The velocity of the object, v = 0.8 c
According to the length contraction formula, the contracted length of the object is given by
\(L = \frac {L_o}{\sqrt {1- \frac {v^2}{c^2}}}\)
On substituting the value, we get
\(L = \frac {15}{\sqrt {1- \frac {(0.8c)^2}{c^2}}}\)
\(\Rightarrow L = \frac {15}{\sqrt {0.36}}\)
\(\Rightarrow L = 25 \: m\)
Things to Remember
- Length contraction is the phenomenon in which the length of a moving object is measured to be shorter than its actual length.
- It is also known as Lorentz contraction or Lorentz-FitzGerald contraction.
- Length contraction occurs only in the direction that the body travels.
- The length contraction formula is given by L = Lo / √(1 – v2/c2).
- It is also given by L = γLo, where γ is known as the Lorentz factor.
Also Read:
Sample Questions
Ques. What is the Proper Length? (1 Mark)
Ans. When an observer measures the distance between two positions when at rest relative to both of them, the length is referred to as proper length Lo.
Ques. Who developed the special theory of relativity? (1 Mark)
Ans. Albert Einstein proposed the special theory of relativity.
Ques. Write the equation that shows energy and mass are interchangeable. (1 Mark)
Ans. The equation that shows energy and mass are interchangeable is given by Einstein’s equation E = mc2.
Ques. A rod has a length of 100 m when the rod is in a satellite. Which is moving with a velocity half of the speed of light with respect to the laboratory frame. What is the length of the road observed by the observer in the laboratory? (3 Marks)
Ans. The actual length for the first part of the question will be 100m.
Lo = 100m
V = \(\frac{c}{2}\)
We need to find the length of the road observed by the observer in the laboratory, given by
\(L = L_o\sqrt{1-\frac{v^2}{c^2}}\)
\(L = 100\sqrt{1-\frac{c^2}{4c^2}}\)
\(L = 100\sqrt{1-0.25}\)
L = 86.6 m
Ques. What will the length of a meter road appear to be for a person traveling parallel to the length of the road at a speed of 0.8c relative to the road? (3 Marks)
Ans. The actual length (Lo) given is 100m and the given velocity (v) is given as 0.8c.
We know that -;
\(L = L_o\sqrt{1-\frac{V^2}{C^2}}\)
\(L = 1\sqrt{1-\frac{(0.8c)^2}{c^2}}\)
\(L =\sqrt{1-0.64}\)
L = 0.6m = 60 cm
Ques. Calculate the percentage contraction in the length of a rod in a frame of reference, moving with velocity 0.8c in a direction parallel to its length. (5 Marks)
Ans. The given velocity is 0.8c. We know that,
\(L = L_o\sqrt{1-\frac{v^2}{c^2}}\)
\(L = L_o\sqrt{1-\frac{(0.8c)^2}{c^2}}\)
\(L = L_o\sqrt{1-0.64}\)
\(L = L_o\sqrt{0.36}\)
L = 0.6Lo
Percentage contraction = \(\frac{L-L_o}{L_o} \times 100\)
Percentage contraction = 40%
Ques. If a space save 50 long were to pass the earth traveling at 2.4 × 108 m/s, what would be its apparent length, assuming a Lorentz-Fitzgerald contraction? (3 Marks)
Ans. The actual length is given as 50m and the velocity is given as 2.4 × 108 m/s
We know that,
\(L = L_o\sqrt{1-\frac{V^2}{C^2}}\)
\(L = 50\sqrt{1-\frac{(2.4 \times 10^8)^2}{(3 \times 10^8)^2}}\)
\(L =50\sqrt{1-0.64}\)
\(L = 50\sqrt{0.36}\)
L = 30.00 m
Ques. A rocket ship is 50m long. When it is on flight, its length appears to be 49.5m to an observer on the ground. Find the speed of the rocket. (3 Marks)
Ans. Lo= 50m
L = 49.5m
\(L = L_o\sqrt{1-\frac{V^2}{C^2}}\)
\(49.5 = 50\sqrt{1-\frac{V^2}{C^2}}\)
\(\frac{49.5}{50} = \sqrt{1-\frac{V^2}{C^2}}\)
(0.99)² = 1 - V²/C²
1 - (0.99)² = V²/C²
V = \(\sqrt{0.0199c^2}\)
v = 0.141c
v = 42.3 × 108 m/s
Ques. A rocket ship is 100m long on the ground. When it is in flight, its length is 99 m to an observer on the ground. What is its speed? (3 Marks)
Ans. L = 99
Lo= 100
We know that,
\(L = L_o\sqrt{1-\frac{V^2}{C^2}}\)
\(\frac{99}{100}= \sqrt{1-\frac{V^2}{C^2}}\)
(0.99) = 1 -v2 /c2
\(\frac{v^2}{c^2} =1-(0.99)^2\)
v² /c² = 0.0199
V = 0.14c
Ques. The length of a rod at rest is 1 m. What will be its length if it is moving at one-third of the speed of light? (3 Marks)
Ans. Lo = 1m
V = c / 3
V = 0.33 c
We know,
\(L = L_o\sqrt{1-\frac{V^2}{C^2}}\)
\(L = 1\sqrt{1-\frac{(0.33c)^2}{C^2}}\)
\(L =1\sqrt{1-(0.33)^2}\)
\(L =1\sqrt{1-0.1089}\)
\(L =\sqrt{0.1089}\)
L = 0.943m
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