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A rod of proper length l0 oriented parallel to the x-axis moves with speed 2c/3 along the x-axis in the. S-frame, where c is the speed of the light in free space. The observer is also moving along the x-axis with speed c/2 with respect to the S-frame. The length of the rod as measured by the observer is
- a)0.35l0
- b)0.48l0
- c)0.87l0
- d)0.97l0
Correct answer is option 'D'. Can you explain this answer?
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A rod of proper length l0 oriented parallel to the x-axis moves with s...
For finding the length of rod firstly finding the relativistic velocity which is equal to
Now the measured length
Now the measured length
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A rod of proper length l0 oriented parallel to the x-axis moves with s...
Explanation:
To solve this problem, we need to apply the principles of special relativity to the given scenario. Let's break down the steps:
Step 1: Define the proper length of the rod (l0):
The proper length of an object is the length of the object as measured in its own rest frame. In this case, the rod is at rest in the S-frame, so the proper length of the rod is l0.
Step 2: Calculate the velocity of the observer (v) in the S-frame:
The observer is moving along the x-axis with a speed of c/2 with respect to the S-frame. Since the S-frame is stationary, the velocity of the observer in the S-frame is also c/2.
Step 3: Apply the Lorentz transformation to find the length of the rod as measured by the observer:
The Lorentz transformation relates the length of an object as measured in one frame of reference (S-frame) to the length of the same object as measured in another frame of reference (observer's frame).
The Lorentz transformation for length is given by:
L = L0 * sqrt(1 - (v^2/c^2))
Where:
L = Length of the rod as measured by the observer
L0 = Proper length of the rod (l0)
v = Velocity of the observer (c/2)
c = Speed of light in free space
Let's substitute the values into the equation:
L = l0 * sqrt(1 - ((c/2)^2/c^2))
L = l0 * sqrt(1 - 1/4)
L = l0 * sqrt(3/4)
L = l0 * sqrt(3)/2
Step 4: Simplify the expression:
To simplify the expression, we can rationalize the denominator:
L = l0 * sqrt(3)/2
L = (l0 * sqrt(3)/2) * (sqrt(2)/sqrt(2))
L = (l0 * sqrt(3) * sqrt(2))/(2 * sqrt(2))
L = (l0 * sqrt(6))/(2 * sqrt(2))
L = (l0 * sqrt(6))/sqrt(4)
L = (l0 * sqrt(6))/2
Step 5: Compare the result with the given answer options:
Comparing the simplified expression with the given answer options, we find that the length of the rod as measured by the observer is 0.97l0, which matches option 'D'.
Therefore, the correct answer is option 'D' - 0.97l0.
To solve this problem, we need to apply the principles of special relativity to the given scenario. Let's break down the steps:
Step 1: Define the proper length of the rod (l0):
The proper length of an object is the length of the object as measured in its own rest frame. In this case, the rod is at rest in the S-frame, so the proper length of the rod is l0.
Step 2: Calculate the velocity of the observer (v) in the S-frame:
The observer is moving along the x-axis with a speed of c/2 with respect to the S-frame. Since the S-frame is stationary, the velocity of the observer in the S-frame is also c/2.
Step 3: Apply the Lorentz transformation to find the length of the rod as measured by the observer:
The Lorentz transformation relates the length of an object as measured in one frame of reference (S-frame) to the length of the same object as measured in another frame of reference (observer's frame).
The Lorentz transformation for length is given by:
L = L0 * sqrt(1 - (v^2/c^2))
Where:
L = Length of the rod as measured by the observer
L0 = Proper length of the rod (l0)
v = Velocity of the observer (c/2)
c = Speed of light in free space
Let's substitute the values into the equation:
L = l0 * sqrt(1 - ((c/2)^2/c^2))
L = l0 * sqrt(1 - 1/4)
L = l0 * sqrt(3/4)
L = l0 * sqrt(3)/2
Step 4: Simplify the expression:
To simplify the expression, we can rationalize the denominator:
L = l0 * sqrt(3)/2
L = (l0 * sqrt(3)/2) * (sqrt(2)/sqrt(2))
L = (l0 * sqrt(3) * sqrt(2))/(2 * sqrt(2))
L = (l0 * sqrt(6))/(2 * sqrt(2))
L = (l0 * sqrt(6))/sqrt(4)
L = (l0 * sqrt(6))/2
Step 5: Compare the result with the given answer options:
Comparing the simplified expression with the given answer options, we find that the length of the rod as measured by the observer is 0.97l0, which matches option 'D'.
Therefore, the correct answer is option 'D' - 0.97l0.
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A rod of proper length l0 oriented parallel to the x-axis moves with speed 2c/3 along the x-axis in the. S-frame, where c is the speed of the light in free space. The observer is also moving along the x-axis with speed c/2 with respect to the S-frame. The length of the rod as measured by the observer isa)0.35l0b)0.48l0c)0.87l0d)0.97l0Correct answer is option 'D'. Can you explain this answer?
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A rod of proper length l0 oriented parallel to the x-axis moves with speed 2c/3 along the x-axis in the. S-frame, where c is the speed of the light in free space. The observer is also moving along the x-axis with speed c/2 with respect to the S-frame. The length of the rod as measured by the observer isa)0.35l0b)0.48l0c)0.87l0d)0.97l0Correct answer is option 'D'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about A rod of proper length l0 oriented parallel to the x-axis moves with speed 2c/3 along the x-axis in the. S-frame, where c is the speed of the light in free space. The observer is also moving along the x-axis with speed c/2 with respect to the S-frame. The length of the rod as measured by the observer isa)0.35l0b)0.48l0c)0.87l0d)0.97l0Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A rod of proper length l0 oriented parallel to the x-axis moves with speed 2c/3 along the x-axis in the. S-frame, where c is the speed of the light in free space. The observer is also moving along the x-axis with speed c/2 with respect to the S-frame. The length of the rod as measured by the observer isa)0.35l0b)0.48l0c)0.87l0d)0.97l0Correct answer is option 'D'. Can you explain this answer?.
A rod of proper length l0 oriented parallel to the x-axis moves with speed 2c/3 along the x-axis in the. S-frame, where c is the speed of the light in free space. The observer is also moving along the x-axis with speed c/2 with respect to the S-frame. The length of the rod as measured by the observer isa)0.35l0b)0.48l0c)0.87l0d)0.97l0Correct answer is option 'D'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about A rod of proper length l0 oriented parallel to the x-axis moves with speed 2c/3 along the x-axis in the. S-frame, where c is the speed of the light in free space. The observer is also moving along the x-axis with speed c/2 with respect to the S-frame. The length of the rod as measured by the observer isa)0.35l0b)0.48l0c)0.87l0d)0.97l0Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A rod of proper length l0 oriented parallel to the x-axis moves with speed 2c/3 along the x-axis in the. S-frame, where c is the speed of the light in free space. The observer is also moving along the x-axis with speed c/2 with respect to the S-frame. The length of the rod as measured by the observer isa)0.35l0b)0.48l0c)0.87l0d)0.97l0Correct answer is option 'D'. Can you explain this answer?.
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