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In the diagram shown, the object is performing SHM according to the equation y2Asin (ot) and the plane mirror is performing SHM according to the equation Y =-Asin(ot- The diagram shows the state of the object and the mirror at time t 0 sec. The minimum time from t 0 sec after which the velocity of the image becomes equal to zero ? (A) π 3ω Object Plane Mirror (B) 3T 2π 3o (D) None of these?
Most Upvoted Answer
In the diagram shown, the object is performing SHM according to the eq...
Analysis of the problem:
- The object is performing simple harmonic motion (SHM) according to the equation y = 2Asin(ωt).
- The plane mirror is also performing SHM according to the equation Y = -Asin(ωt - Φ), where Φ is the phase difference between the object and the mirror.
- At time t = 0 sec, the object and the mirror are in the same position.
- We need to find the minimum time after t = 0 sec when the velocity of the image becomes zero.
Solution:
Step 1: Find the equation of motion for the image:
- The equation of motion for the object is y = 2Asin(ωt).
- The equation of motion for the mirror is Y = -Asin(ωt - Φ).
- At t = 0 sec, both the object and the mirror are in the same position, so we can set y = Y = 0.
- This gives us 2Asin(ωt) = -Asin(ωt - Φ).
Step 2: Solve for the phase difference:
- Divide both sides of the equation by Asin(ωt) to get: 2 = -sin(ωt - Φ)/sin(ωt).
- Using the identity sin(A - B) = sin(A)cos(B) - cos(A)sin(B), the equation becomes: 2 = -[sin(ωt)cos(Φ) - cos(ωt)sin(Φ)]/sin(ωt).
- Simplifying further, we get: 2 = -cos(Φ) + tan(ωt)sin(Φ).
- Rearranging the terms, we get: cos(Φ) = 2 - tan(ωt)sin(Φ).
Step 3: Find the value of sin(Φ):
- Recall that sin^2(Φ) + cos^2(Φ) = 1.
- Squaring both sides of the equation cos(Φ) = 2 - tan(ωt)sin(Φ), we get: cos^2(Φ) = (2 - tan(ωt)sin(Φ))^2.
- Expanding and simplifying, we get: 1 - sin^2(Φ) = 4 - 4tan(ωt)sin(Φ) + tan^2(ωt)sin^2(Φ).
- Rearranging the terms, we get: sin^2(Φ) + tan^2(ωt)sin^3(Φ) - 4tan(ωt)sin(Φ) + 3 = 0.
- This is a cubic equation in sin(Φ). Solving this equation is quite complex and beyond the scope of this explanation.
Step 4: Find the minimum time when the velocity of the image becomes zero:
- Since we are unable to find the exact value of sin(Φ), we cannot determine the exact minimum time when the velocity of the image becomes zero.
- Therefore, the correct answer is (D) None of these.
Conclusion:
- The minimum time after t =
- The object is performing simple harmonic motion (SHM) according to the equation y = 2Asin(ωt).
- The plane mirror is also performing SHM according to the equation Y = -Asin(ωt - Φ), where Φ is the phase difference between the object and the mirror.
- At time t = 0 sec, the object and the mirror are in the same position.
- We need to find the minimum time after t = 0 sec when the velocity of the image becomes zero.
Solution:
Step 1: Find the equation of motion for the image:
- The equation of motion for the object is y = 2Asin(ωt).
- The equation of motion for the mirror is Y = -Asin(ωt - Φ).
- At t = 0 sec, both the object and the mirror are in the same position, so we can set y = Y = 0.
- This gives us 2Asin(ωt) = -Asin(ωt - Φ).
Step 2: Solve for the phase difference:
- Divide both sides of the equation by Asin(ωt) to get: 2 = -sin(ωt - Φ)/sin(ωt).
- Using the identity sin(A - B) = sin(A)cos(B) - cos(A)sin(B), the equation becomes: 2 = -[sin(ωt)cos(Φ) - cos(ωt)sin(Φ)]/sin(ωt).
- Simplifying further, we get: 2 = -cos(Φ) + tan(ωt)sin(Φ).
- Rearranging the terms, we get: cos(Φ) = 2 - tan(ωt)sin(Φ).
Step 3: Find the value of sin(Φ):
- Recall that sin^2(Φ) + cos^2(Φ) = 1.
- Squaring both sides of the equation cos(Φ) = 2 - tan(ωt)sin(Φ), we get: cos^2(Φ) = (2 - tan(ωt)sin(Φ))^2.
- Expanding and simplifying, we get: 1 - sin^2(Φ) = 4 - 4tan(ωt)sin(Φ) + tan^2(ωt)sin^2(Φ).
- Rearranging the terms, we get: sin^2(Φ) + tan^2(ωt)sin^3(Φ) - 4tan(ωt)sin(Φ) + 3 = 0.
- This is a cubic equation in sin(Φ). Solving this equation is quite complex and beyond the scope of this explanation.
Step 4: Find the minimum time when the velocity of the image becomes zero:
- Since we are unable to find the exact value of sin(Φ), we cannot determine the exact minimum time when the velocity of the image becomes zero.
- Therefore, the correct answer is (D) None of these.
Conclusion:
- The minimum time after t =
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In the diagram shown, the object is performing SHM according to the eq...
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In the diagram shown, the object is performing SHM according to the equation y2Asin (ot) and the plane mirror is performing SHM according to the equation Y =-Asin(ot- The diagram shows the state of the object and the mirror at time t 0 sec. The minimum time from t 0 sec after which the velocity of the image becomes equal to zero ? (A) π 3ω Object Plane Mirror (B) 3T 2π 3o (D) None of these?
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In the diagram shown, the object is performing SHM according to the equation y2Asin (ot) and the plane mirror is performing SHM according to the equation Y =-Asin(ot- The diagram shows the state of the object and the mirror at time t 0 sec. The minimum time from t 0 sec after which the velocity of the image becomes equal to zero ? (A) π 3ω Object Plane Mirror (B) 3T 2π 3o (D) None of these? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about In the diagram shown, the object is performing SHM according to the equation y2Asin (ot) and the plane mirror is performing SHM according to the equation Y =-Asin(ot- The diagram shows the state of the object and the mirror at time t 0 sec. The minimum time from t 0 sec after which the velocity of the image becomes equal to zero ? (A) π 3ω Object Plane Mirror (B) 3T 2π 3o (D) None of these? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In the diagram shown, the object is performing SHM according to the equation y2Asin (ot) and the plane mirror is performing SHM according to the equation Y =-Asin(ot- The diagram shows the state of the object and the mirror at time t 0 sec. The minimum time from t 0 sec after which the velocity of the image becomes equal to zero ? (A) π 3ω Object Plane Mirror (B) 3T 2π 3o (D) None of these?.
In the diagram shown, the object is performing SHM according to the equation y2Asin (ot) and the plane mirror is performing SHM according to the equation Y =-Asin(ot- The diagram shows the state of the object and the mirror at time t 0 sec. The minimum time from t 0 sec after which the velocity of the image becomes equal to zero ? (A) π 3ω Object Plane Mirror (B) 3T 2π 3o (D) None of these? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about In the diagram shown, the object is performing SHM according to the equation y2Asin (ot) and the plane mirror is performing SHM according to the equation Y =-Asin(ot- The diagram shows the state of the object and the mirror at time t 0 sec. The minimum time from t 0 sec after which the velocity of the image becomes equal to zero ? (A) π 3ω Object Plane Mirror (B) 3T 2π 3o (D) None of these? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In the diagram shown, the object is performing SHM according to the equation y2Asin (ot) and the plane mirror is performing SHM according to the equation Y =-Asin(ot- The diagram shows the state of the object and the mirror at time t 0 sec. The minimum time from t 0 sec after which the velocity of the image becomes equal to zero ? (A) π 3ω Object Plane Mirror (B) 3T 2π 3o (D) None of these?.
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