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A particle of mass 10 g moves in a straight line with retardation 2x where x us the displacement in si units it's lose of kinetic energy for the above displacement is (10/x)^-n j the value of n?
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A particle of mass 10 g moves in a straight line with retardation 2x w...
Understanding the Problem
The particle has a mass of 10 g, which is equivalent to 0.01 kg. It experiences a retardation (deceleration) that is proportional to its displacement, described by the equation a = -2x.
Key Concepts to Consider
- Retardation: The force acting on the particle causes it to lose kinetic energy. The retardation is given as a function of displacement: a = -2x.
- Kinetic Energy: The kinetic energy (KE) of the particle can be expressed as KE = (1/2)mv^2, where m is mass and v is velocity.
- Work-Energy Principle: The work done on the particle is equal to the change in kinetic energy. The force can be derived from the retardation.
Calculating the Change in Kinetic Energy
1. Force Calculation: The force acting on the particle can be described as F = ma. Here, m = 0.01 kg and a = -2x. Therefore, F = -0.02x.
2. Work Done: The work done by this force over a displacement x is given by W = ∫F dx = ∫(-0.02x) dx, which evaluates to -0.01x^2.
3. Kinetic Energy Loss: The loss of kinetic energy (ΔKE) can be equated to the work done by the force, resulting in ΔKE = -(-0.01x^2) = 0.01x^2.
4. Comparing with Given Equation: The problem states the loss of kinetic energy is (10/x)^-n J. We can express this as 0.01x^2 = (10/x)^-n.
Finding the Value of n
To match both sides of the equation, we rearrange and analyze the terms:
- Setting 0.01x^2 equal to (10/x)^-n, we can conclude that:
0.01x^2 = 10^n / x^n
This implies:
- n must equal 2 for both sides to match (since 0.01 = 10^-2 and x^2 = x^-n).
Final Conclusion
Thus, the value of n is 2.
The particle has a mass of 10 g, which is equivalent to 0.01 kg. It experiences a retardation (deceleration) that is proportional to its displacement, described by the equation a = -2x.
Key Concepts to Consider
- Retardation: The force acting on the particle causes it to lose kinetic energy. The retardation is given as a function of displacement: a = -2x.
- Kinetic Energy: The kinetic energy (KE) of the particle can be expressed as KE = (1/2)mv^2, where m is mass and v is velocity.
- Work-Energy Principle: The work done on the particle is equal to the change in kinetic energy. The force can be derived from the retardation.
Calculating the Change in Kinetic Energy
1. Force Calculation: The force acting on the particle can be described as F = ma. Here, m = 0.01 kg and a = -2x. Therefore, F = -0.02x.
2. Work Done: The work done by this force over a displacement x is given by W = ∫F dx = ∫(-0.02x) dx, which evaluates to -0.01x^2.
3. Kinetic Energy Loss: The loss of kinetic energy (ΔKE) can be equated to the work done by the force, resulting in ΔKE = -(-0.01x^2) = 0.01x^2.
4. Comparing with Given Equation: The problem states the loss of kinetic energy is (10/x)^-n J. We can express this as 0.01x^2 = (10/x)^-n.
Finding the Value of n
To match both sides of the equation, we rearrange and analyze the terms:
- Setting 0.01x^2 equal to (10/x)^-n, we can conclude that:
0.01x^2 = 10^n / x^n
This implies:
- n must equal 2 for both sides to match (since 0.01 = 10^-2 and x^2 = x^-n).
Final Conclusion
Thus, the value of n is 2.
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A particle of mass 10 g moves in a straight line with retardation 2x where x us the displacement in si units it's lose of kinetic energy for the above displacement is (10/x)^-n j the value of n?
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A particle of mass 10 g moves in a straight line with retardation 2x where x us the displacement in si units it's lose of kinetic energy for the above displacement is (10/x)^-n j the value of n? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about A particle of mass 10 g moves in a straight line with retardation 2x where x us the displacement in si units it's lose of kinetic energy for the above displacement is (10/x)^-n j the value of n? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A particle of mass 10 g moves in a straight line with retardation 2x where x us the displacement in si units it's lose of kinetic energy for the above displacement is (10/x)^-n j the value of n?.
A particle of mass 10 g moves in a straight line with retardation 2x where x us the displacement in si units it's lose of kinetic energy for the above displacement is (10/x)^-n j the value of n? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about A particle of mass 10 g moves in a straight line with retardation 2x where x us the displacement in si units it's lose of kinetic energy for the above displacement is (10/x)^-n j the value of n? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A particle of mass 10 g moves in a straight line with retardation 2x where x us the displacement in si units it's lose of kinetic energy for the above displacement is (10/x)^-n j the value of n?.
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