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The potential energy for a conservative system is given by U = ax2 – bx, where aa and bb are positive constants. The law of the force governing the system is
- a)F = constant
- b)F = bx – 2a
- c)F = b – 2ax
- d)F = 2ax
Correct answer is option 'C'. Can you explain this answer?
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The potential energy for a conservative system is given by U = ax2 – ...
Potential Energy:
The potential energy for a conservative system is given by the equation U = ax^2 - bx, where a and b are positive constants.
Force and Potential Energy:
According to the concept of potential energy and force in a conservative system, the force acting on a particle is the negative gradient of the potential energy function. Mathematically, F = -dU/dx, where dU/dx represents the derivative of U with respect to x.
Calculating the Force:
To determine the force in this system, we need to take the derivative of the potential energy function U with respect to x.
dU/dx:
Taking the derivative of U = ax^2 - bx with respect to x, we get:
dU/dx = 2ax - b
Comparing with Given Options:
Now, let's compare the derived expression for dU/dx with the options provided to find the correct law of force governing the system.
Option A: F = constant
This option suggests that the force is constant, which contradicts with the derived expression for dU/dx. Therefore, option A is incorrect.
Option B: F = bx - 2a
This option suggests that the force is directly proportional to x and inversely proportional to a, which does not match with the derived expression. Therefore, option B is incorrect.
Option C: F = b - 2ax
This option matches exactly with the derived expression for dU/dx, indicating that the force is indeed given by F = b - 2ax. Therefore, option C is correct.
Option D: F = 2ax
This option suggests that the force is directly proportional to x, which contradicts with the derived expression. Therefore, option D is incorrect.
Conclusion:
From the comparison, we can conclude that the correct law of the force governing the system is F = b - 2ax, as stated in option C.
The potential energy for a conservative system is given by the equation U = ax^2 - bx, where a and b are positive constants.
Force and Potential Energy:
According to the concept of potential energy and force in a conservative system, the force acting on a particle is the negative gradient of the potential energy function. Mathematically, F = -dU/dx, where dU/dx represents the derivative of U with respect to x.
Calculating the Force:
To determine the force in this system, we need to take the derivative of the potential energy function U with respect to x.
dU/dx:
Taking the derivative of U = ax^2 - bx with respect to x, we get:
dU/dx = 2ax - b
Comparing with Given Options:
Now, let's compare the derived expression for dU/dx with the options provided to find the correct law of force governing the system.
Option A: F = constant
This option suggests that the force is constant, which contradicts with the derived expression for dU/dx. Therefore, option A is incorrect.
Option B: F = bx - 2a
This option suggests that the force is directly proportional to x and inversely proportional to a, which does not match with the derived expression. Therefore, option B is incorrect.
Option C: F = b - 2ax
This option matches exactly with the derived expression for dU/dx, indicating that the force is indeed given by F = b - 2ax. Therefore, option C is correct.
Option D: F = 2ax
This option suggests that the force is directly proportional to x, which contradicts with the derived expression. Therefore, option D is incorrect.
Conclusion:
From the comparison, we can conclude that the correct law of the force governing the system is F = b - 2ax, as stated in option C.
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Community Answer
The potential energy for a conservative system is given by U = ax2 – ...
In a conservative field F = -dU / dr
∴ F = −d / dx(ax2 − bx) = b − 2ax
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The potential energy for a conservative system is given by U = ax2 – bx, where aa and bb are positive constants. The law of the force governing the system isa)F = constantb)F = bx – 2ac)F = b – 2axd)F = 2axCorrect answer is option 'C'. Can you explain this answer?
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