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The potential energy of a particle is de...

The potential energy of a particle is determined by the expression `U=alpha(x^2+y^2)`, where `alpha` is a positive constant. The particle begins to move from a point with coordinates `(3, 3)`, only under the action of potential field force. Then its kinetic energy T at the instant when the particle is at a point with the coordinates `(1,1)` is

A

` 8alpha`

B

` 24alpha`

C

` 16alpha`

D

Zero

Text Solution

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The correct Answer is:
To solve the problem, we need to calculate the kinetic energy of a particle moving under a potential energy field described by the equation \( U = \alpha (x^2 + y^2) \). The particle moves from the point (3, 3) to the point (1, 1). We will use the conservation of mechanical energy, which states that the total mechanical energy (kinetic energy + potential energy) remains constant in a conservative force field. ### Step-by-Step Solution: 1. **Identify Initial and Final Points**: - Initial point: \( (x_i, y_i) = (3, 3) \) - Final point: \( (x_f, y_f) = (1, 1) \) 2. **Calculate Initial Potential Energy**: - Using the potential energy formula \( U = \alpha (x^2 + y^2) \): \[ U_i = \alpha (3^2 + 3^2) = \alpha (9 + 9) = 18\alpha \] 3. **Calculate Final Potential Energy**: - Calculate the potential energy at the final point: \[ U_f = \alpha (1^2 + 1^2) = \alpha (1 + 1) = 2\alpha \] 4. **Calculate Change in Potential Energy**: - The change in potential energy \( \Delta U \) is given by: \[ \Delta U = U_f - U_i = 2\alpha - 18\alpha = -16\alpha \] 5. **Apply Conservation of Mechanical Energy**: - According to the conservation of energy: \[ \Delta T + \Delta U = 0 \] - Rearranging gives: \[ \Delta T = -\Delta U \] - Substituting the value of \( \Delta U \): \[ \Delta T = -(-16\alpha) = 16\alpha \] 6. **Conclusion**: - The kinetic energy \( T \) at the instant when the particle is at point (1, 1) is: \[ T = 16\alpha \] ### Final Answer: The kinetic energy \( T \) at the instant when the particle is at the point (1, 1) is \( 16\alpha \). ---
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