An object moves in the xy-plane under the influence of a single force given by F(x)= - αx^2i+βyj, where α and β are known constants. Find the associated potential energy function and show the force is conservative.
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An object moves in the xy-plane under the influence of a single force given by F(x)= - αx^2i+βyj, where α and β are known constants. Find the associated potential energy function and show the force is conservative.
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- Suppose there is a conservative force with potential energy function U(x, y) =−ax − bx2 + 3cy3, where a, b, and c are positive constants, throughout the entire x-yplane. Find the associated force F~ (x, y). Consider a bead of mass m that is free tomove along a frictionless wire that is fixed to the x axis. If the bead is released fromrest at the origin, in which direction will it start to move? How fast will it be travelingafter it moves a distance L?U(x)=9.0x^2-8.0x^3 Determine the force Fx associated with this potential-energy function Assuming no other forces act on the object, at what positions is this object in equilibrium? Which of these equilibrium positions are stable and which are unstable? (Hint: use the 2ndderivative)Consider a force in R3 defined by f(x,y,z)=(6*x*y^3, 9*x^2*y^2+2*y,0). Find the potential energy U(x,y,z) such that U(0,0,0)=0.
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