In a two-dimensional space, a particle of mass m, located by its Cartesian coordinates x, y, is subject to a potential of the form V (x − 2y).   when x increases by the arbitrary quantity s, y increase by s/2   (a) Use Noether’s theorem to deduce the constant of the motion.

A spring with k = 15 N/m with equilibrium length 30 cm is attached vertically to a surface and then allowed to come to equilibrium, after which a weight with mass 0.2 kg is attached to the top side of the spring. → How much work does the Earth do on the weight via gravity between its initial position and the new equilibrium height? Explicitly compute the work with an integral.

A particle moves along a curved path y(x) = (10 m){1+cos[(0.1 m−1)x]} , from x = 0 to x = 10π m, subject to a tangential force of variable magnitude F(x) = (10 N)sin[(0.1 m−1)x]. How much work does the force do? (Hint: Consult a table of integrals or use a numerical integration program.)

2.