An oscillator of mass m oscillates according to the equation ψ(y) = Ay exp (−y²/a²) for permissible states where E = 0. Represent the potential energy function on a graph.

In a two-star system, two stars of equal mass are separated by a distance of 1.5 × 10¹² m. An asteroid is approaching the stars with negligible speed. What will be its speed at the point when it is twice as close to one star as the other? Consider the mass of the two stars as 3.0 × 10²⁶ kg.

In a science fiction film, a skyscraper is under construction on a planet where the gravitational acceleration is 1.35 m/s². A worker drops his 7.00-kg toolbox from a height of 300 m above the ground. Find the speed of the toolbox just before hitting the ground.

A large hollow ball of outer radius R₁ was put into space. Its inner radius is R₂. The shell of the hollow ball of thickness R₁-R₂ is uniform in mass distribution and has a total mass of m. If another mass m' was placed at a distance of R (R > R1) away from the center of the hollow ball, calculate the gravitational potential energy of this mass m'.

A uniform plank, 5.00 m in length and weighing around 12.0 kg, rests against a smooth surface at an angle of approximately 30.0° to the ground. If the plank is pushed to a vertical position, how much will its gravitational potential energy increase?