A slingshot, consisting of a leather pouch attached to two rubber bands which are tied to the prongs of a wooden Y- shaped frame, is used to shoot a pebble horizontally. When the slingshot is stretched by a distance d, it gives the pebble speed v. What speed does it give the same pebble when it is stretched to distance 3d?

1. A particle moves in one dimension and is subject to a conservative force, whose potential energy function is given by U(x), where a and b are positive constants. U ( x ) = a x 3 − b x                  a)  Find the two equilibrium positions of the particle.                  b)  For each of the two equilibrium positions, determine whether the equilibrium is stable or unstable.

The staples inside a stapler are kept in place by a spring with a relaxedlength of 11.5 cm. The spring constant is 51.0 N/m. How much elasticpotential energy is stored in the spring when its length is 8 cm?

The spring connects two objects.  In this case, it's the object that moves and accelerates, and the wall.  I will take x to be the distance from the particle to the wall -- the length of the spring.  The problem is one-dimensional.  (There are two- and three-dimensional forms of the problem.)      The ideal spring exerts a force (F) on the object, with the following properties: The spring has a length (ℓ) where it exerts no force.  (I usually call it the relaxed length.  I might call it the "zero-force length" or the "equilibrium length".) The compressed spring pushes.  The stretched string pulls.  The force on the block is F = -k(x - ℓ).  (Figure out what the negative sign on k does.) Often, people use x for the difference from equilibrium, and write F = -kx instead. Sometimes, both sides of the spring are at different positions.  The force on the object on the right side is -k(x2 - x1 - ℓ). In the pictured situation, x1 = 0 and x2 = x. Understand what is physically happening with…