A block of mass 0.4 kg on a horizontal surface is attached to a horizontal spring of negligible mass. The other end of the spring is attached to a wall, and there is negligible friction between the block and the horizontal surface. The center of mass of the block is stretched a horizontal distance x=0.25 beyond its equilibrium position and released from rest such that the block-spring system undergoes simple harmonic motion. A graph of the spring potential energy of the block-spring system as a function of the horizontal position of the center of mass of the block is shown. The positive direction is considered to be to the right.

Since the investigative question has two variables, you need to focus on each one separately. thinking only about the first part of the question, mass, what might be a hypothesis that would illustrate the relationship between mass and kinetic energy? use the format of "if…then…because…” when writing your hypothesis.

Aparticle of mass m is projected with an initial velocity v0 in a direction making an angle α with the horizontal level ground as shown in the figure. the motion of the particle occurs under a uniform gravitational field g pointing downward. (a) write down the lagrangian of the system by using the cartesian coordinates (x, y). (b) is there any cyclic coordinate(s). if so, interpret it (them) physically. (c) find the euler-lagrange equations. find at least one constant of motion. (d) solve the differential equation in part (c) and obtain x and y coordinates of the projectile as a function of time. (e) construct the hamiltonian of the system, h, and write down the hamilton’s equations (canonical equations) of motion.

An object weighs 40n in air, weighs 20n when submerged in water and 30n when submerged in a liquid of unknown density. what is the density of the liquid?