PROBLEMS 1.1 For each of the following systems, specify the number of degrees of freedom and a set of good coordinates. (a) A bead constrained to move on a closed circular hoop that is fixed in space. (b) A bead constrained to move on a helix of constant pitch and constant radius. (c) A particle on a right circular cylinder. (d) A pair of scissors on a plane. (e) A rigid rod in 3-space. (f) A rigid cross in 3-space. (g) A linear spring in 3-space. (h) Any rigid body with one point fixed. (i) A hydrogen atom. (j) A lithium atom. (k) A compound pendulum (two pendulums attached end to end). 1.2 Show that a particle constrained to move on a curve of any shape has one degree of freedom.

Note: Because the argument of the trigonometric functions in this problem will be unitless, your calculator must be in radian mode if you use it to evaluate any trigonometric functions. You will likely need to switch your calculator back into degree mode after this problem.A massless spring is hanging vertically. With no load on the spring, it has a length of 0.24 m. When a mass of 0.59 kg is hung on it, the equilibrium length is 0.98 m. At t=0, the mass (which is at the equilibrium point) is given a velocity of 4.84 m/s downward. At t=0.32s, what is the acceleration of the mass? (Positive for upward acceleration, negative for downward)

The y-coordinate of a particle varies at a constant speed of 4.2 m/s. At t=0, the y-coordinate was found to be 2.7 m.  Find an analytic expression for the function y(t).