1. Consider a conical pendulum; that is a ball of mass m at the end of a string of length li moving in a horizontal circle and making an angle of θi with vertical. Imagine the string can be shortened or lengthened by pulling the string through a hole, as discussed in class. The variable r is the radius of the circle the ball moves in.
(a) Find equations for r(θ), period(θ), v(θ), tension(θ), ω(θ), and l(θ). These equations should contain only known quantities and θ as variables (plus of course the other variable, such as r, period, etc.).
(b) Which quantities are large as θ approaches 90°? Which quantities are large as θ approaches 0°?
Notes: Be sure to get the physics correct for the physical situation described in the first two sentences above.
2. A particle of mass m is at rest at the origin of a coordinate system. At t = 0, a force of the form F = Bt2e-λt is applied in the x-direction, where B and λ are constants. Calculate v(t) and x(t). Describe the behavior of the system at large times.
3. The motor of a speed boat is shut off when it has attained a speed of v0. Now the boat is slowed down by a retarding force Fr = C(1 + v2/v02) where C is a constant. Calculate v(t) and x(t). How long will it take for the boat to stop, and how much distance will it travel before stopping?