Reference no: EM13587378

1. Consider a conical pendulum; that is a ball of mass m at the end of a string of length l_i moving in a horizontal circle and making an angle of θ_i 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 F_r = 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?