Use calculus to find an exact expression for the depth d to which a ball of mass M and radius R will sink in liquid of uniform density D in an atmosphere of density G.
I wrote my question to challenge you and to inspire you, so I am not going to simply post the answer. I would be glad to help you if you get stuck or any other calculus question though!
a) A balanced seven-phase (let's say!) "Y"-connected source with a positive sequence measures 200Vrms at an angle 0 degrees from phase A to phase C. What are the approximate voltages and phase angles of each of the seven sources A,B,C,D,E, F and G? b) Is there any advantage to using seven phases?
a) I admit I made this question up out of complete thin air ---- (I am not aware of anyone actually building anything with seven phases). b) ???
At the circus, a trapeze artist of mass M drops onto a teeter-totter (see-saw) at a distance L from the fulcrum from a height H above the fulcrum. If a second acrobat of mass m is standing on the other end at a distance l from the fulcrum and a depth d below it (since the device is resting with that end on the floor) , find an expression for the maximum theoretical height h above the fulcrum the second acrobat should be able to reach. Assume the beam is perfectly stiff and massless. Ignore losses due to air resistance and friction. a) (introductory) Neglect the mass and rotational inertia of the teeter-totter and use conservation of energy to find an expression for the upper limit on the maximum possible height actually reached by any such acrobat. Assume that the waiting acrobat does not expend any energy. b) (advanced) Considering the teeter totter to be a rectangular solid with a width W, thickness t, length L+l, and density rho, and initial orientation find a more restricted expression that takes into account the inelastic collision of the teeter-totter and falling acrobat. c) (project) Choose realistic values for all of the dimensions involved, design a beam that could withstand the maximum forces in this problem. Do not neglect the self-weight of the beam. Assume the forces on the beam are maximal at the moment it is flat. Ensure that neither bending stress nor shear stress exceed the maximum allowable for the material you choose (factor of safety of three).
Let me know what you think the answers are and I will give you feedback on your work or let me know if you have no idea how to proceed and we can talk about any of the principles involved!