Arod of length l coincides with the interval [0, l] on the x-axis. set up the boundary-value problem problem for the temperature u(x, t) in the following scenarios:

(a) the left end is held at temperature u0, and the right end is held at temperature u1. the initial temperature is zero throughout.

(b) there is heat transfer from the left end into a surrounding medium at temperature 20◦ , and the right end is insulated. the initial temperature is f(x) throughout.

(c) the ends are insulated, and there is heat transfer from the lateral surface of the rod into the surrounding medium held at 50◦ . the initial temperature is 100◦ throughout.

10 newton object is placed 3 meters from the fulcrum. at what distance on the other side would you need to place a 15 newton object to balance the lever? show your work!

Your starship, the aimless wanderer, lands on the mysterious planet mongo. as chief scientist-engineer, you make the following measurements: a 2.50-kg stone thrown upward from the ground at 14.0 m/s returns to the ground in 5.70 s ; the circumference of mongo at the equator is 2.20×10^5 km ; and there is no appreciable atmosphere on mongo. (a) the starship commander, captain confusion, asks for the following information: what is the mass of mongo? (b) if the aimless wanderer goes into a circular orbit 2.20×10^4 km above the surface of mongo, how many hours will it take the ship to complete one orbit? (c) an unmanned spacecraft is in a circular orbit around the moon, observing the lunar surface from an altitude of 44.0 km . to the dismay of scientists on earth, an electrical fault causes an on-board thruster to fire, decreasing the speed of the spacecraft by 23.0 m/s . if nothing is done to correct its orbit, with what speed (in km/h) will the spacecraft crash into the lunar surface?

The electric field direction is defined by the direction of the force felt by (select one of the following answers): 1. a negative charge. 2. a positive charge. 3. both positive and negative charges.