Two ropes are attached to a heavy box to pull it along the floor. one rope applies a force of 475 newtons in a direction due west; the other applies a force of 315 newtons in a direction due south. as we will see later in text, force is a vector quantity. how much force should be applied by a single rope and in what direction (relative to due west) if it is to accomplish the same effect as the two forces added together.

Cyclotrons are widely used in nuclear medicine for producing short-lived radioactive isotopes. these cyclotrons typically accelerate h- (the hydride ion, which has one proton and two electrons) to an energy of 5 mev to 20 mev. this ion has a mass very close to that of a proton because the electron mass is negligible—about 1 2000 of the proton’s mass. a typical magnetic field in such cyclotrons is 1.9 t. (a) what is the speed of a 5.0-mev h-? (b) if the h- has energy 5.0 mev and b = 1.9 t, what is the radius of this ion’s circular orbit?