A 2.4 kg copper rod rests on two horizontal rails (see figure below) 2.5 m apart

Question

A 2.4 kg copper rod rests on two horizontal rails (see figure below) 2.5 m apart and carries a current of 35 A from one rail to the other. The coefficient of static friction between rod and rails is 0.56. What is the smallest magnetic field (not necessarily vertical) that would cause the rod to slide? (Based on the bottom picture, define to the right as the +x-direction and up as the +y-direction. Assume the current in the bottom picture is into the page and the magnetic force will cause the rod will slide to the right.)

magnitude 0.0798 T
direction 119 ° counter-clockwise from the +x-axis

http://www.webassign.net/hrw8/tut28-45.gif



pls help I'm just getting it wrong! I will rate!

Explanation / Answer

Place the B field at some unknown angle and split it intohorizontal and vertical components. Note that the verticalcomponent of the B field applies a horizontal force to therod:.
horizontal magnetic force = LIB sin..
vertical magnetic force = LIB cos
Now write the sum of the forces in two dimensions:
Force equilibrium:
horizontal: LIBsin - friction = 0
vertical: LIBcos - mg + normal = 0
or... LIB sin = un mg - LIB cos = n
eliminate n bysubstitution LIB sin = u (mg - LIBcos )

take the derivative, dB/d. Set it equal to zero and solve for. This tells you the value of for which B is minimized. Then plugthis value of back into the function and get B.
2.5 * 35 * B sin = 0.56 * (2.4 *9.8 - 2.5 * 35 * Bcos )
87.5 B sin = 13.2 - 49 Bcos

B = 13.2* (49 cos + 87.5sin)-1

dB/d = 13.2 * (-1) (49 cos +87.5sin)-2 * (-49sin + 87.5 cos).
Set this equal tozero: 0 = -13.2* (-49 sin + 87.5cos) (49cos +87.59 sin)-2

0 = -49 sin + 87.5cos tan = 87.5 / 49
Therefore = 61 degrees measured clockwise
Or 180-61 = 119 degrees measured anti clockwise

Now, for the minimum value of B: at B = 13.2* (49 cos + 87.5sin)-1 ; =61
Therefore B = 0.132 approx

i am pretty sure this value of B is correct. because otherwise we wouldnt have gotten the angle itself!

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