Question

An initially stationary box of sand is to be pulled across a floor by means of a cable in which the tension should not exceed 883 N. The coefficient of static friction between the box and the floor is 0.360. (a) What should be the angle between the cable and the horizontal in order to pull the greatest possible amount of sand, and (b) what is the weight of the sand and box in that situation?

Answer 1

(a)Suppose A is the angle between the cable and the horizontal and T is the tension in the cable.

So, we have the following expression -

Tcos A = 0.36(mg - Tsin A)
Let T take the maximum of 883 N
so -

883* cosA = 3.53m - 317.9 sinA
=> m = 250.14* cos A + 90.06* sin A

to determine the maximum value of A, take derivatives both the sides -

dm/dA = -250.14* sin A + 90.06* cos A
dm/dA = 0 at a maximum
=> 250.14* sin A = 90.06* cos A
=> tan A = 0.36
=> A = tan^-1(0.36) = 19.8 deg

(b) Plugging back in above
m = 250.14* cos 19.8 + 90.06* sin 19.8 = 235.4 + 30.5 = 265.9 kg

So, the weight W = mg = 265.9 x 9.81 = 2608.5 N