An initially stationary box of sand is to be pulled across a floor by means of a

Question

An initially stationary box of sand is to be pulled across a floor by means of a cable in which will break if the tension exceeds 701 N. The coefficient of static friction between the box and the floor is 0.581. The maximum amount of sand that can be pulled without breaking the cable depends on the angle at which the cable is pulled. Find the angle between the cable and horizontal that will allow you to pull the greatest amount of sand and enter the weight of the sand and box (in Newtons) in that situation.

So the answer given was 1.40e3N, just not sure how the book got that answer. Please explain the steps.

Explanation / Answer

let the maximum weight of the sand is W.

tension in the cable=701 N

let the angle be theta.

forces on the system are:

in vertical direction:

weight of the sand=W, downward

vertical component of tension=701*sin(theta)

hence normal reaction=W-701*sin(theta)

(note that W > 701*sin(theta), otherwise the box wont stay put on the ground, it will be lifted off the ground)

horizontal forces:

horinzontal component of tension=701*cos(theta)

friction force=0.581*(W-701*sin(theta))

these two forces should be balanced.

hence 701*cos(theta)=.581*(W-701*sin(theta))

as here are two different variables and only one equation, we need to first find theta for which we can maximize the force on the box and then find value of W from that value of theta.

hence we need to maximize: 701*cos(theta)-0.581*W+0.581*701*sin(theta)

for maximima, the derivative w.r.t. theta =0

hence -701*sin(theta)+0.581*701*cos(theta)=0

theta=30.157 degrees

using this value of theta,

we get W=1395.4 N

or rounding off, 1400 N

(ans)

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