A box weighing 2000 N is to be slowly slid a distance of 20 m along a straight track. The coefficient of friction between the box and the track is 0.2 . A person pulls the box using a chain oriented at an angle with the horizontal. Compute the work done if the person chooses the value of that minimizes the magnitude of the pulling force.

Step-by-step solution

Let F be the pulling force applied at an angle with the horizontal. For the box to slide slowly, the net force on it must be zero. Resolving forces horizontally and vertically: F = N N + F = W Substituting N = W - F into the first equation: F = (W - F ) F ( + ) = W F = W + To minimize the pulling force F , the denominator + must be maximized. Taking the derivative with respect to and equating to zero: - + = 0 = For = , we have = 1 1+ ^2 and = 1+ ^2 . The minimum force is: F_ min = W 1 1+ ^2 + ^2 1+ ^2 = W 1+ ^2 The