Two horizontal forces,  and , are acting on a box, but only  is shown in the drawing.  can point either to the right or to the left. The box moves only along the x axis. There is no friction between the box and the surface. Suppose that  = +4.2 N and the mass of the box is 4.1  kg. Find the magnitude and direction of  when the acceleration of the box is (a) +4.6 m/s2, (b) -4.6 m/s2, and (c) 0 m /s2.

Fill in the missing text that did not get pasted

x^²-5

x&sup5;-5

x&sup0;¹²³-5

x&sup4;-5

x23-5

To find the magnitude and direction of force F2, we can make use of Newton's second law of motion, which states that the net force acting on an object is equal to the product of its mass and acceleration, i.e., F_net = m * a.

In this case, the net force acting on the box is the vector sum of the two horizontal forces, F1 and F2, so we have F_net = F1 + F2.

(a) When the acceleration of the box is +4.6 m/s^2:
To find the magnitude and direction of F2, we need to substitute the given values into the equation F_net = F1 + F2 and solve for F2. Since F1 is given as +4.2 N, F_net can be calculated as (4.1 kg) * (4.6 m/s^2) = 18.86 N using the mass of the box and the given acceleration.

Now, rearranging the equation, we have: F2 = F_net - F1 = 18.86 N - 4.2 N = 14.66 N.

Therefore, when the acceleration of the box is +4.6 m/s^2, the magnitude of force F2 is 14.66 N.

(b) When the acceleration of the box is -4.6 m/s^2:
Using the same approach, we can find F_net as (4.1 kg) * (-4.6 m/s^2) = -18.86 N since the mass remains the same, but the acceleration changes direction.

Again, rearranging the equation, we have: F2 = F_net - F1 = -18.86 N - 4.2 N = -23.06 N.

Therefore, when the acceleration is -4.6 m/s^2, the magnitude of force F2 is 23.06 N, and its direction is opposite to the direction of F1.

(c) When the acceleration of the box is 0 m/s^2:
Since the box is at rest (no acceleration), we can conclude that the net force acting on the box is zero (F_net = 0). Therefore, we have F_net = F1 + F2 = 0.

Rearranging the equation, we find F2 = -F1. Substituting the given value of F1 as +4.2 N, we get F2 = -4.2 N.

Hence, when the acceleration of the box is 0 m/s^2, the magnitude of force F2 is 4.2 N, and its direction is opposite to the direction of F1.