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 = +2.6 N and the mass of the box is 4.7 kg. Find the magnitude and direction of when the acceleration of the box is (a) +6.1 m/s2, (b) -6.1 m/s2, and (c) 0 m /s2.
To find the magnitude and direction of the second horizontal force (F), we can use Newton's second law of motion, which states that the net force acting on an object is equal to its mass times its acceleration.
Let's calculate it step by step for each case:
(a) When the acceleration is +6.1 m/s^2:
Given:
Mass of the box (m) = 4.7 kg
Acceleration (a) = 6.1 m/s^2
First horizontal force (F1) = +2.6 N
Since the box is moving to the right, the second horizontal force must be to the right as well. Let's assume the force F2 is positive (+):
The net force acting on the box is the sum of F1 and F2:
Net force = F1 + F2
Now, use Newton's second law to relate net force, mass, and acceleration:
Net force = mass * acceleration
Plug in the given values:
F1 + F2 = m * a
2.6 N + F2 = 4.7 kg * 6.1 m/s^2
Rearranging the equation to solve for F2:
F2 = (4.7 kg * 6.1 m/s^2) - 2.6 N
F2 = 28.67 N - 2.6 N
F2 = 26.07 N
Therefore, when the acceleration of the box is +6.1 m/s^2, the magnitude of F2 is 26.07 N, and its direction is to the right.
(b) When the acceleration is -6.1 m/s^2:
Since the box now accelerates in the opposite direction, the direction of the second horizontal force will also be in the opposite direction (to the left). Let's assume the force F2 is negative (-).
Using the same approach as in part (a), the net force equation becomes:
F1 + F2 = m * a
2.6 N + F2 = 4.7 kg * (-6.1 m/s^2)
Rearranging the equation to solve for F2:
F2 = (4.7 kg * (-6.1 m/s^2)) - 2.6 N
F2 = -28.67 N - 2.6 N
F2 = -31.27 N
Therefore, when the acceleration of the box is -6.1 m/s^2, the magnitude of F2 is 31.27 N, and its direction is to the left.
(c) When the acceleration is 0 m/s^2:
In this case, the box is not accelerating, so the net force acting on it is zero. Therefore, the sum of the two forces (F1 and F2) must be zero as well.
F1 + F2 = 0
2.6 N + F2 = 0
Rearranging the equation to solve for F2:
F2 = -2.6 N
Therefore, when the acceleration of the box is 0 m/s^2, the magnitude of F2 is 2.6 N, and its direction is to the left.