To determine the acceleration of the box, we need to consider the forces acting on it. The main forces involved are the applied force, the gravitational force, and the frictional force.
First, let's calculate the gravitational force acting on the box. The weight of the box can be calculated using the formula: weight = mass * gravity, where gravity is approximately 9.8 m/s² on Earth. Therefore, the weight of the box is:
weight = 11 kg * 9.8 m/s² = 107.8 N
Next, let's find the vertical component of the applied force. Since the force is given at an angle of 23 degrees above the horizontal, the vertical component can be determined as:
vertical component = applied force * sin(23°) = 51 N * sin(23°) ≈ 21.09 N
The vertical component of the applied force opposes the gravitational force and reduces the normal force on the box. However, since the box is not moving vertically, these forces do not contribute to its acceleration in the horizontal direction.
The horizontal component of the applied force can be found as:
horizontal component = applied force * cos(23°) = 51 N * cos(23°) ≈ 46.12 N
Now, let's calculate the maximum static frictional force. The maximum static frictional force can be determined by multiplying the normal force (equal to the weight of the box in this case) by the coefficient of friction. Therefore:
maximum static frictional force = coefficient of friction * weight maximum static frictional force = 0.20 * 107.8 N ≈ 21.56 N
Since the applied horizontal force (46.12 N) is greater than the maximum static frictional force (21.56 N), the box will experience kinetic friction instead of static friction.
The kinetic frictional force can be determined by multiplying the coefficient of kinetic friction by the normal force. Therefore:
kinetic frictional force = coefficient of friction * weight kinetic frictional force = 0.20 * 107.8 N ≈ 21.56 N
Since the box is in motion, the kinetic frictional force opposes the applied force.
The net force acting on the box in the horizontal direction can be calculated by subtracting the kinetic frictional force from the horizontal component of the applied force:
net force = horizontal component - kinetic frictional force net force = 46.12 N - 21.56 N ≈ 24.56 N
Finally, we can calculate the acceleration of the box using Newton's second law of motion: net force = mass * acceleration. Therefore:
acceleration = net force / mass acceleration = 24.56 N / 11 kg ≈ 2.24 m/s²
Thus, the acceleration of the box is approximately 2.24 m/s².