A box is being lowered down a wall at constant speed by a girl, as shown.
Her hand is pushing on the edge of the box with 50 N of force at a 45° angle
above the horizontal. The box weighs 40 N.
To understand the situation, let's break it down step by step:
Step 1: Determine the force components acting on the box.
The force of 50 N applied by the girl can be broken down into its horizontal and vertical components. Given that the force is applied at a 45° angle above the horizontal, we can use trigonometry to calculate these components.
The horizontal component is calculated as follows:
F_horizontal = F_push * cos(theta)
where F_push is the force applied by the girl (50 N) and theta is the angle (45°).
F_horizontal = 50 N * cos(45°)
F_horizontal = 50 N * (0.707)
F_horizontal = 35.4 N
The vertical component is calculated as follows:
F_vertical = F_push * sin(theta)
where F_push is the force applied by the girl (50 N) and theta is the angle (45°).
F_vertical = 50 N * sin(45°)
F_vertical = 50 N * (0.707)
F_vertical = 35.4 N
Step 2: Determine the net force acting on the box.
The net force acting on the box is the vector sum of all the individual forces. In this case, there are two forces: the force applied by the girl and the weight of the box.
The weight of the box is 40 N, and it acts downward. Therefore, the net vertical force is the difference between the vertical component of the force applied by the girl and the weight of the box.
Net vertical force = F_vertical - weight
Net vertical force = 35.4 N - 40 N
Net vertical force = -4.6 N
Since the box is being lowered at a constant speed, we know that the net force in the vertical direction is zero. Therefore, the force applied by the girl is equal in magnitude and opposite in direction to the weight of the box.
Step 3: Calculate the horizontal acceleration of the box.
Since the net force in the vertical direction is zero, we can conclude that there is no vertical acceleration. Therefore, the box is not moving vertically.
The horizontal acceleration of the box can be calculated using Newton's second law, which states that force equals mass times acceleration (F = ma). In this case, the horizontal force acting on the box is the horizontal component of the force applied by the girl.
F_horizontal = ma
Rearranging the equation:
a = F_horizontal / m
Substituting the values:
a = 35.4 N / 40 kg
a = 0.885 m/s^2
Therefore, the horizontal acceleration of the box is 0.885 m/s^2.
Step 4: Determine the angles of the forces acting on the box.
The force applied by the girl is at an angle of 45° above the horizontal, and the weight of the box acts vertically downward.
Hence, the angle of the force applied by the girl with respect to vertical is 45° (since it is 45° above the horizontal), and the angle of the weight is 90°.
To summarize:
- The horizontal force acting on the box is 35.4 N.
- The vertical force acting on the box is -4.6 N.
- The horizontal acceleration of the box is 0.885 m/s^2.
- The force applied by the girl is at an angle of 45° above the horizontal.
- The weight of the box acts vertically downward at an angle of 90°.
To find the net force acting on the box, we need to analyze the forces acting in different directions.
First, let's break down the force pushing the box into its horizontal and vertical components:
Horizontal component: The force at a 45° above the horizontal creates an angle with the horizontal component. To find its value, we can use trigonometry:
Horizontal component = Force * cos(angle)
Horizontal component = 50 N * cos(45°)
Horizontal component = 50 N * 0.707
Horizontal component ≈ 35.4 N
Vertical component: The vertical component is the force acting perpendicular to the wall. To find its value, we can use trigonometry:
Vertical component = Force * sin(angle)
Vertical component = 50 N * sin(45°)
Vertical component = 50 N * 0.707
Vertical component ≈ 35.4 N
Next, let's consider the weight of the box, which acts in the downward direction:
Weight = 40 N (given)
Since the box is lowered down the wall at a constant speed, the net force in the vertical direction must be zero. The vertical component of the force pushing the box (35.4 N) must balance out the weight of the box (40 N) to maintain a constant speed.
Therefore, the net force in the vertical direction is:
Net force (vertical) = Vertical component of force pushing the box - Weight
Net force (vertical) = 35.4 N - 40 N
Net force (vertical) ≈ -4.6 N
The negative sign indicates that the net force is in the upward direction, opposing the weight of the box. This negative force is necessary to balance out the weight and keep the box moving downward at a constant speed.
In summary, the net horizontal force is 35.4 N and the net vertical force is approximately -4.6 N.