A box rests on a plank that is inclined to the horizontal. As the angle between the plank and the horizontal increases, does the component of the weight of the box parallel to the plank increase, decrease, or remain the same? Explain your answer.
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It increases
As the angle between the plank and the horizontal increases, the component of the weight of the box parallel to the plank will decrease. This can be explained by breaking down the weight force vector into its components.
The weight force vector can be divided into two components: one perpendicular to the plank (normal force) and the other parallel to the plank. As the angle of inclination increases, the normal force component will also increase, as it is directly proportional to the sine of the angle.
Since the weight force remains constant, the increase in the normal force component will lead to a decrease in the parallel component of the weight. This is because the weight force has to be distributed between the normal and parallel components, and as the normal component increases, the parallel component decreases to maintain the total weight force.
Therefore, the component of the weight of the box parallel to the plank will decrease as the angle of inclination increases.
To determine whether the component of the weight of the box parallel to the plank increases, decreases, or remains the same as the angle between the plank and the horizontal increases, we need to consider the forces acting on the box.
There are two forces acting on the box: gravity acting vertically downward, and the normal force exerted by the plank on the box perpendicular to the surface of the plank.
The weight of the box can be resolved into two components: one parallel to the plank and one perpendicular to the plank. The component of the weight parallel to the plank is responsible for causing the box to slide down the plank, while the component perpendicular to the plank is supported by the normal force.
As the angle between the plank and the horizontal increases, the gravitational force acting vertically downward remains constant. However, the component of the weight parallel to the plank increases because the angle between the gravitational force and the plank decreases. This can be explained using trigonometry.
The formula to calculate the component of the weight parallel to the plank is given by:
F_parallel = m * g * sin(theta)
Where:
- F_parallel is the component of the weight parallel to the plank
- m is the mass of the box
- g is the acceleration due to gravity
- theta is the angle between the plank and the horizontal
As theta increases, sin(theta) also increases, resulting in an increase in the component of the weight parallel to the plank. Therefore, as the angle between the plank and the horizontal increases, the component of the weight of the box parallel to the plank also increases.