91Ó°ÊÓ

Hooke's Law describes the relationship between the force exerted by a spring and its displacement from equilibrium. Mathematically, it is represented as \( F = -kx \). The constant \( k \) is the spring constant, indicating the stiffness of the spring, while \( x \) is the displacement. When a spring is compressed or stretched, it exerts a force in the opposite direction of the displacement. In the context of the inclined plane problem, the block compresses the spring by \( 0.1 \, \text{m} \). Using Hooke's Law, we calculate the spring force \( F_s \) as \( -100 \, \text{N} \). The negative sign signifies that this force acts in the opposite direction to the spring's compression, essentially "pushing back" against the block."

Newton's Second Law of Motion states that the acceleration of an object is directly proportional to the net force acting upon it and inversely proportional to its mass. It is mathematically expressed as \( F = ma \), where \( F \) is the net force, \( m \) is the mass, and \( a \) is the acceleration. When applying this law to our problem, after identifying the net force exerted on the block by both the spring and gravity, we can determine the block's resulting acceleration. At initial release, the net force on the block is \( -90.2 \, \text{N} \), leading to an acceleration of \( -45.1 \, \text{m/s}^2 \). The negative sign indicates that the acceleration is directed up the plane, opposing the gravitational pull.

The spring force is the result of a spring being either compressed or extended from its natural position. Hooke's Law introduces this concept, which implies that the force exerted by the spring is proportional to the displacement. With the inclined plane scenario, as the spring is compressed by different amounts, the force varies: - For an initial compression of \( 0.1 \, \text{m} \), the force is \(-100 \, \text{N} \). - When the spring is compressed by \( 0.05 \, \text{m} \), the force is reduced to \( -50 \, \text{N} \). - Ultimately, when the spring returns to its natural position, the force becomes \( 0 \, \text{N} \).The variations in spring force clearly affect the net force and consequently, the acceleration of the block.

Inclined planes are surfaces angled against horizontal surfaces. They are essential in physics for illustrating components of forces, especially when encountering gravitational force. In this exercise, the plane is inclined at \( 30^{\circ} \). Gravitational force acting on the block has two components: parallel and perpendicular to the inclined plane. The component acting down the incline is \( F_g = mg \sin \theta \). This component must be considered when calculating the net force and subsequent acceleration of the block as it moves up and down the plane.By understanding these force components, students can better predict how forces work on inclined planes and analyze motion effectively.