Figure 8-36 shows an 8.00 kg stone at rest on a spring. The spring is compressed 10.0 cm by th

step 1 All right, we have in Part A, Newton's second law on application. So the net force is equal to m

Figure 8-36 shows an 8.00 kg stone at rest on a spring. The...

Figure $8-36$ shows an $8.00 \mathrm{~kg}$ stone at rest on a spring. The spring is compressed $10.0 \mathrm{~cm}$ by the stone. (a) What is the spring constant? (b) The stone is pushed down an additional $30.0 \mathrm{~cm}$ and released. What is the elastic potential energy of the compressed spring just before that release? (c) What is the change in the gravitational potential energy of the stone-Earth system when the stone moves from the release point to its maximum height? (d) What is that maximum height, measured from the release point?

Key Concepts

Equilibrium Position

In a spring-mass system, the equilibrium position is the point at which the net force acting on the mass is zero, meaning the upward spring force balances the downward gravitational force. Understanding the equilibrium position is important because it serves as a reference point for measuring displacements and analyzing energy changes when the system is disturbed from its rest position.

Conservation of Energy

The principle of conservation of energy states that energy cannot be created or destroyed, only converted from one form to another. In the context of spring-mass systems, the elastic potential energy stored in a compressed or extended spring is converted to kinetic energy or gravitational potential energy as the system evolves. This principle is fundamental when analyzing the motion of objects influenced by both spring forces and gravity, and it helps in establishing energy relationships between different system states.

Gravitational Potential Energy

Gravitational potential energy is the energy possessed by an object due to its position in a gravitational field. It is calculated by the expression mgh, where m is mass, g is the acceleration due to gravity, and h is the height above a reference level. This concept is pivotal in analyzing systems where vertical motion occurs, and it plays an integral role in energy conservation problems involving objects moving under the influence of gravity.

Elastic Potential Energy

Elastic potential energy is the energy stored in a deformed elastic object, such as a spring. The energy is given by the formula (1/2)kx², where k is the spring constant and x is the displacement from its equilibrium position. This concept is crucial in problems involving energy conversion, where the energy stored in the spring is later transformed into other forms, such as kinetic or gravitational potential energy.

Hooke's Law

Hooke's Law defines the linear relationship between the force exerted by a spring and its displacement from equilibrium. It states that the force is directly proportional to the displacement, with the proportionality constant being the spring constant (k). This concept is essential for understanding how springs behave under compression or extension and is used to calculate the spring constant when the force and displacement are known.

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