91Ó°ÊÓ

The spring constant, denoted as \( k \), is a measure of a spring's stiffness. In simpler terms, it tells you how hard or difficult it is to compress or stretch the spring. According to Hooke's Law, the relationship between the force exerted by a spring, \( F \), and its displacement, \( x \), is a linear one: \( F = kx \). This means that the force needed to compress or extend the spring by a certain distance is directly proportional to that distance.
To find the spring constant, we need the spring's displacement under a known force. In our scenario, that force is the weight of the stone resting on the spring, calculated using the mass of the stone \( m \) and gravitational acceleration \( g \). By rearranging the formula:\

• \( k = \frac{mg}{x} \)
• Substitute the known values: mass \( m = 8.00 \, \text{kg} \), \( g = 9.81 \, \text{m/s}^2 \), and displacement \( x = 0.10 \, \text{m} \)

This gives us \( k = 784 \, \text{N/m} \).
In essence, this signifies a relatively stiff spring, requiring a force of 784 Newtons to compress it by one meter.

Elastic potential energy is the energy stored in a spring when it is compressed or stretched. Imagine pulling on a rubber band – the more you stretch it, the more energy is stored within its elastic material. The energy stored is determined by how much the spring is compressed or stretched and by how stiff the spring is (i.e., its spring constant).
The formula for elastic potential energy, \( U_s \), is given by:

• \( U_s = \frac{1}{2}kx^2 \)

Where \( k \) is the spring constant and \( x \) is the displacement from its relaxed position.
In this exercise, the total displacement of the spring from its relaxed state before the stone is released is 0.40 meters (0.10 m + 0.30 m). Using the spring constant calculated previously:

• \( U_s = \frac{1}{2} \times 784 \times (0.40)^2 = 62.72 \, \text{J} \)

This calculation shows that 62.72 Joules of energy are stored in the spring in its compressed position, ready to be converted to another form of energy when released.

Gravitational potential energy is a form of potential energy related to the height of an object above a reference point, like the ground. This energy is due to the gravitational force working on that object. It is calculated using the formula:

• \( U_g = mgh \)
• \( m \) is mass, \( g \) is gravitational acceleration, and \( h \) is height above the reference point.

When the stone is released, the elastic potential energy stored in the spring is converted into gravitational potential energy as the stone rises.
The change in gravitational potential energy \( \Delta U_g \) from the release point to the stone's peak height equals the elastic potential energy at release (given as 62.72 J).
This energy conversion from one type to another is a fundamental aspect of energy conservation in physics, illustrating how energy transforms but is never lost.

Determining the maximum height the stone reaches involves calculating how far the gravitational potential energy will carry it upward from the point of release. Since all the initial elastic potential energy converts into gravitational potential energy at the maximum height, we use the equation:

• \( mgh = 62.72 \, \text{J} \)

Here, we solve for the height \( h \):

• \( h = \frac{U_s}{mg} = \frac{62.72}{8.00 \times 9.81} = 0.80 \, \text{m} \)

This calculation shows that the maximum height above the original release point is 0.80 meters.
Keeping these expressions simple shows how gravitational potential energy is directly proportional to the height of an object, illustrating the predictable nature of energy transitions.