91Ó°ÊÓ

Potential energy refers to the stored energy of an object due to its position or state. In this scenario, the climber has potential energy because of his elevation above the ground. This energy is dependent on three factors:

• The mass of the climber (\( m = 86 \, \text{kg} \)).
• The height from which he falls (\( h = 0.750 \text{ m} \)).
• The acceleration due to gravity (\( g = 9.81 \, \text{m/s}^2 \)).

You can calculate this potential energy using the formula:\[PE = mgh\]In our case, this becomes: \[PE = 86 \times 9.81 \times 0.750 \approx 632.55 \, \text{Joules}\] The importance of understanding potential energy in this problem is pivotal. This energy is converted into another form, helping in the calculation of the spring stretch.

Hooke's Law is fundamental to understanding spring mechanics. It describes how the force exerted by a spring is proportional to the amount it stretches or compresses. Expressed mathematically: \[F = kx\]where

• \( F \) is the force exerted by the spring,
• \( k \) is the spring constant, and
• \( x \) is the amount the spring is stretched or compressed from its original position.

In the case of the climber, when he falls, the rope (acting as a spring) stretches until it stops him, converting his potential energy into spring energy. This is pivotal to determining how much the rope will stretch and ensuring it's strong enough to prevent exceeding physical limits.

Gravitational force is the attractive force exerted by the Earth on an object and it directly affects potential energy. It is calculated using:\[F_g = mg\]where:

• \( F_g \) is the gravitational force,
• \( m \) is the mass of the object (climber in this case = 86 kg), and
• \( g \) is the acceleration due to gravity (9.81 m/s² here).

Therefore, this gravitational force contributes to the potential energy but must be counteracted when the climber falls. The role of the rope, behaving as a spring, comes into play to resist this force and bring the climber to a rest safely.

The spring constant (\( k \)) is a measure of a spring's stiffness. It is essential when calculating the energy stored in a spring-like mechanism during stretch or compression. For the climber’s safety rope, a given spring constant of \( 1.20 \times 10^3 \text{ N/m} \) suggests how resistant the rope is to stretching.
This value feeds into Hooke’s Law and is used to determine how much the rope will stretch when the climber falls.
By knowing both the potential energy transferred into the rope and the spring constant, we use the equation:\[PE_{spring} = \frac{1}{2} k x^2\]Here, solving for \( x \) gives insight into how the potential energy is managed and how it affects the rope's properties when it halts the climber’s fall.