An elevator cable breaks when a 920-kg elevator is 24 m above a huge spring (k=2.2 ×10^5 N / m)

step 1 So there is an elevator which is falling and the work done. So we have this elevator is falling

An elevator cable breaks when a 920-kg elevator is 24 m...

An elevator cable breaks when a $920-\mathrm{kg}$ elevator is 24 $\mathrm{m}$ above a huge spring $\left(k=2.2 \times 10^{5} \mathrm{N} / \mathrm{m}\right)$ at the bottom of the shaft. Calculate (a) the work done by gravity on the elevator before it hits the spring, $(b)$ the speed of the elevator just before striking the spring, and $(c)$ the amount the spring compresses (note that work is done by both the spring and gravity in this part).

An elevator cable breaks when a $920-\mathrm{kg}$ elevator is 24 $\mathrm{m}$
above a huge spring $\left(k=2.2 \times 10^{5} \mathrm{N} / \mathrm{m}\right)$ at the bottom of the shaft. Calculate (a) the work done by gravity on the elevator before it hits the spring, $(b)$ the speed of the elevator just before striking the spring, and $(c)$ the amount
the spring compresses (note that work is done by both the spring and gravity in this part).

Key Concepts

Work Done by Gravity

This concept refers to the energy that is transferred when an object moves under the influence of the gravitational force over a certain distance. It is calculated as the product of the weight of the object (mass times gravitational acceleration) and the vertical displacement, assuming the force acts over the full distance. It quantifies how much work is performed by gravity in moving the object along a displacement.

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion. When an object accelerates under the influence of forces such as gravity, its kinetic energy increases as a function of the square of its speed. Understanding kinetic energy is essential for calculating the velocity of an object when energy is converted from potential energy.

Spring Potential Energy and Hooke's Law

Spring potential energy is the energy stored in a spring when it is compressed or stretched. According to Hooke’s Law, this energy is proportional to the square of the displacement from its equilibrium position, characterized by the spring constant. This concept is crucial in determining how much a spring will compress when it absorbs energy from an impacting object.

Conservation of Mechanical Energy

The conservation of mechanical energy principle states that in a closed system with no non-conservative forces (like friction), the total mechanical energy (sum of kinetic and potential energies) remains constant. This principle is used to analyze problems where gravitational potential energy transforms into kinetic energy or spring potential energy, ensuring that energy is neither created nor destroyed in the process.

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