(III) An elevator cable breaks when a 925 -kg elevator is 22.5 m above the top of a huge spring

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(III) An elevator cable breaks when a 925 -kg elevator is...

(III) An elevator cable breaks when a 925 -kg elevator is 22.5$\mathrm { m }$ above the top of a huge spring $( k =$ $8.00 \times 10 ^ { 4 } \mathrm { N } / \mathrm { m } )$ 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 betore striking the spring; $( c )$ the amount the spring compresses (note that here work is done by both the spring and gravity.

Key Concepts

Work Done by Gravity

This concept refers to the energy transferred to an object when gravity acts upon it over a distance. It is calculated as the product of the gravitational force (mass times gravitational acceleration) and the displacement in the direction of the force and can be associated with the conversion of gravitational potential energy to kinetic energy during free fall.

Conservation of Mechanical Energy

This principle states that in the absence of non-conservative forces (or when they are accounted for), the total mechanical energy (the sum of potential and kinetic energies) of a system remains constant. It is essential in analyzing situations where energy transforms from one form to another, such as when gravitational potential energy is converted into kinetic energy or elastic potential energy.

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion. It is dependent on the mass and the square of the velocity of the moving object and is calculated using the formula (1/2)mv². This concept is critical in linking the energy gained from gravitational work to the velocity of the falling object.

Gravitational Potential Energy

This is the energy stored in an object due to its position within a gravitational field. It is determined by the product of the object's mass, the gravitational acceleration, and its height above a reference point. Changes in gravitational potential energy are fundamental to understanding the dynamics of objects in free fall.

Elastic Potential Energy and Hooke’s Law

Elastic potential energy is the energy stored in a spring when it is compressed or stretched. Hooke’s Law states that the force exerted by a spring is proportional to its displacement from equilibrium. This concept is used to determine how much energy is transferred into the spring as it compresses, and it is vital for analyzing systems that involve elastic forces, such as spring-mass systems.

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