Hooke's law describes a certain light spring of unstressed...

Hooke's law describes a certain light spring of unstressed length $35.0 \mathrm{cm} .$ When one end is attached to the top of a door frame and a $7.50-\mathrm{kg}$ object is hung from the other end, the length of the spring is $41.5 \mathrm{cm} .$ (a) Find its spring constant. (b) The load and the spring are taken down. Two people pull in opposite directions on the ends of the spring, each with a force of 190 $\mathrm{N}$ . Find the length of the spring in this situation.

Key Concepts

Equilibrium of Forces

The concept of equilibrium of forces is critical in mechanics where an object is at rest or moving with constant velocity, indicating that the sum of the forces acting on it is zero. In the context of springs, equilibrium is reached when the restoring force provided by the spring (as given by Hooke's Law) exactly balances the external forces applied, such as gravitational force or forces applied from opposite directions. This balance allows for the calculation of unknown quantities, like the spring's extension or contracted length.

Hooke's Law

Hooke's Law is a principle in mechanics stating that the force exerted by an ideal spring is directly proportional to its displacement from its natural, unstressed length. The law is mathematically expressed as F = k?x, where F is the force applied, ?x is the extension or compression of the spring, and k is the spring constant. This law forms the basis for understanding how elastic materials behave under load.

Spring Constant

The spring constant (k) is a parameter that quantifies the stiffness of a spring. It tells us how resistant a spring is to being deformed under an applied force. A higher spring constant means the spring requires more force to achieve the same displacement compared to a spring with a lower spring constant. Determining the value of k is essential when analyzing systems that involve elastic components.

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