A spring whose natural length is 15 cm exerts a force of 45 N when stretched to a length of 20 c

step 1 So first it wants us to find the spring constant, so we'll use Hook's Law, F equals KX, and we'l

A spring whose natural length is 15 cm exerts a force of 45...

A spring whose natural length is $15 \mathrm{cm}$ exerts a force of $45 \mathrm{N}$ when stretched to a length of $20 \mathrm{cm} .$ (a) Find the spring constant (in newtons/meter). (b) Find the work that is done in stretching the spring $3 \mathrm{cm}$ beyond its natural length. (c) Find the work done in stretching the spring from a length of $20 \mathrm{cm}$ to a length of $25 \mathrm{cm} .$

A spring whose natural length is $15 \mathrm{cm}$ exerts a force of
$45 \mathrm{N}$ when stretched to a length of $20 \mathrm{cm} .$
(a) Find the spring constant (in newtons/meter).
(b) Find the work that is done in stretching the spring $3 \mathrm{cm}$
beyond its natural length.
(c) Find the work done in stretching the spring from a length
of $20 \mathrm{cm}$ to a length of $25 \mathrm{cm} .$

Key Concepts

Work Done in Stretching a Spring

The work done in stretching a spring involves the gradual accumulation of energy, as the force exerted by the spring increases with displacement. Since the force is not constant but proportional to the displacement, the work is calculated by integrating the force over the displacement. For a spring obeying Hooke's law, this leads to the formula W = ½ k x² when stretching from the equilibrium position, illustrating how energy is stored in the form of elastic potential energy.

Integration in Work Calculations

Integration is used in physics to sum continuously varying quantities, which is essential when the force applied is not constant over a distance. In the context of stretching a spring, integration allows the calculation of total work done by considering the incremental work over each infinitesimal segment of the displacement, providing a precise measure of the energy stored in the spring.

Hooke's Law

Hooke's Law establishes the linear relationship between the force exerted by a spring and its displacement from equilibrium. It is typically expressed as F = kx, where F is the force applied by the spring, k is the spring constant reflecting its stiffness, and x is the displacement from its natural length. This principle is fundamental in analyzing how springs store and release energy.

Spring Constant

The spring constant, denoted as k, quantifies the stiffness of a spring by indicating the force required to produce a unit displacement from its natural length. Measured in newtons per meter (N/m), a higher spring constant means a stiffer spring that requires more force to stretch or compress by a specific distance.

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