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Revision:Elastic Strings And Springs

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TSR Wiki > Study Help > Subjects and Revision > Revision Notes > Mathematics > Elastic Stings and Springs

These notes are based on the requirements of the M3 A Level mathematics module.

Contents

2.1 - Hooke's law

\displaystyle T = \frac{\lambda x}{L}

  • \displaystyle T = tension in spring or string
  • \displaystyle \lambda = modulus of elasticity of spring or string
  • \displaystyle x = extension/compression of string or spring
  • \displaystyle L = natural length of spring/string


If a particle is attached to a string in equilibrium;

Resolve vertically

\displaystyle T = mg

\displaystyle \frac{\lambda x}{L} = mg

\displaystyle x = \frac{mgL}{\lambda}


If a particle is attached to a string and moves;

When the particle P is in equilibrium, using Hooke's law gives;

\displaystyle T = \frac{\lambda e}{L}

\displaystyle e = \frac{TL}{\lambda}

\displaystyle e = \frac{mgL}{\lambda}


When the particle P is at some point where the extension, x, is greater than mgL/λ (e) then the tension in the string will be greater than the weight of the particle and the particle will move with an acceleration a.


Using the equation of motion,

\displaystyle F = ma

\displaystyle T - mg = ma

\displaystyle \frac{\lambda x}{L} - mg = ma

\displaystyle a = \frac{\lambda x}{mL} - g


The resulting acceleration depends on the extension in the string and is constantly changing.


2.2 - Energy stored in an elastic string or spring

Consider a particle attached to one end of an elastic string whose other end is fixed on a smooth horizontal table. When the string is stretched beyond its natural length by pulling the particle along the table and then released, the particle will move along the table and will gain kinetic energy. As the motion is horizontal there is no change in gravitational potential energy so it follows that when stretched, the string has energy stored in it. This form of potential energy is called the elastic potential energy (E.P.E) of the string.


Work done = area under a Tension-extension graph


Before the elastic limit of the string/spring, the gradient of the Tension-extension graph is a straight line.


So, the area under a Tension-extension graph

\displaystyle = \frac{1}{2}Tx

\displaystyle = \frac{1}{2}\frac{\lambda x^2}{L}

\displaystyle = \frac{\lambda x^2}{2L}


Therefore the work done in stretching an elastic spring with modulus λ from its natural length L to a length

\displaystyle (L+x) is \displaystyle \frac{\lambda x^2}{2L}


if a string (or spring) which has its extension (or compression) increased from x to y will have its elastic energy increased.

This increase is given by;


Increase in E.P.E.

= final E.P.E. - initial E.P.E.

\displaystyle = \frac{\lambda y^2}{2L} - \frac{\lambda x^2}{2L}


2.3 - Problems involving kinetic energy, gravitational potential energy and elastic potential energy

By the work-energy principle;

The total change of the mechanical energies (that is kinetic, gravitational potential and elastic potential energies) of a system is equal to the work done by any external forces acting on a system.


\displaystyle mgh_1 + \frac{1}{2}mu^2 + \frac{\lambda x^2}{2L} - F_Rx = mgh_2 + \frac{1}{2}mv^2 + \frac{λy^2}{2L}.


Comments

Originally written by Widowmaker on TSR forums.

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