Determination of the Spring Constant and the Effective Mass of a Loaded Spring and Hence to Calculate the Rigidity Modulus of the Spring.

Experiment no. : W1 

Name of the experiment: Determination of the spring constant and the effective mass of a loaded spring and hence to calculate the rigidity modulus of the spring.

Abstract

The spring constant and effective mass of a given spring can be determined by recording the vibration of the spring along a vertical line when its one end is loaded with a mass. The rigidity modulus of the given spring can also be determined upon knowing the number of turns in the spring, radius of the spring, radius of the wire of the spring and the spring constant. The spring constant of the given spring was found to be 12262.5 dyne/cm, the effective mass of the loaded spring was found to be 32.9 gm and the rigidity modulus was found to be 7.678*10^11 dyne/sq.cm. with an error of 0.2857%

Introduction

Forces cause objects to move or deform in some way. Newton’s third law states that for every force, there is an equal and opposite force. This is true for springs, which store and use mechanical energy to do work.

Springs are elastic, which means after they are deformed (when they are being stressed or compressed), they return to their original shape. Springs are in many objects we use on a daily basis. They in ball point pens, mattresses, trampolines, and absorb shock in our bikes and cars. According to the Third Law of Motion, the harder your pull on a spring, the harder it pulls back. Springs obey Hooke’s Law, discovered by Robert Hooke in the 17th century. Hooke’s law is described by:

F = -kx

Where F is the force exerted on the spring in Newtons (N),

k is the spring constant, in Newtons per meter (N/m),

and x is the displacement of the spring from its equilibrium position.

The spring constant, k, is representative of how stiff the spring is. Stiffer (more difficult to stretch) springs have higher spring constants. The displacement of an object is a distance measurement that describes that change from the normal, or equilibrium, position.                                [1]

In a real spring–mass system, the spring has a non-negligible mass m. Since not all of the spring's length moves at the same velocity v as the suspended mass M, its kinetic energy is not equal to mv^{2}/2 . As such, m cannot be simply added to M to determine the frequency of oscillation, and the effective mass of the spring is defined as the mass that needs to be added to M to correctly predict the behavior of the system.                  [2]

Modulus of Rigidity - G  - (or Shear Modulus) is the coefficient of elasticity for a shearing force. It is defined as "the ratio of shear stress to the displacement per unit sample length (shear strain)" .

Modulus of Rigidity can be experimentally determined from the slope of a stress-strain curve created during tensile tests conducted on a sample of the material.

Definition of Modulus of Rigidity:

The ratio of shear stress to the displacement per unit sample length (shear strain)                                        [3]