Elasticity and Simple Harmonic Motion

Dewi MuliyatiDepartment of Physics State University of Jakarta

Outline…

Stress, Strain, and Modulus of Elasticity

Hooke’s LawForce Constant of Elastic ObjectsRestoring ForceEquation for Displacement in SHMPeriod of SHMHooke’s Law for Spring ArrangementSeveral Benefits of Springs

!!!

Elasticity is the ability of an object to return to its original shape as soon as the external force which is applied to the object is eliminated (relieved).

Stress, Strain, and Modulus of ElasticityCourse-1

A Pulling Stress, σ

Is defined as the quotient between the pulling force F experienced by the wire and its cross-sectional area (A).

A

F

Strain, e

Is defined as the quotient between the change in length ∆L and its initial length L.

L

Le

Stress-Strain Graph

Modulus of Elasticity, E

The modulus of elasticity, E of a material is defined as the stress-strain ratio experienced by the material.

eE

Table: Modulus of Elasticity of Various Substances

Substance E (N/m2)

IronSteelBronzeConcreteCoalGraniteMarmerGranitWood (pine)NylonYoung Bone

100 x 109

200 x 109

100 x 109

70 x 109

20 x 109

14 x 109

50 x 109

45 x 109

10 x 109

5 x 109

15 x 109

Hooke’s LawCourse-1

Hooke’s Law

If the pulling force does not exceed the spring’s elastic limit, then the spring’s length increase is directly proportional to the pulling force.

xkF

Force Constant of Elastic ObjectsCourse-1

Force Constant of Elastic Objects

E is the modulus of elasticity of material (N/m2)

L is the length of object, no force applied (m)

A is the cross-sectional area (m2)

L

AEk

Restoring ForceCourse-2

Restoring Force

The force whose magnitude is proportional to displacement and always acts in the opposite direction of displacement (position).

The restoring force always causes an object to move back and forth about the equilibrium point (simple harmonic motion).

The restoring force is always opposite to the displacement direction (motion) of the object.

Equation for Displacement in SHMCourse-2

Equation for Displacement in SHM

Displacement equation

θ0 , the initial phase angle is obtained from the initial condition.

A is amplitude. ω is angular frequency.

)sin()( 0 tAtx

Period of SHMCourse-2

Period of SHM

Acceleration of SHM,

Angular Frequency,

Period,

xa 2

m

k

k

mT 2

Hooke’s Law for Spring ArrangementCourse-2

Serial spring arrangement

...1111

321 1

kkkk is

For the special case of two springs with constants k1 and k2,the spring constant of replacement spring, ks:

21

21

kk

kkks

Parallel spring arrangement

For n identical springs in a parallel arrangement, where every spring has a force constant k, the replacement spring constant kp:

...321

1

kkkki

p

nkk p

Several Benefits of SpringsCourse-2

Suspension systems in Motor Vehicles to Damped Shocks

Springs on steering wheels