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2.12 Elasticity

Meaning of Elasticity

 

Elasticity is the property of  a substance which enables it to return to original shape @ size @ length after  an  applied external force( compressive force or stretching force)  is removed.

 

Why is a solid  is elastic ?

clip_image002

The property of elasticity is caused by the existence of two forces between molecules or atoms in the solid material.

The two forces are force of repulsion and force of attraction between molecules.

When a compressive force is applied to the solid,force of repulsion between the molecules pushes the molecules back  to their equilibrium positions.

When a stretching force is applied to the solid force of attraction between the molecules pulls  the molecules back to their equilibrium positions.

In  the absence of an applied external force on the solid, the force of attraction is balanced by the force of repulsion  or the resultant force is zero.

 

 

Graph of force between molecules , F against the distance between molecules, x.

 

clip_image004

 

At distance X1  : is the equilibrium position where the resultance force is zero.

At distance X < X1  : the solid is compressed where force of repulsion > force of attraction.

At distance  X1 < X<X2 : the solid is stretched where force of attraction  > force of repulsion until the force of attraction reaches  a maximum value at X2.  

At distance  X > X2 : the force of attraction will decrease and the molecular layer  will begin to slip and solid  will permanently change its shape. The point where the solid loses its elastic characteristics is call as the elastic limit. After this limit , the solid will not return to its original shape.

 

Restoring Force

clip_image006

Figure(a): The spring is untstretched ,i.e at natural length and exerts zero force on the trolley.

Figure(b): As the spring is stretched to the right, it exerts a force to the left on the trolley. This is called a  restoring force.

Hooke’s Law

 

Hooke’s Law states that the extension of an elastic substance is directly proportional to the stretching force acting on it  provided that the elastic limit is not exceeded.

 

clip_image010

 

@   F   µ  x

       F = k x

 

F =  the acting force or the effort

X = extension

k = the spring constant

Spring constant , k

 

F = kx,

  k = F

        x

      the unit of  k is  Nm-1

 

clip_image010

 

spring constant , k = Gradient of the graph

 

A larger value of k or  gradient  indicates a stiffer spring.

 

clip_image012

P:            stiff spring

Q:            soft spring

 

Graph of Stretching Force, F against Spring extension,

clip_image008

A : Elastic limit

OA : The graph is a straight line passing through the origin. Thus the stretching force is directly proportional  to the extension of the spring and Hooke’s  law is obeyed.

AB: The graph takes the form a curve, that is the stretching force is not vary directly with extension of the spring and Hooke’s law is no applicable.

 

Why does the oscillation of a spring stops?

 

If you leave a mass on a spring oscillating it eventually slows down and stops. Air resistance slows the object down. Energy is lost from the system in overcoming this friction. This effect is called damping.

 

In an ammeter or in a car’s suspension needs to stop the oscillations as quickly as possible .So damping process  should be happen as quickly as possible.

 

Factors affecting the rate of extension or stiffness of a spring.

 

Type of  spring material:  A spring made from a hard material  extending shorter than  a   spring made from a soft material. For example a steel spring extending shorter than a copper spring.

Diameter of spring coil:  A spring coil of a larger diameter  is easier to stretch ( the rate of extension is high) compared to a spring coil of smaller diameter.

Diameter of the wire of the spring :  A spring coil made from a thicker wire is harder ( the rate of extension is low) compared a spring made from a thinner wire.

Spring arrangement:  Stretch of  a spring in series is more easier than stretch of  a spring in parallel.

The original length of the spring: Stretch of  a longer spring   is more easier than stretch of  a shorter spring .

The spring constant , k :  The spring which has a larger value of k is the spring which more stiff(the rate of extension is low)

 

 

Use of Elasticity in Everyday Life:

(1)    Cushion/mattress: The spring in a cushion or mattress undergo many cycles of compression during use and each time the cushion is able to return to its original shape. This is due to the elasticity of the springs.

(2)     Electric meter : Electric meters such as ammeter, voltmeter and galvanometer have spiral springs. The springs are used to stop the pointer at a specific point on the scale or to return the pointer to the zero mark on the scale after a measurement has been taken

(3)     Weighing apparatus:  A  weighing apparatus such as spring balance , a spring is either extended or compressed and it obeys the

Hooke ‘ law and it caused the apparatus has a linear scale.

(4)     Vehicles spring support:

         It enables the passengers in a  vehicle to be seated in a comfortable position   when the vehicle goes on a bumpy road because springs shock absorbers are mounted on the wheels of vehicles to absorb impacts and damp vibrations resulting from movement on the bumpy road or uneven road surface.

(5)     In sports :  The elastic strings of a tennis or a badminton racket enable them to rebound the ball or shuttle.

          The ropes used by rock climbers have elastic properties that can save lives during climbing accidents. The ropes are made of  a continuous-drawn nylon fibre core and a protective textile covering . This reduces the stopping force acting on a falling climber.

          A bow bends or elastic twine of the bow is stretched to store the elastic potential energy used to propel the arrow.

 

Spring Systems

There are two ways to arrange a spring, that is,

(a)       Series arrangement

(b)       Parallel arrangement

       clip_image014

         Series                             Parallel

In series arrangement  same load  is applied to each spring i.e  W

In parallel arrangement the load is shared equally among the springs . i.e clip_image002

 

Example 1

The original length of a spring is 5 cm. With a load of mass 20 g, the length of the spring is extended to  7 cm.

Determine

(a)     the extension of the spring with a load  40 g

(b)     the  length of the spring with a load 60 g.

the load required to extend the spring to 20 cm.

 

Solution

 

 

Example 2

Spring A  extends by  2 cm when it hung with a  10 g weight. Spring B extends by 4 cm when it  hung with a 10g weight. Find the total stretch in each of the spring systems shown in the following figure.

clip_image016

Solution

 

 

Elastic Potential Energy ( Ee)

Elastic potential energy is the energy stored in a elastic matter when it is extended or compressed.

Thus,


 

Ee  =  ½ F x   =  ½ kx2   = Area under the                  

                                          graph  F vs.  x

 

                                F = Force   

                  x = extension   k = spring constant

 

Example 3

            

The original length of a spring is 12 cm. With a load of  20 g  , the length of the spring is extended  to 15 cm.

What is the elastic potential energy stored in the spring?

Solution

 

 

 

 

 

 

Example 4

 

Figure shows a graph of force,F against extension,x for  a spring.

 

clip_image020

What is the potential energy stored  when the spring is extended  by 0.4 m?

 

Solution

 

 

 

 

 

 

Example  5

 

Figure shows a ball of mass 10 g pushed against one end of a spring on a smooth surface. The original length of the spring is 14 cm and its spring constant is 200 N m-1.

 

 

clip_image022

Determine

(a)     the elastic potential energy stored in the spring.

(b)     the maximum velocity reached by the ball after the compressive force on the spring is removed.

 

Solution

One Response

  1. useful!!!thanx!

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