HOOKE’S LAW - Form 2 Physics Notes

• Introduction
• Characteristics of Materials
• Hooke’s Law
  • Factors Affecting Spring Constant
  • Compressing a Spring
  • Work Done in Stretching or Compressing a Spring
• Revision Exercise

Introduction

• The knowledge of stretching materials when forces are applied is important particularly in the construction industry.
• It helps engineers to determine the strength of the materials to be used for specific work.
• This topic deals with study of how materials behave when stretched and the relationship between the extent of stretching and stretching force.
• The pioneer of the topic is the physicist Robert Hooke.

Characteristics of Materials

1. Strength

• It is the ability of a material to resist breakage when under stretching, compressing or shearing force.
• A strong material is one which can withstand a large force without breaking.

2. Stiffness

• Refers to the resistance a material offers to forces which tend to change its shape or size or both.
• Stiff materials are not flexible and resist bending.

3. Ductility

• This is the quality of a material which leads to permanent change of shape and size.
• Ductile materials elongate considerably when under stretching forces and undergo plastic determination until they break e.g.lead, copper, plasticine.

4. Brittleness

• This is the quality of a material which leads to breakage just after elastic limit is exceeded.
• Brittle materials do not undergo extension and break without warning on stretching. E.g.blackboard chalk, bricks, castiron, glass, and dry biscuits.

5. Elasticity

• This is the ability of a material to recover its original shape and size after the force causing deformation is removed.
• The materials with this ability are called elastic e.g. rubber bands, spring, and somewires.
• A material which does not recover its shape but is deformed permanently is called plastic e.g.plasticine.

Hooke’s Law

• Hooke’s law relates the stretching force and extension produced.
• It states that “for a helical spring or any other elastic material, extension is directly proportional to the stretching force,provided elastic limit is not exceeded”
  i.e. F∝e; F=ke,
  Where k is the constant of proportionality called spring constant.
• Sl unit of spring constant is the newton per meter (N/m).
• Spring constant is defined as the measure of stiffness of a spring.
• Graphically, Hooke’s law can be expressed as below.
  
• The graph of stretching force against extension, for material that obeys Hooke’s law, is a straight line through the origin. The gradient(slope) of such a graph gives the spring constant of the spring used.
  Gradient (slope)= ^{change in F}/_{change in e} = spring constant
  S= ^ΔF/_Δe = k
• If the stretching force exceeds a certain value, permanent stretching occurs.
• The point beyond which the elastic material does not obey Hooke’s law is called elastic limit.
• A point beyond which a material loses its elasticity is called yield point.
• Along OE the spring(or elastic material) is said to undergo elastic deformation.
• Along EA the spring is said to undergo plastic deformation

Factors Affecting Spring Constant

1. Type of material making the wire
2. Length of the spring
3. The number of turns per unit length of the spring
4. The diameter (thickness) of the spring
5. The thickness of the wire

Examples

1. A spring stretches by 1.2cm when a 600g mass is suspended on it. What is its spring constant?
   
   Solution
   
   
2. The figure below shows a spring when unloaded, when supporting a mass of 80g and when supporting a stone.Study the diagrams and use them to determine the mass of the stone.
   
   Solution
   
   =0.048kg (this is the mass of the stone)
   
3. A spiral spring produces an extension of 6mm when a force of 0.3N is applied to it. Calculate the spring constant for a system when two such springs are arranged in:
   1. Series
      
   2. Parallel
      
      Since the two springs will share the weight, extension of the system is ¹/₂ x 6mm = 3mm
      Spring constant of the system, k_P is
      k_p = ^F/_e = ^0.3N/_0.003m =100Nm^-1
4. The data below represents the total length of a spring as the load suspended on it is increased.
   
   

   Weight, W (N)	0.5	1.0	1.5	2.0	2.5	3.0
   Total length, L (x10-2m)	7.5	8.0	8.5	9.0	9.5	10.0

   1. Plot a graph of total length (y-axis) against weight.
      
   2. Use the graph to determine
      1. The length of the spring
         The length of the spring is that when force acting on it is zero. From the graph it is 7.1x10^-2m
      2. The spring constant,k.
         

Compressing a Spring

• Compression refers to change in length that occurs when a spring is squeezed from its two ends.
• A sketch of length against compression for a spring which obeys Hooke’s law is as below.
  
• Beyond the point E, the turns of the spring are virtually pressing onto one another and further increase in force achieves no noticeable decrease in length.

Exercise

1. The figure below shows a simple apparatus for studying the behavior of a spring when subjected to forces of compression.
   
   Describe how the apparatus may be used to obtain readings of compression force and corresponding length of spring.
2. In a similar experiment the following readings were obtained.
   
   

   Force of compression, F (N)	0.0	5.0	10.0	15.0	17.5	22.5	25.0	30.0	35.0	40.0	45.0	50.0
   Length of spring, L(cm) compression	14.50	13.00	11.50	10.00	9.25	7.75	7.00	6.50	6.25	6.00	6.00	6.00

   
   Plot a graph of:
   1. Compression forces versus length of the spring and from the graph determine the minimum force that will make the spring coils to just come into contact.
   2. Compression forces versus compression of spring and from the graph determine the spring constant.

Work Done in Stretching or Compressing a Spring

• The area under force versus extension graph represents work done in stretching the spring.
  
  Area under the graph= ¹/₂Fe,
  
  where F is the force applied and e the extension attained.
  
  From Hooke's law, F=ke
  Workdone=¹/₂(ke)e= ¹/₂ke²

Exercise

Two springs of negligible weights and of constants k₁= 50Nm^-1 and k₂=100Nm^-1 respectively are connected end to end and suspended from a fixed point. Determine

1. The total extension when a mass of 2.0kg is hung from the one end
2. The constant of the combination.
3. Work done in stretching each spring (elastic potential energy of each)

Revision Exercise

1. State Hooke’s law
2. Define the following terms
   1. Elasticity
   2. Elastic material
   3. Plastic deformation
   4. Spring constant
   5. Stiffness
   6. A stiff material
   7. Elastic material
   8. Yieldpoint
3. A 60g mass is suspended from a spring. When 1.5g wire is added, the spring stretches by 1.2cm. Given that the spring obeys Hooke’s law, find:
   1. The spring constant
   2. The total extension of the spring
4. A piece of wire of length 12m is stretched through 2.5cm by a mass of 5kg. Assuming that the wire obeys Hooke’s law
   1. Through what length will a mass of 12.5kg stretch it?
   2. What force will stretch it through 4.0cm?
5. The following readings were obtained in an experiment to verify Hooke’s law using a spring.
   

   Mass(g)	0	25	50	75	100	125
   Reading(cm)	10	11.5	12.5	13.5	14.4	16.0
   Force(N)
   Extension(mm)

   1. For each reading, calculate:
      1. The value of the force applied
      2. The extension in mm
   2. Plot a graph of extension against force. Does the spring obey Hooke’s law?
   3. From the graph determine:
      1. The elastic limit(mark on graph)
      2. The spring constant
      3. The weight of a bottle of ink hung from the spring if the reading obtained is 12cm
      4. The extension in mm when a force of 0.3N is applied
      5. The scale reading in cm for a mass of 0.02kg