Elasticity: Hooke's Law and Young's Modulus | Jamb(UTME)

Jamb(utme) key points on elastic limit; yield point; breaking Point; Hookes' Law; Young's Modulus

Elastic Limit

1. The elastic limit is the maximum stress a material can handle and still return to its original shape.
2. When stress is removed below the elastic limit, no permanent deformation occurs.
3. Exceeding the elastic limit causes the material to deform permanently (plastic deformation).
4. It represents the boundary between elastic behavior and plastic behavior.
5. The elastic limit varies from material to material (e.g., rubber has a higher elastic limit than brittle materials like glass).
6. Engineers consider the elastic limit when designing materials to ensure they don't deform permanently under stress.
7. Beyond the elastic limit, the material enters the plastic region.
8. In daily life, a rubber band being overstretched shows an elastic limit in action.
9. Stress-strain graphs show the elastic limit as the point where the curve stops being linear.
10. Materials operating below their elastic limit are safe and maintain their original properties.
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Yield Point

11. The yield point is the stress level at which a material starts to deform plastically.
12. After the yield point, even when the stress is removed, the material doesn’t return to its original shape.
13. The yield point is slightly above the elastic limit for most materials.
14. It is marked by a small plateau or sudden drop in stress on a stress-strain curve.
15. Materials like mild steel have distinct upper and lower yield points.
16. The yield point defines the material's yield strength.
17. Yield strength is the maximum stress the material can take without permanent deformation.
18. For brittle materials, the yield point and elastic limit are nearly the same.
19. Yield point testing ensures materials can withstand intended loads in practical applications.
20. Exceeding the yield point is acceptable in some designs, as long as safety factors are applied.
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Breaking Point

21. The breaking point is where a material fails and fractures under stress.
22. It is also known as the fracture point.
23. On a stress-strain curve, the breaking point is the last point before the material snaps.
24. Brittle materials, like glass, break suddenly with minimal deformation.
25. Ductile materials, like copper or aluminum, stretch significantly before breaking.
26. The breaking point depends on the material’s tensile strength.
27. Tensile strength is the maximum stress a material can handle before breaking.
28. The breaking point is higher for materials like steel and lower for fragile materials like ceramics.
29. Environmental conditions, such as temperature, affect the breaking point.
30. Knowing the breaking point is essential for ensuring safety in construction and manufacturing.
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Hooke’s Law

31. Hooke’s Law states that the force applied to a material is directly proportional to its extension, within the elastic limit.
32. The formula for Hooke’s Law is:
    
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    $F = kx$
    where:
    • $F$ = Force,
    • $k$ = Spring constant (stiffness),
    • $x$ = Extension or compression.
33. Hooke’s Law only applies within the material’s elastic limit.
34. The spring constant measures how stiff the spring is.
35. Stiffer springs have a larger spring constant $(k)$.
36. Hooke’s Law explains the behavior of elastic materials like springs, rubber bands, and trampoline fabric.
37. The relationship between force and extension is shown as a straight line on a graph, up to the elastic limit.
38. The law is widely used in designing mechanical systems like shock absorbers and measuring devices.
39. When force is removed within the elastic limit, the material regains its original shape.
40. Hooke’s Law helps calculate how much energy a material can store as elastic potential energy.
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Young’s Modulus

41. Young’s Modulus is a measure of a material’s stiffness, defined as the ratio of stress to strain.
42. The formula for Young’s Modulus is:
    
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    $E = \frac{Stress}{Strain}$ where:
    • Stress = Force per unit area $(\sigma = \frac{F}{A})$,
    • Strain = Relative deformation $(\epsilon = \frac{\Delta L}{L})$.
43. A higher Young’s Modulus means the material is stiffer and less likely to deform under stress.
44. It is represented by the slope of the straight-line portion of a stress-strain curve.
45. Young’s Modulus is measured in pascals (Pa) in the SI system.
46. For materials like steel, Young’s Modulus is very high, indicating great stiffness.
47. For materials like rubber, Young’s Modulus is low, showing flexibility.
48. Young’s Modulus helps engineers choose the right materials for construction, machinery, and medical devices.
49. It is crucial in determining how materials stretch or compress under load.
50. Young’s Modulus applies only to the elastic region of the stress-strain curve.
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Jamb(utme) key points on the spring balance as a device for measuring force; work done per unit volume in spring and elastic strings

Spring Balance as a Device for Measuring Force

1. A spring balance is a device used to measure force.
2. It consists of a spring attached to a hook and a calibrated scale.
3. The spring stretches when a force is applied, and the scale measures the amount of stretching.
4. It works based on Hooke's Law, which states that the extension of a spring is proportional to the applied force.
5. The formula for force measured by a spring balance is $F = kx$, where:
   • $F$ is the force,
   • $k$ is the spring constant,
   • $x$ is the extension of the spring.
6. The spring constant $(k)$ is a measure of the stiffness of the spring.
7. The scale of a spring balance is marked in units of force, such as newtons (N).
8. A spring balance can measure the weight of objects, which is a force due to gravity.
9. When measuring weight, the spring balance must be vertical for accurate readings.
10. The hook at the end is used to hang objects whose force is to be measured.
11. Spring balances are commonly used in laboratories, kitchens, and industries.
12. They are portable and easy to use.
13. The accuracy of a spring balance depends on the quality of the spring and proper calibration.
14. Overstretching the spring can damage the balance or make it inaccurate.
15. The maximum force a spring balance can measure is limited by the spring's elasticity.
16. Spring balances are calibrated such that they read zero when no load is attached.
17. When overloaded, the spring balance may permanently deform and lose its accuracy.
18. A spring balance can measure forces acting in one direction at a time.
19. Friction within the device can slightly affect its accuracy.
20. Digital versions of spring balances exist, providing precise force measurements.
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Work Done Per Unit Volume in Springs and Elastic Strings

21. When a spring is stretched or compressed, work is done to change its shape.
22. This work is stored as elastic potential energy in the spring.
23. The formula for the elastic potential energy stored in a spring is:
    
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    $U = \frac{1}{2}kx^2$ where:
    • $U$ is the potential energy,
    • $k$ is the spring constant,
    • $x$ is the extension or compression.
24. The work done per unit volume is the energy stored divided by the volume of the spring or string.
25. The formula for work done per unit volume is:
    
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    ${Work per unit volume} = \frac{1}{2}kx^2$ / $V$ where $V$ is the volume of the spring.
26. Elastic potential energy is propo rtional to the square of the deformation $(x^2)$.
27. Springs with higher stiffness constants $(k)$ store more energy for the same deformation.
28. Elastic strings work similarly, storing energy when stretched within their elastic limits.
29. Elastic potential energy depends on the material and dimensions of the spring or string.
30. For a spring or string, exceeding the elastic limit results in permanent deformation, and energy storage becomes inefficient.
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Properties of Springs and Elastic Strings

31. Springs and elastic strings obey Hooke's Law within the elastic limit.
32. Elastic strings stretch linearly with force until the yield point is reached.
33. Both springs and strings store energy that can be used to perform work, like in slingshots or bows.
34. The ability to store energy makes springs useful in mechanical clocks, mattresses, and vehicles.
35. The amount of energy stored depends on both the material's elasticity and its dimensions.
36. Work done on a spring is recoverable as long as it remains within the elastic range.
37. Thin, long elastic strings tend to have a lower stiffness constant and stretch more under force.
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Applications of Work Done in Springs and Strings

38. Springs are used in shock absorbers to store and release energy efficiently.
39. Elastic potential energy in strings is utilized in sports equipment like bows and catapults.
40. Work done in springs is fundamental to energy storage in mechanical toys.
41. In weighing scales, the energy stored in springs is used to calculate applied force.
42. Elastic strings are used in exercise bands for strength training.
43. Springs store energy in watches, releasing it gradually to power movements.
44. Work done per unit volume in materials helps engineers design springs for durability.
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Key Concepts for Understanding Work Done

45. The energy stored in a spring depends on how far it is stretched or compressed from its equilibrium position.
46. The volume of the spring affects the energy density—smaller springs have higher energy densities for the same force.
47. Energy density is a critical factor in designing compact and efficient mechanical systems.
48. Understanding the work done per unit volume helps in material selection for springs and strings.
49. Exceeding the elastic limit of a spring or string results in permanent deformation, reducing energy efficiency.
50. Properly maintained springs and strings retain their elasticity and can store and release energy reliably over time.
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