Elasticity

1. Elasticity •Hooke’s Law •Shear •Elasticity of Volume •Springs

2. Elasticity The property of a body by which it experiences a change in size or shape whenever a deforming force acts on the body. When the force is removed, the body returns to its original size and shape. Applies to all solids.

3. Atomic Nature of Solids Solids have lattice structure (3-D grid of atoms). Held together due to the interactions of their electrons.

4. Atomic Nature of Solids In equilibrium (no outside forces being applied), the net force of interactions is zero. The attractive and repulsive forces are balanced. The lattice is stable.

5. Atomic Nature of Solids If we stretch or compress the material, we move the atoms away from their equilibrium positions. The displacement is small but billions of atoms are displaced. This results in macroscopic lengthening or shortening. Once the force is removed, atoms return to equilibrium position and the material returns to its original length.

6. Elasticity – Factors Involved Force. Directly Proportional.

7. Elasticity – Factors Involved Cross-sectional area. The larger the area, the more atoms the force has to act over. Inversely proportional.

8. Elasticity – Factors Involved Original length. If there are twice as many atoms to shift, the change in length will be twice as much. Directly proportional.

9. Elasticity – Factors Involved Putting it all together:

10. Elasticity – Factors Involved Commonly written as: Ratio of applied force to cross-sectional area is call the STRESS on the wire. Ratio of the change in length to the original length of the wire is called the STRAIN. Stress is what is applied. Strain is the result.

11. Types of Stresses Tensile Stress: the bar, wire, pole, etc. is being stretched. Forces pull the ends of the object in opposite directions. Compressive Stress: the bar, wire, pole, etc. is being compressed. Forces act inwardly on the body.

12. Elasticity – Factors Involved To make the ratio into an equation, we must add a constant. Young’s modulus of elasticity (Y) is a measure of the stiffness of the material. The resulting equation:

13. Hooke’s Law This equation of elasticity is called Hooke’s law of elasticity. Equation states that stress is proportional to strain. Applying twice the force results in twice the stretch. (We touched on this with the pendulum!)

14. Elastic Limit We cannot stretch material indefinitely and stay in the elastic regime. Excess stress will permanently move atoms away from equilibrium. The material will experience some permanent stretching.

15. Elastic Limit Elastic limit: permanent stretching, object will not return to original length, stress and strain no longer proportional Ultimate stress: highest point on stress-strain curve, greatest stress the material can bear, break shortly after Breaking point

16. Example #1 – Hooke’s Law A steel wire 1.00 m long with a diameter d = 1.00 mm has a 10.0-kg mass hung from it. (Young’s modulus for steel is 21 x 1010 N/m2.) (a) How much will the wire stretch? (b) What is the stress on the wire? (c) What is the strain?

17. Types of Stresses Tensile Stress: the bar, wire, pole, etc. is being stretched. Forces pull the ends of the object in opposite directions. Compressive Stress: the bar, wire, pole, etc. is being compressed. Forces act inwardly on the body. Shear Stress: the object has equal and opposite forces applied across its opposite faces (parallel to the surfaces).

18. Elastic Deformation of Shape - Shear In addition to changing of length, elasticity applies to changing of shape. Shear stress is the application of equal and opposite forces parallel to the face of the object. Force results in torque but body is not free to rotate. Force applied to the body causes the layers of atoms to be displaced sideways. Body changes shape (shear).

19. Elastic Deformation of Shape - Shear Angle of shear ( ): a measure of the deformation of the body (radians!). Shear modulus measures resistance to shear. Hooke’s law for elastic deformation of shape: Tangential force divided by area of the top of the shape equals the shear modulus times the angle of shear (in radians).

20. Types of Stresses Tensile Stress: the bar, wire, pole, etc. is being stretched. Forces pull the ends of the object in opposite directions. Compressive Stress: the bar, wire, pole, etc. is being compressed. Forces act inwardly on the body. Shear Stress: the object has equal and opposite forces applied across its opposite faces (parallel to the surfaces). Pressure: uniform force is exerted on all sides of an object.

21. Elasticity of Volume Another variety of elastic deformation is elasticity of volume. Uniform force is exerted on all sides of an object. Example: object is submerged in water. Volume of object decreases. Hooke’s law for elasticity of volume includes the bulk modulus.

22. Table of Constants

23. Example #2 – Elasticity of Volume A solid copper sphere of 0.500-m3 volume is placed 100 feet below the ocean surface where the pressure is 3.00 x 105 N/m2. What is the change in volume of the sphere? (The bulk modulus of copper is 14 x 1010 N/m2.)

24. Types of Stresses

25. Springs Springs display the properties of Hooke’s law of elasticity, as well. Springs are considered massless and frictionless. Springs (like the pendulum) are examples of simple harmonic oscillators!

26. Springs as SHOs Equilibrium position: position of object if no force has been applied to the system. Displacement: distance object is pulled from equilibrium. Amplitude: maximum displacement. Restoring force: force from spring on object which tries to bring object back to equilibrium position.

27. Hooke’s Law for Springs F = -kx. k is the spring constant in units of [N/m]. It is a measure of the stiffness of the spring. x is the displacement in units of [m]. F is the restoring force of the spring on the object in units of [N]. Proportional to the displacement. Opposite direction of the displacement. The force you must apply to stretch the spring is then F = kx.

28. Natural Frequency If you stretch a spring to any amplitude (within the elastic regime), it will oscillate with its natural frequency. Angular frequency: Frequency/period: Like all SHOs, we want to be able to describe the motion of the object using a sinusoidal wave:

29. Springs and Work Your hand must do work to compress or stretch a spring. In the past, work = force x distance. Problem: our force is not constant! It varies with distance. We cannot use our old work equation.

30. Springs and Work The answer to this problem is calculus! ? In reality, work is the integral of force with respect to displacement.

31. Springs and Work First, let’s look at the situation where the force is constant. The force in this situation can be represented by a constant value F.

32. Springs and Work Now the situation of the spring where the force is not constant. In the case of the spring, F = kx.

33. Springs and Work For the spring, the work your hand does to displace the object is . Also, remember that the integral of a function gives the area under the curve. Using simple geometry, the area under the constant force curve is force x distance. Using simple geometry, the area under the F = kx force curve is (½)(base)(height) = (½)(x)(F) = (½)(x)(kx) = ½kx2. With this knowledge, we could find the work done by any force over any distance.

34. Springs and Potential Energy If we graph the work function, we end up with a parabola representing the energy of the spring as it oscillates. This function is typical of all simple harmonic oscillators. Amplitude give total energy of the system (all potential energy). System oscillates between kinetic and potential energy with the total energy remaining the same.

35. Example #3 - Springs A ball of mass m = 2.60 kg, starting from rest, falls a vertical distance h = 55.0 cm before striking a vertical coiled spring. The ball compresses the spring y = 15.0 cm. Determine the spring constant of the spring.