2054Lecture1 - Hooke's Law & Deformations Topics of...

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Unformatted text preview: Hooke's Law & Deformations Topics of Discussion Hooke's Law Simple Harmonic Motion Pendulums Elastic, Shear & Volume Deformations Elastic Potential Energy Hooke's Law Empirically, the magnitude of the restoring force F is directly proportional to the displacement x, that is F x. Hooke's Law says that the restoring force always opposes the displacement, and is given by F = - kx where x is the displacement of the spring of mass m and k is the spring constant in kg/s2. Simple Harmonic Motion Simple Harmonic Motion is an approximation to periodic motion when the net force obeys Hooke's Law. The period T and angular frequency for simple harmonic motion are given by the following equations. T = 2 m k = k m Pendulums The periodic motion of a pendulum is described by simple harmonic motion. The period T is given by T = 2 L g where L is the length of the pendulum and g is the gravitational acceleration. Three Types of Deformation Elastic Deformations Shear Deformations Bulk Deformations Elastic Deformations An elastic deformation is the stretching and compression produced by forces that act along the displacement of the material. Elastic deformations of a material obey Hooke's Law, and their motion can be approximated by simple harmonic motion. Empirically, the magnitude of the applied forces is directly proportional to the fractional increase in length, that is L F L0 . Young's Modulus A physical constant, called Young's modulus Y, which describes the elastic deformations of a material. The unit of Young's modulus is N/m2. Young's Law is given by L F= Y A L0 where F is the applied force, A is the crosssectional area, L is the change in the original length L0. Shear Deformations A shear deformation is the stretching and compression by the forces that DO NOT act along the displacement of the material. The amount of shear deformation of a material is described by the shear modulus S. The unit of shear modulus is N/m2. Young's Law for shear deformations is X F= S A L0 where F is the applied force, A is the crosssectional area, X is the amount of shear, L0 is the original length. Pressure The pressure P is the magnitude of a force F that acts perpendicular to the crosssectional area A, and is given by F P= A The unit of pressure is N/m2 or the pascal (Pa). Volume Deformations A volume deformation is the stretching and compression by the forces that change the volume of a material. The amount of volume deformation of a material is described by the bulk modulus B. The unit of bulk modulus is N/m2. Young's Law for volume deformations is V P= B V0 where F is the applied force, V is the change in volume of the material, V0 is the original volume. Stress & Strain Stress is the ratio of the magnitude of the applied force to the cross-sectional area, and has the units of pressure. Strain is the fractional change in the length, amount of shear or volume, and has NO units. Stress is proportional to Strain, that is F L X V , , A L0 L0 V0 . Elastic Potential Energy The energy stored during the compression or expansion of a spring or material is called elastic potential energy of the spring or material, and is given by 1 2 PE = kx . 2 ...
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