Physics 153 Chapter 10 Notes PDF

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Simple Harmonic Motion and Elasticity (Vibrations): Chapter 10 (allsections)10 The Ideal Spring and Simple Harmonic Motion ● Experiment reveals that for relatively small displacements, the force F​x applied required to stretch or compress a spring is directly proportional to the displacement x,○■ Th...

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Simple Harmonic Motion and Elasticity (Vibrations): Chapter 10 (all sections) 10.1 The Ideal Spring and Simple Harmonic Motion ● Experiment reveals that for relatively small displacements, the force Fx applied required to stretch or compress a spring is directly proportional to the displacement x, ○ ■ The constant k is called the spring constant (units is N/m) ■ A spring that behaves like this is called an ideal spring ● To stretch or compress a rping, a force must be applied to it ○ In accord with Newton’s 3rd law, the spring exerts an oppositely directed force of equal magnitude ● Hooke’s Law* Restoring Force of an Ideal Spring ○ The restoring force of an ideal spring is ■ ● Where k is the spring constant and x is the displacement of the spring from its unstrained length. The minus sign indicates that the restoring force always points in a direction opposite to the displacement of the spring from its unstrained length. ■ Can contribute to the net external force ● When the restoring force has the mathematical form given by Fx =-kx, the type of friction-free motion is designated to be simple harmonic motion ● Amplitude A is the maximum excursion from equilibrium ● The restoring force also leads to simple harmonic motion when the object is attached to a vertical spring, just as it does when the spring is horizontal ○ When the spring is vertical, however, the weight of the object causes the spring to stretch, and the motion occurs with respect to the equilibrium position of the object on the stretched spring ■ The amount of initial stretching d0 due to the weight can be calculated by equating the weight to the magnitude of the restoring force that supports it ● mg=kd0 and thus d0=mg/k

10.2 Simple Harmonic Motion and the Reference Circle ● Simple harmonic motion can be described in terms of displacement, velocity, and acceleration ● A small ball attached to the top of a rotating turntable ○ The ball is moving in uniform circular motion, on a path known as the reference circle ■ As the ball moves, its shadow falls on a strip of film, which is moving upward at a steady rate recording where the shadow is ● Displacement ○ ○ For any object in simple harmonic motion, the time required to complete one cycle is the period T ■ Depends on the angular speed of the ball; for one cycle:

● ○ Frequency f is the number of cycles of the motion per second

■ ● One cycle per second is known as one hertz (Hz) ○ Relating angular speed to frequency

■ ○ Because w is directly proportional to the frequency, w is often called the angular frequency ● Velocity ○ ○

○ The velocity is not constant, but varies between maximum and minimum values as time passes ○ When the shadow passes through the x=0 m position, the velocity has a maximum magnitude of Aq, since the sine of an angle is between +1 and -1 ■ ● Acceleration ○ The velocity is not constant in simple harmonic motion ■ Thus there must be acceleration ● ■ Maximum magnitude of the acceleration ● ● Frequency of Vibration ○

○ 10.3 Energy and Simple Harmonic Motion ● Elastic potential energy is the potential energy when the spring is stretched or compressed ○ Because of the elastic potential energy, a stretched or compressed spring can do work on an object that is attached to the spring

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● Definition of Elastic Potential Energy ○ The elastic potential energy PEelastic is the energy that a spring has by virtue of being stretched or compressed. For an ideal spring that  nd is stretched or compressed by an has a spring constant k a amount x r elative to its unstrained length, the elastic potential energy is

■ ● Total mechanical energy E

○ ■ The total mechanical energy is conserved when external nonconservative forces do no net work; that is, when Wnc=0J. ● Then, Ef=E0 10.4 The Pendulum ● A simple pendulum consists of a particle of mass m , attached to a  nd negligible mass frictionless pivot P by a cable of length L a ● Gravity causes the back-and-forth rotation about the axis at P ○ The rotation speeds up as the particle approaches the lowest point and slows down on the upward part of the swing ○ Eventually the angular speed is reduced to zero, and the particle swings back ○ Gravitational force mg produces this torque ■ (The tension T in the cable creates no torque, because it points directly at the pivot P and therefore has a zero lever arm) ■ The minus sign is included since the torque is a restoring torque; it acts to reduce the angle theta

● The angle theta is positive (counterclockwise), while the torque is negative (clockwise) ■ Gravitational torque: ●

■ For small angles:

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● ● Physical pendulum is a rigid extended object 10.5 Damped Harmonic Motion ● In the presence of energy dissipation, the oscillation amplitude decreases as time passes, and the motion is no longer simple harmonic motion ○ This is damped harmonic motion ■ The decrease in amplitude being called “damping” ● The smallest degree of damping that completely eliminates the oscillations is termed “critical damping” ● Overdamped is when the damping exceeds critical value ● Underdamped is when the damping is less than critical level 10.6 Driven Harmonic Motion and Resonance ● Driven harmonic motion is when the additional force (driving force) drives or controls the behavior of the object to a large extent ● Natural frequency is the frequency at which the spring system naturally oscillates

○ ● Since the driving force and the velocity always have the same direction, positive work is done on the object at all times, and the total mechanical energy of the system increases ○ As a result, the amplitude of the vibration becomes larger and will increase without limit if there is no damping force to dissipate the energy being added by the driving force ■ This is known as resonance ● Resonance: Resonance is the condition in which a time-dependent force can transmit large amounts of energy to an oscillating object, leading to a large-amplitude motion. In the absence of damping, resonance occurs when the frequency of the force matches a natural frequency at which the object will oscillate. 10.7 Elastic Deformation ● Experiments have shown that the magnitude of the force can be expressed by the following relation, provided that the amount of stretch or compression is small compared to the original length of the object

○ ■ Delta L to L0 is parallel ● Shear deformation occurs because of the combined effect of the force F applied to the top of the book and the force F applied to the bottom of the book ○ In general, shearing forces cause a solid object to change its shapes

○ ■ Delta X to L0 are perpendicular

■ The constant of proportionality S is called shear modulus (units is N/m^2) ● Volume deformation and bulk modulus ○ When a compressive force is applied along one dimension of a solid, the length of that dimension decreases ○ It is also possible to apply compressive forces so that the size of every dimension (length, width, and depth) decreases, leading to a decrease in volume ■ The forces acting in such situations are applied perpendicular to every surface ● The magnitude of the perpendicular force per unit area is called the pressure P

○ ■ B i s the bulk modulus and units is N/m^2 ■ The minus sign occurs because an increase in pressure (positive delta P) always creates a decrease in volume (delta V is negative), and B is a positive quantity 10.8 Stress, Strain, and Hooke’s Law ● The ratio of the magnitude of the force to the area is called stress ● Strain results from stress ● As long as stress remains proportional to strain, a plot of stress versus strain is a straight line ○ The point on the graph where the material begins to deviate from the straight-line behavior is called the “proportionality limit” ■ Beyond this, stress and strain are no longer directly proportional ○ However, if the stress does not exceed the “elastic limit” of the material, the object will return to its original size and shape once the stress is removed ○ The “elastic limit” is the point beyond which the object no longer returns to its original size and shape when the stress is removed; the object remains permanently deformed

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