Chapter 16 Vibrations Motion

Vibrations/Oscillations Object at the end of a spring Object at the end of a spring Tuning fork Tuning fork Pendulum Pendulum String of a violin String of a violin Atoms in a crystal Atoms in a crystal Source of a wave Source of a wave

Periodic Motion Motion repeats itself over and over again Motion repeats itself over and over again Examples of periodic motion Examples of periodic motion

Period and Frequency The period, T, is the time that it takes for the object to complete one complete cycle of motion The period, T, is the time that it takes for the object to complete one complete cycle of motion From x = A to x = - A and back to x = AFrom x = A to x = - A and back to x = A The frequency, ƒ, is the number of complete cycles or vibrations per unit time The frequency, ƒ, is the number of complete cycles or vibrations per unit time ƒ = 1 / Tƒ = 1 / T Frequency is the reciprocal of the periodFrequency is the reciprocal of the period Unit: Hertz, 1 Hz = 1 cycle/sUnit: Hertz, 1 Hz = 1 cycle/s

Hooke’s Law F = k x F = k x F is the stretch forceF is the stretch force k is the spring constantk is the spring constant It is a measure of the stiffness of the spring It is a measure of the stiffness of the spring A large k indicates a stiff spring and a small k indicates a soft springA large k indicates a stiff spring and a small k indicates a soft spring x is the displacement of the object from its equilibrium positionx is the displacement of the object from its equilibrium position x = 0 at the equilibrium position x = 0 at the equilibrium position Force exerts by the spring F sForce exerts by the spring F s

Hooke’s Law Force The force F s always acts toward the equilibrium position The force F s always acts toward the equilibrium position It is called the restoring forceIt is called the restoring force The direction of the restoring force is such that the object is being either pushed or pulled toward the equilibrium position The direction of the restoring force is such that the object is being either pushed or pulled toward the equilibrium position

Hooke’s Law Applied to a Spring – Mass System When x is positive (to the right), F s is negative (to the left) When x is positive (to the right), F s is negative (to the left) When x = 0 (at equilibrium), F s is 0 When x = 0 (at equilibrium), F s is 0 When x is negative (to the left), F s is positive (to the right) When x is negative (to the left), F s is positive (to the right)

Motion of the Spring-Mass System Assume the object is initially pulled to a distance A and released from rest Assume the object is initially pulled to a distance A and released from rest As the object moves toward the equilibrium position, F s and a decrease, but v increases As the object moves toward the equilibrium position, F s and a decrease, but v increases At x = 0, F s and a are zero, but v is a maximum At x = 0, F s and a are zero, but v is a maximum The object’s momentum causes it to overshoot the equilibrium position The object’s momentum causes it to overshoot the equilibrium position

Motion of the Spring-Mass System, cont The force and acceleration start to increase in the opposite direction and velocity decreases The force and acceleration start to increase in the opposite direction and velocity decreases The motion momentarily comes to a stop at x = - A The motion momentarily comes to a stop at x = - A It then accelerates back toward the equilibrium position It then accelerates back toward the equilibrium position The motion continues indefinitely The motion continues indefinitely

Simple Harmonic Motion Motion that occurs when the net force along the direction of motion obeys Hooke’s Law Motion that occurs when the net force along the direction of motion obeys Hooke’s Law The force is proportional to the displacement and always directed toward the equilibrium positionThe force is proportional to the displacement and always directed toward the equilibrium position The motion of a spring mass system is an example of Simple Harmonic Motion The motion of a spring mass system is an example of Simple Harmonic Motion

Simple Harmonic Motion, cont. Not all periodic motion over the same path can be considered Simple Harmonic motion Not all periodic motion over the same path can be considered Simple Harmonic motion To be Simple Harmonic motion, the force needs to obey Hooke’s Law To be Simple Harmonic motion, the force needs to obey Hooke’s Law

Amplitude Amplitude, A Amplitude, A The amplitude is the maximum position of the object relative to the equilibrium positionThe amplitude is the maximum position of the object relative to the equilibrium position In the absence of friction, an object in simple harmonic motion will oscillate between the positions x = ±AIn the absence of friction, an object in simple harmonic motion will oscillate between the positions x = ±A

Period and Frequency The period, T, is the time that it takes for the object to complete one complete cycle of motion The period, T, is the time that it takes for the object to complete one complete cycle of motion From x = A to x = - A and back to x = AFrom x = A to x = - A and back to x = A The frequency, ƒ, is the number of complete cycles or vibrations per unit time The frequency, ƒ, is the number of complete cycles or vibrations per unit time ƒ = 1 / Tƒ = 1 / T Frequency is the reciprocal of the periodFrequency is the reciprocal of the period Unit: Hertz, 1 Hz = 1 cycle/sUnit: Hertz, 1 Hz = 1 cycle/s

Acceleration of an Object in Simple Harmonic Motion Newton’s second law will relate force and acceleration Newton’s second law will relate force and acceleration The force is given by Hooke’s Law The force is given by Hooke’s Law F s = - k x = m a F s = - k x = m a a = -kx / m !!!a = -kx / m !!! The acceleration is a function of position The acceleration is a function of position Acceleration is not constant and therefore the uniformly accelerated motion equation cannot be appliedAcceleration is not constant and therefore the uniformly accelerated motion equation cannot be applied

Elastic Potential Energy A compressed spring has potential energy A compressed spring has potential energy The compressed spring, when allowed to expand, can apply a force to an objectThe compressed spring, when allowed to expand, can apply a force to an object The potential energy of the spring can be transformed into kinetic energy of the objectThe potential energy of the spring can be transformed into kinetic energy of the object

Elastic Potential Energy, cont The energy stored in a stretched or compressed spring or other elastic material is called elastic potential energy The energy stored in a stretched or compressed spring or other elastic material is called elastic potential energy E p,sp = ½kx 2E p,sp = ½kx 2 The energy is stored only when the spring is stretched or compressed The energy is stored only when the spring is stretched or compressed Elastic potential energy can be added to the statements of Conservation of Energy and Work-Energy Elastic potential energy can be added to the statements of Conservation of Energy and Work-Energy

Energy in a Spring Mass System A block sliding on a frictionless system collides with a light spring A block sliding on a frictionless system collides with a light spring The block attaches to the spring The block attaches to the spring The system oscillates in Simple Harmonic Motion The system oscillates in Simple Harmonic Motion

Energy Transformations The block is moving on a frictionless surface The block is moving on a frictionless surface The total mechanical energy of the system is the kinetic energy of the block The total mechanical energy of the system is the kinetic energy of the block

Energy Transformations, 2 The spring is partially compressed The spring is partially compressed The energy is shared between kinetic energy and elastic potential energy The energy is shared between kinetic energy and elastic potential energy The total mechanical energy is the sum of the kinetic energy and the elastic potential energy The total mechanical energy is the sum of the kinetic energy and the elastic potential energy

Energy Transformations, 3 The spring is now fully compressed The spring is now fully compressed The block momentarily stops The block momentarily stops The total mechanical energy is stored as elastic potential energy of the spring The total mechanical energy is stored as elastic potential energy of the spring

Energy Transformations, 4 When the block leaves the spring, the total mechanical energy is in the kinetic energy of the block When the block leaves the spring, the total mechanical energy is in the kinetic energy of the block The spring force is conservative and the total energy of the system remains constant The spring force is conservative and the total energy of the system remains constant

Energy Conservation

Graphical Representation of Motion When x is a maximum or minimum, velocity is zero When x is a maximum or minimum, velocity is zero When x is zero, the velocity is a maximum When x is zero, the velocity is a maximum When x is a maximum in the positive direction, a is a maximum in the negative direction When x is a maximum in the positive direction, a is a maximum in the negative direction

Period of a SHM It can be shown that It can be shown that independent of amplitude A independent of amplitude A Valid for vertical spring-mass system Valid for vertical spring-mass system

Example A body with a mass of 5.0 kg is suspended by a spring, which stretches 10 cm when the body is attached. The body is then pulled downward an additional 5 cm and released. Find k, T, f, E, v max and a max.

Simple Pendulum The simple pendulum is another example of simple harmonic motion The simple pendulum is another example of simple harmonic motion The force is the component of the weight tangent to the path of motion The force is the component of the weight tangent to the path of motion F t = - m g sin θF t = - m g sin θ

Simple Pendulum, cont In general, the motion of a pendulum is not simple harmonic In general, the motion of a pendulum is not simple harmonic However, for small angles, it becomes simple harmonic However, for small angles, it becomes simple harmonic In general, angles < 15° are small enoughIn general, angles < 15° are small enough

Period of Simple Pendulum This shows that the period is independent of the amplitude This shows that the period is independent of the amplitude The period depends on the length of the pendulum and the acceleration of gravity at the location of the pendulum The period depends on the length of the pendulum and the acceleration of gravity at the location of the pendulum Measure g Measure g

Example Period of a simple pendulum is 2s on Earth. What would be the period of the same pendulum on the moon? (g moon =1.67 m/s^2)

Forced Vibrations and Resonance Shaking (vibration) as specific frequency Shaking (vibration) as specific frequency Pushing child on swing Pushing child on swing glass glass Tacoma Narrows Bridge Tacoma Narrows Bridge ! Every object has its natural frequencies ! Every object has its natural frequencies

SHM

More on SHM Displacement Displacement Velocity Velocity Acc. Acc. Units Units

Example A 30-g mass undergoes simple harmonic motion at the end of a spring. The period of motion is 2.1 s. the mass oscillates back and forth through a total distance of 20 cm. Find the maximum speed and acceleration of the mass. A 30-g mass undergoes simple harmonic motion at the end of a spring. The period of motion is 2.1 s. the mass oscillates back and forth through a total distance of 20 cm. Find the maximum speed and acceleration of the mass.

Example, continue If the mass start in the equilibrium position at t=0. What are the displacement, velocity and acc. At t= s?