Download Chapter 13 Periodic Motion Simple Harmonic Motion Amplitude

CHAPTER 13
 Periodic Motion
 Simple Harmonic Motion
 Amplitude, Period, Frequency
 Energy Conservation in Oscillatory Motion
 Mass on Spring
 Pendulum
 Resonance
 Periodic motion – oscillation or vibration that repeats itself, back and forth, over the same path
 Mass on a spring, swinging pendulum, …
 Cause of oscillations: when an object is displaced from a position of stable equilibrium, it
experiences a restoring force (directed back toward equilibrium position)
 Restoring force is provided by intermolecular forces
 Periodic Motion Examples:
 An object oscillating on the end of a coil spring
 Mass m slides without friction on the horizontal surface, mass of the spring can be ignored
 Simple pendulum
 Amplitude, A – magnitude of the maximum displacement (x) of an object from its equilibrium position
 In the absence of friction, an object in simple harmonic motion will oscillate between ±A on each
side of the equilibrium position
 Period, T, - the time that it takes for the object to complete one full cycle of a periodic motion
 From x = A to x = - A and back to x = A
 Unit: s (seconds/cycle)
 Frequency, ƒ, - the number of complete cycles or oscillations per unit time
 Unit: Hz (Hertz)
 1 Hz = 1 cycle/second = 1/s
 ex1
If the processing speed of a personal computer is 1.80 GHz, how much time is required for one
processing cycle?
 ex2
A tennis ball is hit back and forth between two players warming up for a match. If it takes 2.31 s for the
ball to go from one player to the other, what are the period and the frequency of the ball’s motion?
7/31/2017
1
 Simple Harmonic Motion (SHM) is any vibrating system for which the restoring force F is directly
proportional to the negative of displacement x (Hooke’s Law type of force)
 The motion of a spring mass system is an example of SHM
 Hooke’s Law:
 Magnitude of the restoring force F is directly proportional to the displacement x
F=-kx
 F is the restoring force
 k is the spring constant
 It is a measure of the stiffness of the spring
 A large k indicates a stiff spring and a small k indicates a soft spring
 x is the displacement of the object from its equilibrium position (at equilibrium x = 0)
 The negative sign indicates that the force is always directed opposite to the displacement
 Mass Attached to a Spring:
 The force always acts toward the equilibrium position (restoring force F)
 A: When x = 0 (at equilibrium), F = 0
 B, D: When x is to the right, F is negative (to the left)
 C: When x is to the left, F is positive (to the right)
 Motion of the Spring-Mass System
 Let the object be initially pulled to x = A and released
 As it moves toward the equilibrium position, force and acceleration decrease, but velocity increases
 At x = 0, F =0 and a = 0, but v = max
 The object’s momentum causes it to overshoot the equilibrium position
 As it moves away from the equilibrium position, force and acceleration increase, and velocity decreases
 The motion continues indefinitely
 Force is not constant, acceleration is not constant – cannot use the equations for constant acceleration
 Acceleration of an Object in Simple Harmonic Motion
 Newton’s second law will relate force and acceleration (a = F/m)
 Force is given by Hooke’s Law (F = - k x)
a = - kx / m
Acceleration is a function of position
 In simple harmonic motion acceleration is directly proportional to the displacement and is in the
opposite direction
7/31/2017
2
 Sinusoidal Nature of SMH
 The spring mass system oscillates in simple harmonic motion
 The attached pen traces out the sinusoidal motion
 Connections Between Uniform Circular Motion and SHM
 A ball is attached to the rim of a turntable of radius A
 The focus is on the shadow that the ball casts on the screen
 When the turntable rotates with a constant angular speed, the shadow moves in simple harmonic
motion
 Connections Between Uniform Circular Motion and SHM
 SHM can be visualized as the projection of uniform circular motion onto x-axis
 Uniform Circular Motion:
radius A,
angular velocity ω
 SHM:
amplitude A,
angular frequency ω
ω = 2πf
 Motion as a Function of Time
 x = A cos (ωt)
 x - position at time t , varies between +A and –A
 A - amplitude
 ω = 2πf
 v = - v0 sin (ωt)
 v - velocity at time t
 v0 - maximum velocity (at x = 0)
 v0 = Aω
 a = - a0 cos (ωt)
 a - acceleration at time t
 a0 - maximum acceleration (acceleration at x = A)
 a0 = Aω2
 Graphical Representation of Motion
 When x is a maximum or minimum, velocity is zero
7/31/2017
3
 When x is zero, the velocity is a maximum
 When x is maximum in the positive direction, acceleration is maximum in the negative direction
 ex3
An air-track cart attached to a spring completes one oscillation every 2.4 s.
At t = 0 the cart is
released from rest at a distance of 0.10 m from its equilibrium position. What is the position of the cart
at (a) 0.30 s;
(b) 0.60 s?
What is the first time the cart is at the position x = -5.0 cm?
 ex4
An air-track cart attached to a spring completes one oscillation every 2.4 s.
At t = 0 the cart is
released from rest at a distance of 0.10 m from its equilibrium position. What are the velocity and
acceleration of the cart at (a) 0.30 s; (b) 0.60 s?
What is the first time the velocity of the cart is +26 cm/s?
 ex5
An airplane hit by a turbulence is moving up and down with an amplitude of 30.0 m and a maximum
acceleration of 1.8g. Treating the up-and-down movement as simple harmonic, find (a) the time
required for one complete oscillation and (b) the plane’s maximum vertical speed.
 ex6
A red delicious apple floats in a barrel of water. If you lift the apple 2.00 cm above its floating level and
release it, it bobs up and down with a period of 0.750 s. Assuming the motion simple harmonic, find the
position, velocity, and acceleration of the apple at the times (a) T/4, and (b) T/2.
The maximum kinetic energy of this apple is 0.00388 J. What is its mass?
 Angular Velocity, Frequency, Period of a Mass on a Spring
 Angular Velocity ω
 Frequency f
 Period T
 ex7
When a 0.22 kg air-track cart is attached to a spring, it oscillates with a period of 0.84 s.
What is the spring constant of the spring?
 ex8
A 0.120 kg mass attached to a spring oscillates with an amplitude of 0.0750 m and a maximum speed of
0.524 m/s. Find (a) the spring constant and (b) the period of the motion.
What is the maximum acceleration of the mass?
 ex9
7/31/2017
4
When a 0.420 kg mass is attached to a spring, it oscillates with a period of 0.350 s. If, instead, a different
mass m2 is attached to the same spring, it oscillates with a period of 0.700 s. Find the (a) spring constant
and (b) the mass m2.
The maximum speed of the second mass is 0.787 m/s. What is its amplitude of motion?
 ex10
A 0.260 kg mass is attached to a vertical spring. When the mass is put into motion, its period is 1.12 s.
How much does the mass stretch the spring when it is at rest in its equilibrium position?
 Energy in the Simple Harmonic Oscillator
 The energy stored in a stretched or compressed spring or other elastic material is called elastic potential
energy
 The energy is stored only when the spring is stretched or compressed
 The 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 object
 Example:
Energy Transformations, 1
 The block is moving on a frictionless surface
 The total mechanical energy of the system is the kinetic energy of the block
 Example:
Energy Transformations, 2
 The spring is partially compressed
 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
E = ½ mv2 + ½ k x2
 Example:
Energy Transformations, 3
 The spring is now fully compressed
 The block momentarily stops
 The total mechanical energy is stored as elastic potential energy of the spring
 Example:
Energy Transformations, 4
 When the block leaves the spring, the total mechanical energy is in the kinetic energy of the block
 ex11
A 0.980 kg block slides on a frictionless horizontal surface with a speed of 1.32 m/s. The block
encounters an un-stretched spring with spring constant of 245 N/m.
(a) How far is the spring compressed when the block comes to rest?
7/31/2017
5
(b) How long is the block in contact with the spring before it comes to rest?
 ex12
A bullet of mass m embeds itself in a block of mass M, which is attached to a spring of spring constant k.
If the initial speed of the bullet is vo, find (a) the maximum compression of the spring and (b) the time
for the bullet-block system to come to rest.
 Simple Pendulum
 Consists of a small object (pendulum bob) suspended from a cord (assume: cord does not stretch and its
mass can be ignored)
 Force is the component of the weight tangent to the path of motion
F = - m g sin θ
 In general, the motion of a pendulum is not simple harmonic
 For small angles, it becomes simple harmonic
 Angles < 15° are small enough sin θ = θ
 F = - m g θ = -mg(s/L)
 This force obeys Hooke’s Law
 Period of Simple Pendulum
 The period is independent of the amplitude and mass of the bob
 The period depends on the length of the pendulum and the acceleration of gravity at the location of the
pendulum
 ex13
The pendulum in a grandfather clock is designed to take one second to swing in each direction, that is
2.00 s for a complete cycle. Find the length of the pendulum.
 Simple Pendulum Compared to a Spring-Mass System
 Damped Oscillations
 Only ideal systems oscillate indefinitely, in real systems, friction retards the motion
 Friction reduces the total energy of the system and the oscillation is said to be damped
 Forced Vibrations and Resonance
 A system with an external force with its own frequency will force a vibration at that frequency
 When the frequency of the external (driving) force equals the natural frequency of the system, the
system is said to be in resonance
 An Example of Resonance
 Pendulum A is set in motion
 The others begin to vibrate due to the vibrations in the flexible beam
7/31/2017
6
 Pendulum C oscillates at the greatest amplitude since its length, and therefore frequency, matches that
of A
 Other Examples of Resonance
 Child being pushed on a swing
 Shattering glasses
 Tacoma Narrows Bridge collapse due to oscillations by the wind
7/31/2017
7