Chapter 15: Oscillations

COURSE THEME: NEWTON’S LAWS OF MOTION!

• Chs. 5 - 13: Methods to analyze dynamics of objects in

Translational & Rotational Motion using Newton’s Laws! Chs. 5 & 6: Newton’s Laws using Forces (translational motion)

Chs. 7 & 8: Newton’s Laws using Energy & Work (translational motion)

Ch. 9: Newton’s Laws using Momentum (translational motion)

Chs. 10 & 11: Newton’s Laws (rotational language; rotating objects).

Ch. 14: Newton’s Laws for fluids (fluid language)

NOW

• Ch. 15: Methods to analyze the dynamics oscillating objects.

First, need to discuss (a small amount of) VIBRATIONAL LANGUAGE

Then, Newton’s Laws in Vibrational Language!

Simple Harmonic Motion• Vibration Oscillation = back & forth motion of an object

• Periodic motion: Vibration or Oscillation that regularly repeats itself. Motion of an object that regularly returns to a given position after a fixed time interval.

• A special kind of periodic motion occurs in mechanical systems when the force acting on the object is proportional to the position of the object relative to some equilibrium position– If the force is always directed toward the equilibrium position, the motion is

called simple harmonic motion

• Simplest form of periodic motion: Mass m attached to an ideal spring, spring constant k (Hooke’s “Law”: F = -kx) moving in one dimension (x).

Contains most features of more complicated systems

Use the mass-spring system as a prototype

for periodic oscillating systems

Spring-Mass System• Block of mass m attached to ideal spring of

constant k. Block moves on a frictionless horizontal surface

• When the spring is neither stretched nor compressed, block is at the

Equilibrium Position x = 0• Assume the force between the mass & the

spring obeys “Hooke’s Law” Hooke’s Law: Fs = - kx

x is the displacement

Fs is the restoring force. Is always directed toward the equilibrium position

• Therefore, it is always opposite the displacement from equilibrium

Possible Motion:

(a) Block displaced right of x = 0– Position x is positive (right)

– Restoring force Fs to the left

(b) Block is at x = 0– Spring is neither stretched nor

compressed

– Force Fs is 0

(c) Block is displaced left of x = 0– Position x is negative (left)

– Restoring force Fs to the right

Block Motion

Terminology & Notation• Displacement = x(t)

• Velocity = v(t)

• Acceleration = a(t) Constant!!!!!!!

Constant acceleration equations from Ch. 2

DO NOT APPLY!!!!!!!!!!!• Amplitude = A (= maximum displacement)

• One Cycle = One complete round trip

• Period = T = Time for one round trip• Frequency = f = 1/T = # Cycles per second Measured in Hertz (Hz)

A and T are independent!!!• Angular Frequency = ω = 2πf = Measured in radians/s.

• Repeat: Mass-spring system contains most features of more complicated systems Use mass-spring system as a prototype for periodic

oscillating systems.

• Results we get are valid for many systems besides this prototype system.

• ANY vibrating system with restoring force proportional to displacement (Fs = -kx) will exhibit simple harmonic motion (SHM) and thus is a simple harmonic oscillator (SHO).

(whether or not it is a mass-spring system!)

Acceleration• Force described by Hooke’s Law is the net force in

Newton’s 2nd Law

The acceleration is proportional

to the block’s displacement

• Direction of the acceleration

is opposite the displacement• An object moves with simple harmonic motion whenever

its acceleration is proportional to its position and is oppositely directed to the displacement from equilibrium

Hooke Newton

x

x

F F

kx ma

ka x

m

NOTE AGAIN! • The acceleration is not constant!

– So, the kinematic equations can’t be applied!

From previous slide: - kx = max – If block is released from position

x = A, initial acceleration is – kA/m

– When block passes through equilibrium position, a = 0

– Block continues to x = - A where acceleration is + kA/m

Block Motion• Block continues to oscillate between –A and +A

– These are turning points of the motion

• The force is conservative, so we can use the appropriate potential energy U = (½)kx2

• Absence of friction means the motion will continue forever!– Real systems are generally subject to friction, so

they do not actually oscillate forever

Simple Harmonic MotionMathematical Representation

• Block will undergo simple harmonic motion model• Oscillation is along the x axis• Acceleration

• Define:

• Then a = -2x. To find x(t), a function that satisfies this equation is needed– Need a function x(t) whose second derivative is the same as the original function with

a negative sign and multiplied by 2

– The sine and cosine functions meet these requirements

2

2

d x ka x

dt m

2 k

m

Simple Harmonic Motion Graphical Representation

• A solution to this is

x(t) = A cos (t + • A, are all constants• A cosine curve can be

used to give physical significance to these constants

2

2

d x ka x

dt m

Definitionsx(t) = A cos (t +

• A the amplitude of the motion– This is the maximum position of the particle in either the positive or negative

direction

is called the angular frequency– Units are rad/s

A & : Determined by position & velocity at t = 0If the particle is at x = A at t = 0, then = 0

• The phase of the motion is (t + )• x (t) is periodic and its value is the same each time t

increases by 2 radians

Period & Frequency• Period, T time interval required for mass to go

through one full cycle of its motion– Values of x & v for the mass at time t equal the values of x & v

at t + T

• Frequency, f Inverse of period. – Number of oscillations that mass m

undergoes per unit time interval

• Units: cycles per second = hertz (Hz)

2T

1ƒ

2T

Summary Period & Frequency• Frequency & period equations can be rewritten to solve for

Period & frequency can also be written:

• Frequency& period depend only on mass m of block & the spring constant k

• They don’t depend on the parameters of motion A, v

• The frequency is larger for a stiffer spring (large values of k) & decreases with increasing mass of the block

22 ƒ

T

1

2 ƒ2

m kT

k m

x(t), v(t), a(t) for Simple Harmonic Motion

• Simple harmonic motion is one-dimensional& so directions can be denoted by + or – sign

• Obviously, simple harmonic motion is not uniformly accelerated motion. CANNOT USE kinematic equations from earlier.

22

2

( ) cos ( )

sin( t )

cos( t )

x t A t

dxv A

dt

d xa A

dt

NOTE ONE MORE TIME! The acceleration is not constant!

That is, the 1 dimensional kinematic equations for constant acceleration (from Ch. 2) DO NOT APPLY!!!!!!

THESE ARE WRONG AND WILL GIVE YOU WRONG ANSWERS!!

f iv v at 1 2

2f i ix x v t at

2 2 2f iv v a x f ix x vt

TO EMPHASIZE THIS! THROW THESE AWAY FOR THIS CHAPTER!!

THESE ARE WRONG AND WILL GIVE YOU WRONG ANSWERS!!

f iv v at 1 2

2f i ix x v t at

2 2 2f iv v a x f ix x vt

Maximum Values of v & ax(t) = Acos(t +

v(t) = - Asin(t + a(t) = - Acos(t +

• Sine & cosine functions oscillate between 1, so we can easily find the maximum values of velocity and acceleration for an object in SHM

max

2max

kv A A

mk

a A Am

Graphs: General

• The graphs show (general case)(a) Displacement x(t) as a function of time

x(t) = Acos(t + (b) Velocity v(t) as a function of time

v(t) = - Asin(t +

(c ) Acceleration a(t) as as a function of time

a(t) = - Acos(t + • The velocity is 90o out of phase with the displacement

and the acceleration is 180o out of phase with the displacement

Example 1x(t) = Acos(t +

v(t) = - Asin(t + a(t) = - Acos(ωt + )

• Initial conditions at t = 0:

x(0) = A; v(0) = 0• This means = 0

x(t) = Acos(t v(t) = - Asin(t• Velocity reaches a maximum of A at x = 0a(t) = - Acos(t• Acceleration reaches a maximum

of 2A at x = A

x(t) = Acos(t + v(t) = - Asin(t +

a(t) = - Acos(ωt + )• Initial conditions at t = 0

x(0) = 0; v(0) = vi

• This means = /2 x(t) = Asin(t v(t) = Acos(ta(t) = - Asin(t• The graph is shifted one-quarter cycle to

the right compared to the graph of x (0) = A

Example 2