Simple Harmonic Motion

Things that vibrate

§ 14.1–14.3

Hooke’s Law

• Force Law: F = –kx

• F = force exerted by the spring

• k = spring constant (characteristic of the particular spring)

• x = distance spring is displaced from equilibrium

Conditions of Motion

• Newton’s second law: F = ma

• Force F depends on position by Hooke’s law: F = –kx

–kx = ma

• tells how motion changes at each position

• second-order ordinary differential equation

–kx = md2xdt2

Think Question

The net force on a Hooke’s law object is

A. zero at the top and bottom

B. maximum at the top and bottom

C. minimum but not zero at the top and bottom

Think Question

The acceleration of a Hooke’s law object is

A. zero at the top and bottom

B. maximum at the top and bottom

C. minimum but not zero at the top and bottom

Think Question

The speed of a Hooke’s law object is

A. maximum at the equilibrium position

B. maximum at the top and bottom

C. maximum midway between equilibrium and top or bottom

Poll Question

All other things being equal, if the mass on a Hooke’s law oscillator increases, its period

A. decreases.

B. does not change.

C. increases.

Poll Question

All other things being equal, if the stiffness k of a Hooke’s law spring increases, its period

A. decreases.

B. does not change.

C. increases.

Uniform Circular Motion

• Centripetal force F = mv2/r inwards

• Constant magnitude F0; direction depends on position

F0

F0

F0

F0

F0

F0

• Force in y-direction is proportional to –y

Uniform Circular Motion

• Angle changes at a steady rate.

• Projection on y-axis has Hooke’s law force.

• So, projection on y-axis must have Hooke’s law motion too!

• What is the projection of an angle on the y-axis?

Board Work

Verify that

if x = A cos (t + ),

where• A and are any real constants

–kx = md2xdt2

• =   k/m

Equations of Motion

x(t) = A cos (t + ), • Amplitude A • Angular frequency • Initial phase angle • v = dx/dt• a = dv/dt = d2x/dt2

Another form

x(t) = A cos (t + ) = C cos (t) + S sin (t)

where

C = A cos () and S = –A sin ()

and

If C > 0, = arctan(–S/C)

If C < 0, = arctan(–S/C) +

A = C2 + S2

Period and Frequency

• Period T– time of one cycle (units: s)

• Frequency f– cycles per unit time (units: cycles/s = Hz)– f = 1/T

• Angular frequency – radians per unit time (units: 1/s or rad/s)– = 2f = 2/T– 2 =  k/m

Think Question

The potential energy of an oscillating mass is greatest

A. at its extreme positions.

B. at its equilibrium (middle) position.

C. between the middle and an extreme position.

Think Question

The kinetic energy of an oscillating mass is greatest

A. at its extreme positions.

B. at its equilibrium (middle) position.

C. between the middle and an extreme position.

Energy

• Potential energy of a stretched spring :

PE = kx212

• Conservation of energy:

(This of course ignores the sullen reality of energy dispersal by friction and drag. We’ll get to that.)

PE + KE = constant = kA212

where A is the oscillation amplitude.

Initial Conditions

Given m, k, x0, and v0, how does one find the equations of motion?• m and k give .

– 2 = k/m

• x0, v0, and give A.

– 1/2 kA2 = 1/2 kx02 + 1/2 mv0

2

Initial Conditions

Given m, k, x0, and v0, how does one find the equations of motion?

• x0/A and v0/(A) give .

– x0 = A cos()

– v0 = –A sin ()

– tan() = –v0/x0

– if x0 > 0, = arctan(–v0/x0)

– if x0 < 0, = + arctan(–v0/x0)

Effect of Gravity

• Less than you might expect:

• Changes equilibrium position x = 0

• Does not change k

Spring + Gravity

position

forc

e

0

0

spring alone

gravity

Spring + Gravity

position

forc

e

0

0

spring alone

gravity

0

spring + gravity

Spring + Gravity

position

net

forc

e

0

0

different equilibrium position

same k