Waves and Modern Physics PHY 123 - Spring 2012

Prof. Antonio Badolato ���Office: BL 158

Email: badolato@pas.rochester.edu

Course Web Site   http://www.pas.rochester.edu/~badolato/PHY_123/PHY_123.html  

 Main Book:   D. C. Giancoli, Physics for Scientists and Engineers with Modern Physics, Vol. I, II, and III - (Prentice Hall) 4th edition.

Waves and Modern Physics, PHY 123 (Spring 2012)

Recitations:

Starting Today

Waves and Modern Physics, PHY 123 (Spring 2012)

Monday: 3.25 - 5.25 Imran (HYLAN 202), Tuesday: 3.25 - 5.25 Andrew (HYLAN 206),

4.50 - 7.50 Josh (B&L 270),

6:15 - 8.15 Andrew (B&L 269), Wednesday: 6.15 - 8.15 Andrew (B&L 315).

Figure 14.1

14-1 and 14-2

Oscillations and Simple Harmonic Motion

Figure 14.5

Figure 14.7

Figure 14.8

� = 0

x(t) = A cos(!t + �)

v =d

dt

x(t) = �!A sin(!t + �)

a =

d

2

dt

2x(t) = �!

2A cos(!t + �)

14-3 Energy in the SHM

E =12mv

2 +12kx

2

Example:

A 200 g block connected to a spring is free to oscillate on a frictionless horizontal surface. The spring constant is k = 5.00 N/m. The block is displaced 5.00 cm from the equilibrium and released from rest. Find

the displacement, x the period, T

the maximum speed, |vmax|.

Figure 14.12

The SHM has a simple relationship to a particle rotating in a circle with uniform speed.

⇢x(t) = A cos(!t + �)

y(t) = A sin(!t + �)

The projections of the uniform circular motion onto the x and y axes are SHMs.

14-4

Figure 14.14

14-5 The Simple Pendulum

! "!

Approximation of Trigonometric Functions

Using the theory of Taylor series we can show that the following identities hold for all real numbers x (where x are angles in radians)

sin x = (!1)n x2n+1

(2n +1)!n=0

"

# = x !x3

3!+x5

5!!x7

7!! ! " < x < "; n $"

cos x = (!1)n x2n

(2n)!n=0

"

# = 1! x2

2!+x4

4!!x6

6!! ! " < x < "; n $"

These identities are sometimes taken as the definitions of the sine and cosine function. They are often used as the starting point in a rigorous treatment of trigonometric functions and their applications, since the theory of infinite series can be developed from the foundations of the real number system, thus independent of any geometric considerations.

The sine function (blue) is closely approximated by its Taylor polynomial of degree 7 (pink) for a full cycle centered on the origin.

One of the most important applications of trigonometric series is for situations involving very small angles (x<<1). For such angles, the trigonometric functions can be

The sine function (blue) is closely approximated by its Taylor polynomial of degree 7 (pink) for a full cycle centered on the origin.

! #!

approximated by the first term in their series. This gives the useful small angle approximations:

x << 1 !sin x " xcos x " 1

#$%

&%

The approximation sin x = x reaches a 1% error at about 14°:

14! = !180

14! rad sin !180

14! " 0.2443sin !180

14! # 0.2443

sin !180

14!" 0.01

Examples of small angle approximation are in the calculation of

- the period of a simple pendulum,

- in most of the common expressions of geometrical optics that are built on the concept of paraxial approximation and surface power for lenses,

- the calculation of the intensity minima in single slit diffraction.

! "!

Approximation of Trigonometric Functions

Using the theory of Taylor series we can show that the following identities hold for all real numbers x (where x are angles in radians)

sin x = (!1)n x2n+1

(2n +1)!n=0

"

# = x !x3

3!+x5

5!!x7

7!! ! " < x < "; n $"

cos x = (!1)n x2n

(2n)!n=0

"

# = 1! x2

2!+x4

4!!x6

6!! ! " < x < "; n $"

These identities are sometimes taken as the definitions of the sine and cosine function. They are often used as the starting point in a rigorous treatment of trigonometric functions and their applications, since the theory of infinite series can be developed from the foundations of the real number system, thus independent of any geometric considerations.

The sine function (blue) is closely approximated by its Taylor polynomial of degree 7 (pink) for a full cycle centered on the origin.

One of the most important applications of trigonometric series is for situations involving very small angles (x<<1). For such angles, the trigonometric functions can be

As long as the cord can be considered massless and the amplitude is small, the period does not depend on the mass.

14-5 The Simple Pendulum

A physical pendulum is any real extended object that oscillates back and forth.

The torque about point O is:

Substituting into Newton’s second law gives:

14-6 The Physical Pendulum

Damped harmonic motion is harmonic motion with a frictional or drag force. If the damping is small, we can treat it as an “envelope” that modifies the undamped oscillation.

If

then

14-7 Damped Harmonic Motion

This gives

If b is small, a solution of the form (underdamped)

will work, with

14-7 Damped Harmonic Motion

If b2 > 4mk, ω’ becomes imaginary, and the system is overdamped (C).

For b2 = 4mk, the system is critically damped (B) —this is the case in which the system reaches equilibrium in the shortest time.

14-7 Damped Harmonic Motion

There are systems in which damping is unwanted, such as clocks and watches.

Then there are systems in which it is wanted, and often needs to be as close to critical damping as possible, such as automobile shock absorbers and earthquake protection for buildings.

14-7 Damped Harmonic Motion