PHYSICS 1 Solution by DF Sozinando Mechanical Eng. student at the Vaal University of Technology

2- The displacement of a vibrating string versus position along the string is shown in the figure. The periodic waves havea speed of 10cm/s. A and B are two points on the string. The wave is m oving in the positive x direction.a) What is the amplitude of the wave?b) What is the wavelength of the wave?c) What is the frequency of the wave?d) What is the equation of this wave?

DATA:

v 10cms

:=

SOLUTION:

a) A 4cm=

b) λ 13.5cm 7.5cm- 6 cm=:=

c) fvλ

:= f 1.667 Hz=

d) y A sin 2 π f t2 π xλ

-

= calculator in radian mode

PHYSICS 1 Solution by DF Sozinando Mechanical Eng. student at the Vaal University of Technology

3- The periodic wave is produced on a stretched string. Which one of the following properties is not related to thespeed of the wave?

SOLUTION:

Amplitude

4- The speed of soud in a certain metal block is 3.00x103 m/s. The graph shows the amplitude (in meters) of a wavetraveling through the block versus time (in milliseconds). What is the wavelength of this wave?

DATA:

v 3 103ms

:=

SOLUTION:

T 2.5 10 3- s 0.5 10 3- s- 2 10 3- s=:=

λ v T:= λ 6 m=

PHYSICS 1 Solution by DF Sozinando Mechanical Eng. student at the Vaal University of Technology

10- A wave traveling at 5.5m/s on a string with a linear density of 0.082kg/m. What is the tension in the string?

DATA:

v 5.5ms

:= ρ 0.082kgm

:=

SOLUTION:

vFmL

= ρmL

=

vFρ

=solve F,

explicitρ v2

F ρ v2:= F 2.481 N=

11- The intensity of a spherical wave 4.0 m from the source is 120 W/m2. What is the intensity at a point 9.0 m awayfrom the source?

DATA:

r1 4m:= I1 120W

m2:= r2 9m:=

SOLUTION:

P I1 4 π r12= I2 4 π r2

2=

I1 4 π r12 I2 4 π r2

2=solve I2,

explicit

I1 r12

r22

I2I1 r1

2

r22

:= I2 23.704W

m2=

PHYSICS 1 Solution by DF Sozinando Mechanical Eng. student at the Vaal University of Technology

12- The decibel level of a jackhammer is 130 dB relative to the threshold of hearing. Determine the sound intensityproduced by the jackhammer.

dB 1

DATA:

β 130dB:= Io 1 10 12-W

m2:= constant value

SOLUTION:

β 10dB( ) logIIo

=solve I,

explicit10

β10 dB Io

I 10

β10 dB Io:= I 10

W

m2=

14- A wave causes a displacement y that is given in meters according to y=0.67sin(πt-5πx), where t and x areexpressed in seconds and meters, respectively. Determine the Amplitude, frequency, period, wavelength, and speedof the wave. Is the wave traviling in the positive or negative x-direction?

DATA:

y 0.67 sin π t 5 π x-( )=

SOLUTION:

y A sin ω t k x-( )=

A 0.67:= ω πrads

:= k 5 π m 1-:=

ω 2 π f=solve f,

explicitω

2 π

fω

2 π:= f

12

Hz=

PHYSICS 1 Solution by DF Sozinando Mechanical Eng. student at the Vaal University of Technology

T1f

:= T 2 s=

k2 πλ

=solve λ,

explicit2 π

k

λ2 π

k:= λ

25

m=

v λ f:= v 0.2ms

=

PHYSICS 1 Solution by DF Sozinando Mechanical Eng. student at the Vaal University of Technology

Conceptual questions:

3- Suppose that a longitudinal wave moves along a Slinky at a speed of 5m/s. Does one coil of the Slinky movethrough a distance of 5m in one second? Justify your answer.

SOLUTION:

No it will not. The coil will only move backwards and forwards a little bit, not 5 m unless it is a wave with a very largeamplitude. Typically, the energy will travel down the wave at 5 m/s, but the coil will not.

9- In a traveling sound wave, are there any particles that are always at rest as the wave passes by? Justify youranswer.

SOLUTION:

No. Sound waves in air are longitudinal waves. In a traveling longitudinal wave, each particle executes sinusoidal motionwith time in a direction parallel to the direction of propagation of the wave. See first source. Only in a standing wave arethere places where the particles of air are stationary.

PHYSICS 1 Solution by DF Sozinando Mechanical Eng. student at the Vaal University of Technology

Problems:1- Light is an electromagnetic wave and travels at a speed of 3.00x108m/s. The human eye is most sensitive toyellow-green light, which has a wavelength of 5.45x10-7m. What is the frequency of this light?

DATA:

v 3 108ms

:= λ 5.45 10 7- m:=

SOLUTION:

v λ f=solve f,

explicitvλ

fvλ

:= f 5.505 1014 Hz=

2- A woman is standing in the ocean, and she notices that after a wave crest passes, five more crests pass in a timeof 50.0s. The distance between two successive crests is 32m. Determine, if possible, the wave's.(a) period, (b) frequency, (c) wavelength, (d) speed, and (e) amplitude. If is not possble to determine any of thesequantities, then so state.

DATA:

k 5:= t 50s:= λ 32m:=

SOLUTION:

a( ) Ttk

:= T 10 s=

b( ) f1T

:= f 0.1 Hz=

c( ) λ 32m:= λ 32 m=

d( ) v λ f:= v 3.2ms

=

e( ) A "amplitude which is not possible":=

PHYSICS 1 Solution by DF Sozinando Mechanical Eng. student at the Vaal University of Technology

2- The speed of a transverse wave on a string is 450m/s, and the wavelength is 0.18m. The amplitude of the wave is2.0 mm. How much time is required for a particle of the string to move through a total distance of 1.0 km?

DATA:

v 450ms

:= λ 0.18m:= A 2mm:= Dist_string 1km:=

SOLUTION:

If the amplitude of the wave2.0mm during one cycle a pointon the string will move from crestto trough back to crest for a total

cycle_Dist 4 2 mm:= cycle_Dist 8 mm=

v λ f=solve f,

explicitvλ

fvλ

:= f 2500 Hz=

during each second a point on thestring will move

vstring cycle_Dist f:= vstring 20ms

=

Dist vstring t=solve t,

explicitDist

vstring

tDist_string

vstring:= t 5 101 s=

PHYSICS 1 Solution by DF Sozinando Mechanical Eng. student at the Vaal University of Technology

12- A wire is stretched between two posts. Another wire is stretched between two posts that are twice as far apart.The tension in the wires is the same, and they have the same mass. A transverse wave travels on the shorter wire aspeed of 240 m/s. What would be the speed of the wave on the longer wire?

DATA:

v1 240ms

:=

SOLUTION:

v1FmL1

=Fρ1

=

ρ1mL1

=

ρ2mL2

=

ρ1 ρ2=

m2 L1

mL2

=

12ρ1 ρ2=

v2FmL2

=Fρ2

=2 Fρ1

= 2Fρ1

= 2 v1=

v2 2 v1:= v2 339.411ms

=

PHYSICS 1 Solution by DF Sozinando Mechanical Eng. student at the Vaal University of Technology

13- The middle C string on pioano is under a tension of 944N. The period and wavelength of ware on this string are3.82 ms and 1.26 m, respectively. Find the linear density of the string.

DATA:

F 944N:= T 3.82 10 3- s:= λ 1.26m:=

SOLUTION:

vFmL

=Fρ

=

vλT

=

v v=

Fρ

λT

=solve ρ,

explicitF T2

λ2

ρF T2

λ2:= ρ 8.68 10 3-

kgm

=

PHYSICS 1 Solution by DF Sozinando Mechanical Eng. student at the Vaal University of Technology

14- The mass of string is 5.0x10-3 kg, and it is stretched so that the tension in it is 180N. A transverse ware travelingon this string has a frequency of 260Hz and a wavelegth of 0.60m. What is the length of the string?

DATA:

mo 5 10 3- kg:= F 180N:= f 260Hz:= λ 0.60m:=

SOLUTION:

vF

moL

=

v λ f=

v v=

FmoL

λ f=solve L,

explicit

λ2 f2 mo

F

Lλ2 f2 mo

F:= L 0.676 m=

PHYSICS 1 Solution by DF Sozinando Mechanical Eng. student at the Vaal University of Technology

15- Two wires are parallel, and one is directly above the other. Each has a length of 50.0 m and a mass per unitlength of 0.020kg/m. Hower, the tension in wire A is 6.00x102 N, and the tension in wire B is 3.00x102 N. Transversewave pulses are generated simultaneously,one at the left end of wire A and one at right end of wire B. The pulsestravel toward each other. How m uch time does it take until the pulses pass each other?

DATA:

λ 50m:= ρ 0.020kgm

:= FA 6 102N:= FB 3 102N:=

SOLUTION:

vFρ

=

vtλT

=solve T,

explicitλvt

Tλvt

=

Tλ

vA vB+=

Tλ

FAρ

FBρ

+

:= T 0.17 s=

PHYSICS 1 Solution by DF Sozinando Mechanical Eng. student at the Vaal University of Technology

22- The displacement (in meters) of a wave is given accord to y= 0.26 sin (π-3.7πx), where t is in seconds and x is inmeters.(a) Is the wave traveling in the +x or -x direction?(b) What is the displacement y when t= 38 s and x= 13 m?

DATA:

y 0.26 sin π 3 π x-( )=

A 0.26m:= ω πrads

:= k 3.7- π m 1-:=

SOLUTION:

a) y 0.26 sin π 3 π x-( )= wave travel in the +x direction

b) t 38s:= x 13m:=

ω 2 π f=solve f,

explicitω

2 π NB: y A sin ω t k x-( )= wave travel in the +x direction

y A sin ω t k x+( )= wave travel in the -x direction

fω

2 π:= f

12

Hz=

λ2 π sec

k:= λ 0.541- m s=

y x t, ( ) A sin k x ω t-( ):=

y 13m 38s, ( ) 0.08- m=

or

y A sin k x ω t-( ):= y 0.08- m=

PHYSICS 1 Solution by DF Sozinando Mechanical Eng. student at the Vaal University of Technology

23- A wave traveling in the +x direction has an amplitude of 0.35 m, a speed of 5.2m/s, and a frequency of 14 Hz. Writethe equation of the wave in form given by either Equation 16.3 or 16.4 .

DATA:

A 0.35m:= v 5.2ms

:= f 14Hz:=

SOLUTION:

y A sin ω t k x-( )=

ω 2 π f:= ω 87.965rads

=

λvf

:= λ 0.371 m=

k2 πλ

:= k 16.916 m 1-=

y 0.35m sin 87.965rads

t 16.916 m 1- x-

=

PHYSICS 1 Solution by DF Sozinando Mechanical Eng. student at the Vaal University of Technology

24- The drawing shows two graphs that represent a transverse wave on a string. The wave is moving in the x direc tion.Using the information contained in these graphs, write the mathematical expression (similar to Equation 16.3 or 16.4)for the wave.

SOLUTION:

y A sin ω t k x-( )=

From the graph

A 0.010m:= λ 0.040m:= T 0.20s:=

f1T

:= f 5 Hz=

ω 2 π f:= ω 10 πrads

=

k2 πλ

:= k 50 πm 1-=

y 0.010m sin 2 π f t 50 πm 1- x-( )=

PHYSICS 1 Solution by DF Sozinando Mechanical Eng. student at the Vaal University of Technology

58- A middle-aged man typically has poorer hearing than a middle-aged woman. In one case a woman can just beginto hear a musical tone, while a man can just begin to hear the tone only when its intensity level in increased by 7.8 dBrelative to that for the woman. What is the ratio of the sound intensity just detected by the man to that just detected bythe woman?

DATA:

β 7.8dB:=

SOLUTION:

Let the man hear Intensity I1 and the woman I2

β 10 dB logI1I2

=

The ratio of the sound intensity is the I1I2

β 10 dB log Ratio( )=solve Ratio,

explicit10

β10 dB

Ratio 10

β10 dB:= Ratio 6.026=

PHYSICS 1 Solution by DF Sozinando Mechanical Eng. student at the Vaal University of Technology

59- An amplif ied guitar has a sound intensity level that is 14 dBgreater than the same unam plified sound. What is theratio of theamplified intensity to the unamplified intensity?

DATA:

β 14dB:=

SOLUTION:

β 10 dB logI1I2

=

The ratio of the sound intensity is the I1I2

β 10 dB log Ratio( )=solve Ratio,

explicit10

β10 dB

Ratio 10

β10 dB:= Ratio 25.119=

PHYSICS 1 Solution by DF Sozinando Mechanical Eng. student at the Vaal University of Technology

61- When a person wears a hearing aid, the sound intensity level increases by 30.0 dB. By what factor does the soundintensity increase?

DATA:

β 30dB:=

SOLUTION:

β 10 dB logIIo

=solve I,

explicit10

β10 dB Io

β 10 dB log Ratio( )=solve Ratio,

explicit10

β10 dB

Ratio 10

β10 dB:= Ratio 1000=

PHYSICS 1 Solution by DF Sozinando Mechanical Eng. student at the Vaal University of Technology

Tutorial questions:

1- The thunder from a lightning flash is heard 8.0 s after the flash is seen. How far away is the lightning? Thespeed of sound is 343 m/s.

DATA:

T 8s:= v 343ms

:=

SOLUTION:

f1T

:= f18

Hz=

λ v T:= λ 2744 m=

2- A certain loudspeaker has a circular opening with an area of 30 cm2. Assume that the sound i t emits is uniformand through this entire opening. If the sound intensity at the opening is 2x10-4 W/m2, how much power is beingradiated as sound?

DATA:

A 30cm2:= I 2 10 4-W

m2:=

SOLUTION:

IPA

=solve P,

explicitA I

P A I:= P 6 10 7- W=

PHYSICS 1 Solution by DF Sozinando Mechanical Eng. student at the Vaal University of Technology

3- A beam of sound has an intensity of 3x106 W/m2. What is the intensity level in decibels?

DATA:

I 3 106W

m2:= Io 1 10 12-

W

m2:=

SOLUTION:

β 10dB logIIo

:= β 184.771 dB=

4- What is the intensity level in decibels for a sound that has an intensity of 0.70 W/m2 ?

DATA:

I 0.70W

m2:= Io 1 10 12-

W

m2:=

SOLUTION:

β 10dB logIIo

:= β 118.451 dB=

PHYSICS 1 Solution by DF Sozinando Mechanical Eng. student at the Vaal University of Technology

5- What is the sound intensity at a location where the sound level is 23dB ?

DATA:

β 23dB:= Io 1 10 12-W

m2:=

SOLUTION:

β 10dB logIIo

=solve I,

explicit10

β10 dB Io

I 10

β10 dB Io:= I 2 10 10-

W

m2=

PHYSICS 1 Solution by DF Sozinando Mechanical Eng. student at the Vaal University of Technology

6- A certain string has a linear mass density of 0.25 kg.m -1 and is stretched with a tension of 25 N. One end isgiven a sinusoidal motion with a frequency of 5Hz and amplitude 0.01 m. At time t=0 end has zero displacementand is moving in the +x-direction. (a) Find the wave speed, period and wavelength of this wave. (b) write a functiondescribing the wave (i.e. a mathematical description of the wave).

DATA:

ρ 0.25kgm

:= F 25N:= f 5Hz:= A 0.01m:=

SOLUTION:

vF

moL

=

vFρ

:= v 10ms

=

T1f

:= T 0.2 s=

λ v T:= λ 2 m=

y A sin ω t k x-( )=

ω 2π f:= ω 2 π5rads

=

k2πλ

:= k 2π2

m 1-=

y 0.01 sin 2π 5 t2π x

2-

=