Unformatted text preview: ources can be found from
P
(Equation 16.9), where P is the power output of a source and r is the distance
I=
4π r 2
between the source and point A. The sound intensities are equal at point A, so we have that I1 = I 2 = P
1
4π r12 = P2
4π r2 2 (1) Additionally, we know that the power P1 emitted by source 1 is four times greater than the
power P2 emitted by source 2: P = 4 P2
1 (2) SOLUTION In Equation (1), r1 = x, and r2 = d – x when A is taken as the point of equal
sound intensity. Alternatively, we could say that x is the position of point B relative to the
origin (source 1), so that the distance r2 between source 2 and point B would be r2 = x – d. It Chapter 16 Problems 869 does not make a difference which we choose, since r2 is squared in Equation (1), and
r22 = ( d − x ) = ( x − d ) . In fact, because Equation (1) is quadratic, either choice will yield
two values of x, one for position A and one for position B. Substituting Equation (2), r1 = x,
and r2 = d − x into Equation (1) yields
2 2 4 P2
4π x 2 = P2
4π ( d − x ) 4
1
=
2
x
( d − x )2 or 2 (2) Crossmultiplying Equation (2) and taking the square root of both sides, we obtain x2 = 4 (d − x) 2 or x = ±2 ( d − x ) (3) Solving Equation (3) separately for the positive root and the negative root gives
Positive root
x = +2 ( d − x ) = 2 d − 2 x x = −2 ( d − x ) = −2d + 2 x 3 x = 2d − x = −2 d x = 2 d 3 = 2(123 m) 3 = 82.0 m 61. Negative root x = 2 d = 2(123 m) = 246 m P
, according to
A
Equation 16.8. The area is given directly, but the power is not. Therefore, we need to recast
this expression in terms of the data given in the problem. Power is the change in energy per
unit time, according to Equation 6.10b. In this case the energy is the heat Q that causes the
Q
temperature of the lasagna to increase. Thus, the power is P = , where t denotes the time.
t
As a result, Equation 16.8 for the intensity becomes
SSM REASONING Intensity I is power P divided by the area A, or I = I= PQ
=
A tA (1) According to Equation 12.4, the heat that must be supplied to increase the temperature of a
substance of mass m by an amount ∆T is Q = cm∆T, where c is the specific heat capacity.
Substituting this expression into Equation (1) gives
I= Q cm∆T
=
tA
tA (2) 870 WAVES AND SOUND SOLUTION Equation (2) reveals that the intensity of the microwaves is
I= cm∆T 3200 J/ ( kg ⋅ C° ) ( 0.35 kg ) ( 72 C° ) =
= 7.6 × 103 W/m 2
−2
2
tA
( 480 s ) 2.2 × 10 m ( ) ______________________________________________________________________________
62. REASONING Since the sound is emitted from the rocket uniformly in all directions, the
energy carried by the sound wave spreads out uniformly over concentric spheres of
increasing radii r1, r2, r3, . . . as the wave propagates. Let r1 represent the radius when the
measured intensity at the ground is I (position 1 for the rocket) and r2 represent the radius
when the measured intensity at the ground is 1
3 I (position 2 of the rocket). The time required for the energy to spread out over the sphere of radius r1 is t1 = r1 / vs , where vs is
the speed of sound. Similarly, the time required for the energy to spread out over a sphere
of radius r2 is t2 = r2 / vs . As the sound wave emitted at position 1 spreads out uniformly,
the rocket continues to accelerate upward to position 2 with acceleration ay for a time t12.
Therefore, the time that elapses between the two intensity measurements is
t = t2 − t1 + t12 (1) From Equation 3.3b, the time t12 is
t12 = ( v2 − v1 )
ay (2) where v1 and v2 are the speeds of the rocket at positions 1 and 2, respectively. Combining
Equations (1) and (2), we obtain
t = t2 − t1 + ( v2 − v1 )
ay (3) 2
The respective values of v1 and v2 can be found from Equation 3.6b ( v 2 = v0 y + 2a y y ) with
y v0 y = 0 m/s, and y = r . Once values for v1 and v2 are known, Equation (3) can be solved
directly.
SOLUTION From the data given in the problem statement, r1 = 562 m . We can find r2 using the following reasoning: from the definition of intensity, I1 = P /(4π r12 ) and I 2 = P /(4π r22 ) . Since I1 = 3I2 , we have Chapter 16 Problems P /(4π r12 ) = 3P /(4π r22 ) or 871 r2 = r1 3 = 973 m The times t1 and t2 are t1 = r1
562 m
=
= 1.64 s
vs 343 m/s t2 = r2 and
vs = 973 m
= 2.84 s
343 m/s Then, taking up as the positive direction, we have from Equation 3.6b,
v1 = 2a y r1 = 2(58.0 m/s 2 )(562 m) = 255 m/s
and
v2 = 2a y r2 = 2(58.0 m/s 2 )(973 m) = 336 m/s
Substituting these values into Equation (3), we have
t = 2.84 s − 1.64 s + ( 336 m/s − 255 m/s ) = 2.6 s
58.0 m/s 2
______________________________________________________________________________
63. REASONING AND SOLUTION The sound intensity level β in decibels (dB) is related to
the sound intensity I according to Equation 16.10, β = (10 dB) log ( I / I 0 ) , where I0 is the
reference intensity. If β1 and β2 represent two different sound level intensities, then, I2 I I2 / I0 I = 10 dB ) log 2 − (10 dB ) log 1 = (10 dB ) log I...
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 Spring '13
 CHASTAIN
 Physics, Pythagorean Theorem, Force, The Lottery, Right triangle, triangle

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