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wariber [46]
3 years ago
6

A horizontal wire is stretched with a tension of 98.4 N, and the speed of transverse waves for the wire is 406 m/s. What must th

e amplitude of a traveling wave of frequency 60.5 Hz be for the average power carried by the wave to be 0.391 W? Give an answer in mm. Pay attention to the number of significant figures.
Physics
1 answer:
Anika [276]3 years ago
4 0

Answer:

Amplitude = 4.725 mm

Explanation:

We have got the following values :

T = 98.4 N

T is the wire tension

v = 406 m/s

v is the transverse waves speed

f = 60.5 Hz

f is the frequency

ΔU = 0.391 W

ΔU is the average power carried by the wave

If μ is  the mass per unit length of the wire ⇒ μ = \frac{T}{v^{2}}

If ω is the angular frequency of the wire ⇒ ω = 2π.f

So ω = 2π(60.5 Hz) and  ω units are s^{-1}

The equation that relates all is :

ΔU = (1/2).(T/v^2).(ω^2).(A^2).v

Where A is the wave amplitude

0.391 W=\frac{1}{2}.\frac{98.4 N}{(406\frac{m}{s})^{2}}.(2.pi.(60.5Hz))^{2}.A^{2}.406\frac{m}{s}

This will give us A in meters

A=4.7253(10^{-3})m\\A=4.725 mm

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a) 893 N

b) 8.5 m/s

c) 3816 W

d) 69780 J

e) 8030 W

Explanation:

a)

The net force acting on Bolt during the acceleration phase can be written using Newton's second law of motion:

F_{net}=ma

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m is Bolt's mass

a is the acceleration

In the first 0.890 s of motion, we have

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a=9.50 m/s^2 (acceleration)

So, the net force is

F_{net}=(94.0)(9.50)=893 N

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b)

Since Bolt's motion is a uniformly accelerated motion, we can find his final speed by using the following suvat equation:

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a=9.50 m/s^2 (acceleration)

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v=0+(9.50)(0.890)=8.5 m/s

c)

First of all, we can calculate the work done by Bolt to accelerate to a speed of

v = 8.5 m/s

According to the work-energy theorem, the work done is equal to the change in kinetic energy, so

W=K_f - K_i = \frac{1}{2}mv^2-0

where

m = 94.0 kg is Bolt's mass

v = 8.5 m/s is Bolt's final speed after the first phase

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where

t = 0.890 s is the time elapsed

Substituting,

P=\frac{3396}{0.890}=3816 W

d)

First of all, we need to find what is the average force exerted by Bolt during the remaining 8.69 s of motion.

In the first 0.890 s, the force exerted was

F_1=893 N

We know that the average force for the whole race is

F_{avg}=820 N

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F_{avg}=\frac{0.890 F_1 + 8.69 F_2}{0.890+8.69}

And solving for F_2, we find the average force exerted by Bolt on the ground during the second phase:

F_{avg}=\frac{0.890 F_1 + 8.69 F_2}{0.890+8.69}\\F_2=\frac{(0.890+8.69)F_{avg}-0.890F_1}{8.69}=812.5 N

The net force exerted by Bolt during the second phase can be written as

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F_{net}=ma

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a=\frac{v-u}{t} is the acceleration in the second phase, with

u = 8.5 m/s is the initial speed

v = 12.4 m/s is the final speed

t = 8.69 t is the time elapsed

Substituting,

a=\frac{12.4-8.5}{8.69}=0.45 m/s^2

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D=F_2-F_{net}=F_2-ma=812.5 - (94.0)(0.45)=770.2 N

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\Delta E=W=Ds

where

d is Bolt's displacement in the second part, which can be found by using suvat equation:

s=\frac{v^2-u^2}{2a}=\frac{12.4^2-8.5^2}{2(0.45)}=90.6 m

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\Delta E=Ds=(770.2)(90.6)=69780 J

e)

The power that Bolt must expend just to voercome the drag force is given by

P=\frac{\Delta E}{t}

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t is the time elapsed

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\Delta E=69780 J

t = 8.69 s is the time elapsed

Substituting,

P=\frac{69780}{8.69}=8030 W

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