Answer:
The answer to the question is
4433.416 kJ
See explanation below
(3-y)²+r² = 3² or
6y-y² = r²
r =√(6y-y²)
The volume of a small section of height Δy = Δy ×(√(6y-y²))²×π
For water with density of 1000 kg/m³, the mass of the slice
= 1000×Δy ×(√(6y-y²))²×π and since force = mass × acceleration we have
1000×Δy ×(√(6y-y²))²×π ×g = 1000×Δy ×(√(6y-y²))²×π ×9.81
The work done to move a unit height of y+1 = Force × Distance
W = 1000×Δy ×(√(6y-y²))²×π ×9.81 × (y+1)
Integrating the entire section of the sphere that is 2×r high, or from 0 to 6 we get
W =
= ![9810*\pi \int\limits^6_0 {5y^{2} -y^{3} +6y \, dy](https://tex.z-dn.net/?f=9810%2A%5Cpi%20%5Cint%5Climits%5E6_0%20%7B5y%5E%7B2%7D%20-y%5E%7B3%7D%20%20%2B6y%20%5C%2C%20dy)
![= 9810*\pi *[\frac{5y^{3} }{3} -\frac{y^{4} }{4} +3y^{2} ]^{6} _{0}](https://tex.z-dn.net/?f=%3D%209810%2A%5Cpi%20%2A%5B%5Cfrac%7B5y%5E%7B3%7D%20%7D%7B3%7D%20-%5Cfrac%7By%5E%7B4%7D%20%7D%7B4%7D%20%2B3y%5E%7B2%7D%20%5D%5E%7B6%7D%20_%7B0%7D)
=9810×π×144 =4433416 J
Answer:
a) The velocity of the car is 7.02 m/s and the car is approaching to the police car as the frequency of the police car is increasing.
b) The frequency is 1404.08 Hz
Explanation:
If the police car is a stationary source, the frequency is:
(eq. 1)
fs = frequency of police car = 1200 Hz
fa = frequency of moving car as listener
v = speed of sound of air
vc = speed of moving car
If the police car is a stationary observer, the frequency is:
(eq. 2)
Now,
fL = frequecy police car receives
fs = frequency police car as observer
a) The velocity of car is from eq. 2:
![1250=1200(\frac{v+v_{c} }{v-v_{c} } )\\1250(v-v_{c} )=1200(v+v_{c} )\\v_{c} =\frac{50*344}{2450} =7.02m/s](https://tex.z-dn.net/?f=1250%3D1200%28%5Cfrac%7Bv%2Bv_%7Bc%7D%20%7D%7Bv-v_%7Bc%7D%20%7D%20%29%5C%5C1250%28v-v_%7Bc%7D%20%29%3D1200%28v%2Bv_%7Bc%7D%20%29%5C%5Cv_%7Bc%7D%20%3D%5Cfrac%7B50%2A344%7D%7B2450%7D%20%3D7.02m%2Fs)
b) Substitute eq. 1 in eq. 2:
![f_{L} =(\frac{v+v_{p} }{v-v_{c} } )(\frac{v+v_{c} }{v-v_{p} } )f_{s} =(\frac{344+20}{344-7.02} )(\frac{344+7.02}{344-20} )*1200=1404.08Hz](https://tex.z-dn.net/?f=f_%7BL%7D%20%3D%28%5Cfrac%7Bv%2Bv_%7Bp%7D%20%7D%7Bv-v_%7Bc%7D%20%7D%20%29%28%5Cfrac%7Bv%2Bv_%7Bc%7D%20%7D%7Bv-v_%7Bp%7D%20%7D%20%29f_%7Bs%7D%20%3D%28%5Cfrac%7B344%2B20%7D%7B344-7.02%7D%20%29%28%5Cfrac%7B344%2B7.02%7D%7B344-20%7D%20%29%2A1200%3D1404.08Hz)
Answer:
Final temperature, ![T_f=21.85^{\circ}](https://tex.z-dn.net/?f=T_f%3D21.85%5E%7B%5Ccirc%7D)
Explanation:
Given that,
Mass of silver ring, m = 4 g
Initial temperature, ![T_i=41^{\circ}C](https://tex.z-dn.net/?f=T_i%3D41%5E%7B%5Ccirc%7DC)
Heat released, Q = -18 J (as heat is released)
Specific heat capacity of silver, ![c=0.235\ J/g\ C](https://tex.z-dn.net/?f=c%3D0.235%5C%20J%2Fg%5C%20C)
To find,
Final temperature
Solution,
The expression for the specific heat is given by :
![Q=mc\Delta T](https://tex.z-dn.net/?f=Q%3Dmc%5CDelta%20T)
![Q=mc(T_f-T_i)](https://tex.z-dn.net/?f=Q%3Dmc%28T_f-T_i%29)
![T_f=\dfrac{-Q}{mc}+T_i](https://tex.z-dn.net/?f=T_f%3D%5Cdfrac%7B-Q%7D%7Bmc%7D%2BT_i)
![T_f=\dfrac{-18}{4\times 0.235}+41](https://tex.z-dn.net/?f=T_f%3D%5Cdfrac%7B-18%7D%7B4%5Ctimes%200.235%7D%2B41)
![T_f=21.85^{\circ}](https://tex.z-dn.net/?f=T_f%3D21.85%5E%7B%5Ccirc%7D)
So, the final temperature of silver is 21.85 degrees Celsius.
infiltration and evaporation
Answer:![0.8\ mm](https://tex.z-dn.net/?f=0.8%5C%20mm)
Explanation:
Given
length of wire ![l=75\ cm](https://tex.z-dn.net/?f=l%3D75%5C%20cm)
change in length ![\Delta l=1.85\ mm](https://tex.z-dn.net/?f=%5CDelta%20l%3D1.85%5C%20mm)
mass of wire ![m=10\ kg](https://tex.z-dn.net/?f=m%3D10%5C%20kg)
Young's modulus for silver ![E=7.9\times 10^{10}\ N/m^2](https://tex.z-dn.net/?f=E%3D7.9%5Ctimes%2010%5E%7B10%7D%5C%20N%2Fm%5E2)
load on wire ![F=mg](https://tex.z-dn.net/?f=F%3Dmg)
![F=10\times 9.8=98\ kg](https://tex.z-dn.net/?f=F%3D10%5Ctimes%209.8%3D98%5C%20kg)
change in length is given by
![\Delta l=\dfrac{Pl}{AE}](https://tex.z-dn.net/?f=%5CDelta%20l%3D%5Cdfrac%7BPl%7D%7BAE%7D)
Where A=area of cross-section
![A=\dfrac{Pl}{\Delta lE}](https://tex.z-dn.net/?f=A%3D%5Cdfrac%7BPl%7D%7B%5CDelta%20lE%7D)
![A=\dfrac{98\times 0.75}{1.85\times 10^{-3}\times 7.9\times 10^{10}}](https://tex.z-dn.net/?f=A%3D%5Cdfrac%7B98%5Ctimes%200.75%7D%7B1.85%5Ctimes%2010%5E%7B-3%7D%5Ctimes%207.9%5Ctimes%2010%5E%7B10%7D%7D)
![A=\dfrac{73.5}{14.615\times 10^{7}}](https://tex.z-dn.net/?f=A%3D%5Cdfrac%7B73.5%7D%7B14.615%5Ctimes%2010%5E%7B7%7D%7D)
![A=5.029\times 10^{-7}\ m^2](https://tex.z-dn.net/?f=A%3D5.029%5Ctimes%2010%5E%7B-7%7D%5C%20m%5E2)
also wire is the shape of cylinder so cross-section is given by
![A=\dfrac{\pi d^2}{4}=5.029\times 10^{-7}\ m^2](https://tex.z-dn.net/?f=A%3D%5Cdfrac%7B%5Cpi%20d%5E2%7D%7B4%7D%3D5.029%5Ctimes%2010%5E%7B-7%7D%5C%20m%5E2)
![\Rightarrow d^2=\dfrac{5.029\times 10^{-7}\times 4}{\pi }](https://tex.z-dn.net/?f=%5CRightarrow%20d%5E2%3D%5Cdfrac%7B5.029%5Ctimes%2010%5E%7B-7%7D%5Ctimes%204%7D%7B%5Cpi%20%7D)
![\Rightarrow d^2=64.02\times 10^{-8}](https://tex.z-dn.net/?f=%5CRightarrow%20d%5E2%3D64.02%5Ctimes%2010%5E%7B-8%7D)
![\Rightarrow d=8\times 10^{-4}\ m](https://tex.z-dn.net/?f=%5CRightarrow%20d%3D8%5Ctimes%2010%5E%7B-4%7D%5C%20m)
![\Rightarrow d=0.8\ mm](https://tex.z-dn.net/?f=%5CRightarrow%20d%3D0.8%5C%20mm)