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Question: Fill in the blanks A.)\(1\;kgm^2/s^2=……….gcm^2/s^2\) B.)1m=………. ly C.)\(3.0\;ms^{-2}=……….kmh^{...

Fill in the blanks
A.)1  kgm2/s2=……….gcm2/s21\;kgm^2/s^2=……….gcm^2/s^2
B.)1m=………. ly
C.)3.0  ms−2=……….kmh−23.0\;ms^{-2}=……….kmh^{-2}
D.)G=6.67×10−11  Nm2(kg)−2=……….(cm)3s−2g−1G=6.67\times10^{-11}\;Nm^2(kg)^{-2}=……….(cm)^{3}s^{-2}g^{-1}

Explanation

Solution

Hint: The given values are in SI units and we need to change them with the respective units by applying unit conversions for different properties viz. length, mass, force and time.

Complete step by step solution
In this question, we need to convert kilograms into grams, meters into centimeters while the unit of seconds will not change.
We know that 1kg = 1000 g and 1 m = 100 cm. Using this, we can write the given value 1  kgm2/s2=(1×1000)g×(1×100)2  cms21\;kgm^2/s^2=\dfrac{(1\times1000)g\times(1\times100)^2\;cm}{s^2}
  ⟹  1  kgm2/s2=(1000)g×(100)2cms2=103×104s2  gcm2/s2=107  gcm2/s2\implies1\;kgm^2/s^2=\dfrac{(1000)g\times(100)^{2}cm}{s^2}=\dfrac{10^3\times10^4}{s^2}\;gcm^2/s^2=10^7\; gcm^2/s^2
Hence, 1  kgm2/s2=107gcm2/s21\;kgm^2/s^2=10^7gcm^2/s^2
In this question, we need to convert meters in light years.

We know that Distance=Speed×Time…….(i)Distance=Speed\times Time…….(i)

As we know that one light year is the distance travelled by light in one year. Thus, with speed 3×108  m/s3\times10^8\;m/s and in one year time, that is in (365×24×60×60)(365\times24\times60\times60) seconds.

Upon incorporating the two values in equation (i)(i), we will get the 1 light year distance as, 1ly=(3×108)  m/s×(365×24×60×60)=9.46×1015  m1 ly=(3\times10^8)\;m/s\times(365\times24\times60\times60)=9.46\times10^{15}\;m
Therefore, 1  ly=9.46×1015  ly1\;ly=9.46\times10^{15}\;ly

Now, we can write that 1  m=19.46×1015  ly=1.06×10−16  ly1\;m=\dfrac{1}{9.46\times10^{15}}\;ly=1.06\times10^{-16}\;ly
Hence, 1  m=1.06×10−16  ly1\;m=1.06\times10^{-16}\;ly

In this question, the SI unit of acceleration has been given that is in meters and seconds and we need to convert them in kilometers and hours.

So, we know that 1 km = 1000 m and 1 hour = 3600 seconds. Using these, we can write that 3.0  ms−2=3m1s2=(3/1000)  km(1/3600)2  h2=3.88×104  kmh−23.0\;ms^{-2}=\dfrac{3m}{1s^{2}}=\dfrac{(3/1000)\;km}{(1/3600)^{2}\;h^2}=3.88\times10^4\;kmh^{-2}
Hence, 3.0  ms−2=3.88×104  kmh−23.0\;ms^{-2}=3.88\times10^4\;kmh^{-2}

We have been given G=6.67×10−11  Nm2(kg)−2G=6.67\times10^{-11}\;Nm^2(kg)^{-2} and here we need to convert NN into Kgms−2Kgms^{-2}, meters into centimeters and kilograms into grams while the unit of time will remain same, that is seconds

We know that 1  N=1  kgms−21\;N=1\;kgms^{-2}, where 1  kg=1000  g1\;kg=1000\;g and 1  m=100  cm1\;m=100\;cm. Now, upon putting these values, we get G=6.67×10−11  Nm2(kg)−2=6.67×10−11×(1  kgms−2)(1m2)(1kg−2)G=6.67\times10^{-11}\;Nm^2(kg)^{-2}=6.67\times10^{-11}\times(1\;kgms^{-2})(1m^2)(1kg^{-2})

  ⟹  G=6.67×10−11×(1  kg−1×1  m3×1  s−2)\implies G=6.67\times10^{-11}\times(1\;kg^{-1}\times1\;m^3\times1\;s^{-2})
Now, we will convert units, that is G=6.67×10−11×(103  g−1)×(102  cm)3×(1  s−2)=6.67×10−8  cm3s−2g−1G=6.67\times10^{-11}\times(10^3\;g^{-1})\times(10^2\;cm)^{3}\times(1\;s^{-2})=6.67\times10^{-8}\;cm^3s^{-2}g^{-1}

Hence, G=6.67×10−11  Nm2(kg)−2=6.67×10−8  cm3s−2g−1G=6.67\times10^{-11}\;Nm^2(kg)^{-2}=6.67\times10^{-8}\;cm^3s^{-2}g^{-1}

Note: The most common mistakes that may happen while unit conversions are
(i)Confusion in division or multiplication while converting larger units to a smaller one or vice versa
(ii)While breaking units like newtons into kilograms, meters and seconds.