Question

Fill in the blanks
A.)$1\;kgm^2/s^2=……….gcm^2/s^2$
B.)1m=………. ly
C.)$3.0\;ms^{-2}=……….kmh^{-2}$
D.)$G=6.67\times10^{-11}\;Nm^2(kg)^{-2}=……….(cm)^{3}s^{-2}g^{-1}$

Answer 1

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 $1\;kgm^2/s^2=\dfrac{(1\times1000)g\times(1\times100)^2\;cm}{s^2}$
$\implies1\;kgm^2/s^2=\dfrac{(1000)g\times(100)^{2}cm}{s^2}=\dfrac{10^3\times10^4}{s^2}\;gcm^2/s^2=10^7\; gcm^2/s^2$
Hence, $1\;kgm^2/s^2=10^7gcm^2/s^2$
In this question, we need to convert meters in light years.

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

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

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

Now, we can write that $1\;m=\dfrac{1}{9.46\times10^{15}}\;ly=1.06\times10^{-16}\;ly$
Hence, $1\;m=1.06\times10^{-16}\;ly$

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 $3.0\;ms^{-2}=\dfrac{3m}{1s^{2}}=\dfrac{(3/1000)\;km}{(1/3600)^{2}\;h^2}=3.88\times10^4\;kmh^{-2}$
Hence, $3.0\;ms^{-2}=3.88\times10^4\;kmh^{-2}$

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

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

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

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

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.