Question

If the mass of the Sun were ten times smaller and the universal gravitational constant were ten times larger in magnitude, which of the following is not correct?

A. The time period of a simple pendulum on the Earth would decrease
B. Raindrops will fall faster
C. ‘g’ on the earth will not change
D. Walking on the ground would become more difficult

Answer 1

Hint: The equation for gravity is \[g = \dfrac{{GM}}{{{R^2}}}\] . If the value of \[G\] then the value of \[g\] would increase. Now the force on raindrops and us humans would be \[F = mg\] , so the speed of raindrops increases, and walking for us becomes more difficult. The time period of a simple pendulum is \[T = 2\pi \sqrt {\dfrac{l}{g}} \] and if the value of \[g\] were to increase the time period would decrease.

Complete answer:
For the time period of a simple pendulum on earth, the speed of raindrops, the value of ‘g’ on earth, and our walking on earth the mass of the sun do not have any effect but the value of universal gravitational constant will greatly affect our answer as shown below.
We know that the equation for gravity is
\[g = \dfrac{{GM}}{{{R^2}}}\]
Here, \[g = \] The acceleration due to gravity
\[G = \] Universal gravitational constant
\[M = \] The mass of the earth
\[R = \] The radius of the earth
Now, as we can see from the equation, if we were to increase the universal gravitational constant ten times the gravity would increase ten times.
Now the force on raindrops will be as follows
\[F = mg\]
If gravity were to increase then the speed of raindrops would increase.
Similarly, the gravitational force that we feel would also increase, making walking much more difficult.
Now, the time period of a simple pendulum is
\[T = 2\pi \sqrt {\dfrac{l}{g}} \]
Here, \[l = \] The length of the pendulum
\[g = \] The acceleration due to gravity
So, if gravity were to increase we would see that the time period of a simple pendulum would decrease.

So, the correct answer is “Option C”.

Note:
In this problem, we ignored the effect of the smaller mass of the sun, in reality, if the sun were to somehow really get ten times smaller, it would have some effect on the speed of raindrops, time period of a simple pendulum on earth and us walking on the ground, but the effect would be almost negligible, which is why we can ignore its effect.