Question

An Object is weighted at the North Pole by a beam balance and a spring balance giving readings of ${{W}_{B}}$ and ${{W}_{S}}$, respectively. It is again weighed in the same manner at the equator, giving readings of $W_{B}^{'}$ and $W_{S}^{'}$, respectively. Assume that the acceleration due to gravity is the same and that the balances are quite sensitive. (This question has multiple correct options)
$A.{{W}_{B}}={{W}_{S}}$
$B.W_{B}^{'}=W_{S}^{'}$
$C.{{W}_{B}}=W_{B}^{'}$
$D.W_{S}^{'}={{W}_{S}}$

Answer 1

We should start solving this problem by applying the difference in concepts of beam balance and spring balance at poles and equator. The acceleration due to gravity is less at the equator than at the poles. The concept of effect of rotation of earth should be applied in this particular problem for accurate results.
Formula used:
In this particular problem we are going to use following two formulae for poles and equator respectively,
$W=mg$and $W'=m{{(g-{{\omega }^{2}}R)}^{{}}}$

Complete answer:
We know that the workings of both beam balance and spring balance are different from each other. But before that we will know the meanings of symbols used,
${{W}_{B}}=$Weight of an object measured by beam balance at the North Pole
${{W}_{S}}=$Weight of an object measured by spring balance at the North Pole
$W_{B}^{'}=$Weight of an object measured by beam balance at the equator
$W_{S}^{'}=$Weight of an object measured by spring balance at the equator
$m=$ Mass of an object
$g=$Acceleration due to gravity
$R=$Radius of earth
$\omega =$ Rotational effect of the earth
Analysing both the devices we get,
Beam Balance is a device which is used to measure the weight of an object without considering the rotational effect of the earth. It means that Weight measured by beam balance is always constant. But, Spring Balance is a device which is used to measure the weight of an object by considering the rotational effect of the earth. It means that weight measured by spring balance is variable at different places.
At poles rotational effect is zero, Therefore,
${{W}_{b}}=mg$ and ${{W}_{s}}=mg$
So,${{W}_{b}}={{W}_{s}}$, therefore option (a) is correct.
Weight measured by beam balance is constant everywhere. So,
${{W}_{B}}=W_{B}^{'}$’, therefore option (c) is correct.
At Equator, there is rotational effect Weight measured by spring balance at equator will be given as,
$W_{s}^{'}=m(g-{{\omega }^{2}}R)$
Whose value is less than ${{W}_{S}}$.

So, the correct answer is “Option A and C”.

Note:
We should not be confused with the difference in workings of beam balance and spring balance. We should analyse each option correctly in these types of questions as we have to select out correct options and eliminate wrong ones. Correct relations for poles and equators should be used without any confusion.