Question

A car moves at constant speed on a road as shown in the figure. The normal force by the road on the car is when it is at point A and B respectively. Information to decide the relation of
                      

A. \[{{N}_{A}}={{N}_{B}}\]
B. \[{{N}_{A}}>{{N}_{B}}\]
C. \[{{N}_{A}}<{{N}_{B}}\]
D. insufficient

Answer 1

Hint: In this question we are being asked to find the relation between \[{{N}_{A}}\] and \[{{N}_{B}}\]. From the figure, it is clear that radius of curve a is less than radius of curve b. \[{{N}_{A}}\] and \[{{N}_{B}}\] are the normal forces at point A and B. We know that, formula for calculating the normal force on a curved road, states the relation between velocity and the radius of the curve. The velocity is not given as constant at both points.

Formula used:
\[N=mg-\dfrac{m{{v}^{2}}}{r}\]
Where,
Mg is the weight of the car
V is the velocity
And r is the radius of the curve

Complete step-by-step answer:
From the figure, we can say that radius of curve a say \[{{r}_{A}}\]is less than radius of curve B say \[{{r}_{B}}\]. We know that, weight of the car will remain constant at all times. The velocity at point a and point b is given as constant
Now,
We know from formula,
\[N=mg-\dfrac{m{{v}^{2}}}{r}\]
We can say that the normal force N is directly proportional to square of velocity v and inversely proportional to the radius of curve r.
Now, the normal force at point a say \[{{N}_{A}}\] can be given as
\[{{N}_{A}}=mg-\dfrac{m{{v}_{A}}^{2}}{{{r}_{A}}}\] ……………. (1)
Similarly, the normal force at point b say\[{{N}_{B}}\] can be given as
\[{{N}_{B}}=mg-\dfrac{m{{v}_{B}}^{2}}{{{r}_{B}}}\] ………………………….. (2)
Now let us assume that \[{{N}_{A}}\] = \[{{N}_{B}}\]
Therefore, from (1) and (2)
We can say that,
\[mg-\dfrac{m{{v}_{A}}^{2}}{{{r}_{A}}}=mg-\dfrac{m{{v}_{B}}^{2}}{{{r}_{B}}}\]
On solving we get,
\[\dfrac{{{v}_{A}}^{2}}{{{r}_{A}}}=\dfrac{{{v}_{B}}^{2}}{{{r}_{B}}}\]
But we know that velocity is constant
Therefore,
\[\dfrac{1}{{{r}_{A}}}=\dfrac{1}{{{r}_{B}}}\]
But from the diagram given, we know that \[{{r}_{A}}\] < \[{{r}_{B}}\]
Also, we know that the normal force N is inversely proportional to radius r
Therefore, we can say that our assumption is wrong.
Since \[{{r}_{A}}\] < \[{{r}_{B}}\], we can now say that
\[{{N}_{A}}>{{N}_{B}}\], since the normal force is inversely proportional to r

So, the correct answer is “Option B”.

Note: The normal force is said to be the support force. For a normal force to act on a body the body must be in contact or lying on the surface. The normal force for static objects on a horizontal plane is usually taken as the weight of the object i.e. mg. In such cases normal force is counteracting the weight of the object.