A car while travelling at a speed of 72 km / hr . Passes through a curved portion of road in the form of an arc of a radius 10m . If the mass of the car is 500 kg the reaction of the car at the lowest point P is

The correct option is 25 kN.

Step-by-Step Explanation:

1. Identify the given values:
Speed of the car, $v=72\text{km/hr}$
Radius of the curved portion (arc), $r=10\text{m}$
Mass of the car, $m=500\text{kg}$
Acceleration due to gravity, $g=10{\text{m/s}}^2$

2. Convert speed into SI units (m/s):
To convert speed from $\text{km/hr}$ to $\text{m/s}$, multiply by $\frac{5}{18}$:
$v=72\times \frac{5}{18}=20\text{m/s}$

3. Analyze the forces at the lowest point P of the curve:
When the car passes through the lowest point $P$ of a curved valley, it is in a circular path where the net centripetal force is directed vertically upwards towards the center of the arc.
The forces acting on the car at this point are:
1. The normal reaction force $R$ acting vertically upwards.
2. The weight of the car $mg$ acting vertically downwards.

The equation of motion for the centripetal acceleration is:
$R-mg=\frac{m{v}^2}{r}$

Rearranging the equation to solve for the normal reaction force $R$:
$R=mg+\frac{m{v}^2}{r}=m\left(g+\frac{v^2}{r}\right)$

4. Substitute the given values to calculate R:
$R=500\times \left(10+\frac{{20}^2}{10}\right)$
$R=500\times \left(10+\frac{400}{10}\right)$
$R=500\times \left(10+40\right)$
$R=500\times 50$
$R=25000\text{N}$

Converting the unit into kilonewtons ($\text{kN}$):
$R=25\text{kN}$

Thus, the reaction force of the road on the car at the lowest point P is 25 kN.