Question

An aircraft executes a horizontal loop of radius $1.00km$ with a steady speed of $900km/h$. Compare its centripetal acceleration with the acceleration due to gravity.

Answer 1

Hint: The centripetal acceleration of a body is the acceleration produced by a centripetal force. It happens when the body is moving on a circular path. It is always directed towards the center of the circle and helps the body to continue on the circular path by constantly changing its direction. It depends on the square of the instantaneous velocity of the body and the radius of the circular path.

Formula used:
${{a}_{c}}=\dfrac{{{v}^{2}}}{R}$
Where ${{a}_{c}}$ is the centripetal acceleration,$v$ is the speed of the body on a circular path and $R$ is the radius of the circular path

Complete step-by-step answer:

When a body moves or tries to move on a circular path, it is helped in doing so by forces acting in the direction of the line joining the body to the center of the circle. These forces are known as centripetal forces and the acceleration produced by the force on the body is known as centripetal acceleration.
The centripetal acceleration constantly tends to change the velocity of the body by changing its direction and making it move in the circular path. It does not change the speed of the body, since it acts perpendicular to the direction of initial velocity of the body.
For a body moving in a circular path of radius $R$ with a constant speed $v$ the centripetal acceleration ${{a}_{c}}$ is given by the mathematical expression
${{a}_{c}}=\dfrac{{{v}^{2}}}{R}$---(1)
Now, let us analyse the question.
Here the aircraft is executing a loop with a uniform speed, and we have to find the centripetal acceleration and compare it with the acceleration due to gravity $g$.
Hence,
radius of the loop $\left( R \right)=1.00km=1000m$ --$\left( \because 1km=1000m \right)$ --(2)
Speed of the aircraft $\left( v \right)=900km/hr=900\times \dfrac{5}{18}=250m/s$ --$\left( \because 1km/hr=\dfrac{5}{18}m/s \right)$ --(3)
Therefore, putting (2) and (3) in (1), we get the centripetal acceleration as,
${{a}_{c}}=\dfrac{{{250}^{2}}}{1000}=\dfrac{62500}{1000}=62.5m/{{s}^{2}}$ --(4)
Now, comparing this value with the acceleration due to gravity $g=9.8m/{{s}^{2}}$, we get
$\dfrac{{{a}_{c}}}{g}=\dfrac{62.5}{9.8}\approx 6.38$
Hence, the centripetal acceleration is about $6.38$ times the acceleration due to gravity.

Note: Centripetal acceleration is not a separate force like frictional force or normal force. It is just the name given to a force that is helping the body move on a circular path by pointing towards the center of the circle. Thus, any force can become a centripetal force. In fact, for cars going on a corner in the road, the frictional force becomes the centripetal force.
There is a difference between centripetal force and centrifugal force and many students get confused between the two. Centripetal force is the actual force that helps a body move on a circular path, however centrifugal force is the pseudo force experienced by the body in the frame of reference moving along the circular path. Since the body has inertia it does not want to turn to follow the circular path but wants to keep on going on the line of its initial velocity. This inertial resistance, in essence is felt by the body as the centrifugal force.
For example, when a car takes a very sharp turn on the road, passengers are thrown to the opposite side and collide with the door. This is the centrifugal force experienced by them. The normal force from the door helps them to move along with the car on the circular path, so it acts as the centripetal force.