Table of Contents
CENTRIPETAL FORCE
Centripetal force is known as the force responsible for maintaining an object’s circular motion by directing it towards the center of the circle. It is essential for continuously changing the object’s direction without changing its speed. Depending on the circumstances, this force can be exerted by gravity, tension, friction, or other forces. For instance, in planetary orbits, gravity serves as the centripetal force, while in a car navigating a turn, it is the friction between the tires and the road. In the absence of centripetal force, objects would adhere to Newton’s First Law of Motion and move in a straight line.
UNIFORM CIRCULAR MOTION
We know that the axis of rotation of a rotating particle passes through the center of the circle and is perpendicular to the plane of rotation. In case of uniform motion, the particle always covers equal distance in equal time, i.e. the velocity of the particle is constant.
Definition: If the angular velocity of a particle moving in a circular path is constant, its motion is called uniform circular motion.
In uniform circular motion the direction of the particle’s trajectory changes instantaneously. Hence the direction of the linear velocity v of the cone also changes at every instant. But since the velocity is constant, the value of linear velocity remains constant in this case. Since linear velocity is a vector quantity, it has both magnitude and direction.
So, in this case, the linear velocity of the particle cannot be said to be constant. That is, circular motion is an example of symmetry, but not an example of momentum.
The direction of the particle’s linear velocity at each point on the circular path is along the tangent to the circle drawn at that point. This is why when spinning a rock tied to a string, if the string breaks, the rock moves tangentially.
As we know, the rate of change of velocity of an object is the acceleration of the object. An object rotating at uniform circular motion has a variable linear velocity, i.e. the object has an acceleration.
CALCULATION OF CENTRIPETAL ACCELERATION
Let us say that a particle of mass m is moving with velocity v in a circular path of radius r. In a very small interval t, the particle moves from point A to point B and thereby makes a very small angle θ (=∠AOB) at the center O of the circle. So, the angular velocity of the particle, ω = θ/t.
The linear velocity v of the particle at point A is along the tangent AP. The component of this velocity along the radius AO is zero, because AO and AP are perpendicular to each other. The velocity v of the particle at point B is in the tangential direction BQ. This velocity is divided into two perpendicular components. The component of this velocity in the direction BR parallel to AP is vcosθ and the component of this velocity in the direction BS parallel to AO is vsinθ.
If θ is very small then sinθ = θ and cosθ = 1. Again, if the ∠AOB is very small, then the straight lines BR and BS almost coincide with the straight lines AP and AO respectively.
So, initial velocity of particle in AP direction = v, final velocity = vcosθ = v
That is, change in velocity = v – v = 0 or, acceleration = change in velocity / time = 0
Therefore, in the AP direction, i.e. the direction tangent to the circle, the particle has no acceleration.
Again, initial velocity of particle in direction AO = 0
Final velocity = vsinθ = vθ
Therefore, Change in velocity = vθ – 0 = vθ
or, acceleration = change in velocity / time = vθ/t = vω = ωr · ω = ω²r = (v/r)² · r = v²/r
Hence, the particle has an acceleration in the direction AO (i.e. along the radius r again towards the centre) whose value is ω²r or v²/r. This acceleration is called centripetal acceleration.
Definition: A particle moving in uniform circular path has an acceleration towards the center of the circle called centripetal acceleration.
WHAT IS CENTRIPETAL FORCE?
As we have seen in the previous para, the value of centripetal acceleration in uniform circular motion is ω²r or v²/r. So, if an object has mass m on the object,
Acting force = mass X acceleration = mω²r = mv²/r
Since the acceleration is centripetal, the acting force is also centripetal. This force is the centripetal force.
According to Newton’s first law of motion, we know that if no external force is applied on the object, the object remains at rest or in uniform motion, that is, the object cannot be in circular motion. Therefore, an external force must act on the object to make it rotate in a circular path. Acceleration of an object is always centripetal when moving in a circular path. Hence, the force acting on the object is also centripetal i.e. acts towards the center along the radius. This outward force is the centripetal force. So, we can say that centripetal force is what makes an object move in a circular path by deviating it from a state of uniform motion.
Definition: The force that causes an object to rotate in a circular path, acting perpendicular to the velocity of the object and towards the center of the circular path is called centripetal force.
If the mass of the object is m, the radius of the orbit is r, the linear velocity of the object is v and the angular velocity is ω,
Centripetal Force = mω²r = mv²/r
It may be mentioned that centripetal force is called no-work force. This force does not do any work as there is no displacement of the object in the direction of the centripetal force.
SOME PRACTICAL EXAMPLES
- When a stone is tied to a string and rotated in a circular path, the string pulls the stone towards the center. That tension is the centripetal force.
- The Sun’s gravitational pull on any planet provides the centripetal force necessary to rotate the planet around the Sun.
- The mutual electric force of attraction between the positively charged nucleus and the negatively charged electrons within the atom provides the centripetal force necessary to spin the electrons around the nucleus.
- When riding a bicycle in a circular path, the friction between the road and the bicycle wheel provides the centripetal force necessary to keep the bicycle moving in a circular path.
CONCLUSION
In summary, centripetal force is essential for maintaining the circular motion of objects by pulling them inward towards the center of the circle, preventing them from veering off in a straight line. In the absence of centripetal force, objects would continue along a tangential path due to their inertia. This force is observable in various occurrences, such as the orbits of planets around stars and the maneuvering of cars around bends. Understanding centripetal force is key to comprehending the equilibrium between velocity and the force necessary to sustain objects in circular motion, making it a fundamental concept in both physics and engineering.
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