Proof of the equation of centripretal force

In summary, when applying the parallelogram law to two velocities acting on a single body, the velocity in either direction remains the same. However, when one of the velocities is acceleration, such as in projectile motion, the situation becomes more complex. Circular motion also presents a challenge as the body is moving at a uniform velocity while accelerating towards the center. The parallelogram law can still be applied, as seen in the proof of the equation of centripetal force, which can also be done using pure geometry instead of vectors. The use of calculus may be necessary when considering motion in a vertical circle.
  • #1
batballbat
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When we apply the parallelogram on two velocities acting on a single body we see that the velocity in anyone of the two directions remains the same. ( the velocities act on different directions)

When one is velocity and the other acceleration or in other words accelerating a body in some direction other than the body's uniform velocity the situation gets complex. In projectile motion we assume the acceleration to act downwards (rather than the centre) in all the points of its trajectory so that the downward forces are parallel in all the points of its trajectory. This makes the problem easy. And it is legitimate considering the size of earth.

But when we consider circular motion, that's when i get confused. Because here we are dealing with bodies moving in a uniform velocity and at the same time accelerating toward the centre. SO i don't know how to manipulate such conditions. Here the parallelogram law is not applied. I have seen a proof of the equation of centripretal force using pure geometry. And it is intuitive than the vectorial proof. Can it be done for motion in a vertical circle? or is calculus necessary?
 
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  • #2
hi batballbat! :smile:
batballbat said:
But when we consider circular motion, … Here the parallelogram law is not applied.

yes it is!

consider the velocities at small angles ±θ from some direction …

they're equal in magnitude, so draw two lines of equal length from the same point, at angles ±θ …

the line joining them, to make the diagonal of the parallelogram, is at 90° - θ to both velocities :wink:
 

1. What is the equation for centripetal force?

The equation for centripetal force is F = mv^2/r, where F is the force, m is the mass of the object, v is the velocity, and r is the radius of the circular motion.

2. How is centripetal force related to circular motion?

Centripetal force is the force that keeps an object moving in a circular path. It acts towards the center of the circle and is necessary to maintain the object's constant speed and direction.

3. What is the unit of measurement for centripetal force?

The unit of measurement for centripetal force is Newton (N), which is the SI unit for force. It can also be expressed in other units such as pound-force (lbf) or dyne (dyn).

4. Can centripetal force be negative?

No, centripetal force cannot be negative. It always acts towards the center of the circle, so it is always a positive value. A negative value would indicate that the force is acting in the opposite direction of the motion.

5. How does the equation for centripetal force relate to other laws of motion?

The equation for centripetal force is derived from Newton's second law of motion, which states that the net force on an object is equal to its mass multiplied by its acceleration. In the case of circular motion, the acceleration is directed towards the center of the circle, resulting in the equation F = ma = mv^2/r.

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