LEARNING OBJECTIVE 3.A.1.1 - Express the motion of an object using narrative, mathematical, and graphical representations. AP MECHANICS -C LEARNING OBJECTIVE INT 2.B -Explain how a net force in the centripetal direction can be a single force, more than one force, or even components of forces that are acting on an object moving in circular motion.
So far, we have covered all the mathematics required to understand the circular motion. We defined and related all the variables we will need to form our equations for analysis. Now it’s time to look at the physics of uniform circular motion.
Let’s start with one of the most important Kinematical variables; acceleration. We have observed mathematically that when an object moves in a uniform circular motion it experiences an acceleration called centripetal acceleration, which is directed towards the centre of the circle and whose magnitude is given by a simple equation $v^2/r$. We now have the key to understand the uniform circular motion.
Clearly, Newton’s law tells us that if there is an acceleration it must have been because of the presence of some force. This force must have two characteristics so that we can have a uniform circular motion. Firstly, the force must be acting in the direction such that it results in impacting the body a centripetal acceleration and if there is more than one force acting on the body the net force is imparting the body the centripetal acceleration. If then the force ceases to exist we will not have circular motion anymore. Secondly, the force shouldn’t change the value of v, otherwise, it will not be a uniform circular motion anymore, which means the force must act perpendicular to the velocity of the object at all times. This indeed means the force must keep on changing its direction such that it is always perpendicular to the velocity or in case of circular motion, always towards the centre.
Which are those forces that can enable such a motion? Indeed, in several cases, we can indicate such as force. For example, a ball tied to to string rotating in a circle the tension of the string providing the force. For planets rotating about the Sun, it is the gravitational force of attraction. Let’s call this force Centripetal force because it imparts Centripetal acceleration. Of course, if the Centripetal force ceases to act ( for example the string breaks ) the Centripetal acceleration also vanishes ( F= ma ).
Using Newton’s law we can now derive an useful equation which will allow us to solve and analyse circular motion problems.
$F_{Centripetal} = m \times a_{centripetal} = \frac{mv^2}{r}$
Thus, for a body to move uniformly in a circle, it must be acted upon by a net force equal to the product of the mass of the body and the square of its velocity, divided by the radius of the circle. From the above equation, we can now immediately draw some useful conclusions.
- For a given velocity v, to execute a circular motion with a smaller radius we will need a larger force compared with a larger radius. For example, for a given velocity of a car driving around a circular bend, the ground must exert a larger force on the wheels of the car, the smaller the radius of curvature of the bend.
- Pay attention to the fact that in the equation the velocity appears in the second power. This means that as the velocity of the circular motion of a given radius increases, the force required to maintain such a motion increases rapidly.
