Circular Motion Physics And Maths Tutor
Tony Phan
Circular motion is related to Simple Harmonic Motion (SHM) as both involve periodic oscillatory motion.
In more detail, circular motion and Simple Harmonic Motion (SHM) are interconnected concepts in physics. Both involve objects moving in a periodic, oscillatory manner. However, the nature of the motion is different. In circular motion, an object moves along the circumference of a circle at a constant speed. In SHM, an object oscillates back and forth about a fixed point, also known as the equilibrium position.
The connection between these two types of motion becomes apparent when you consider a particle moving in a circle at a constant speed. If you were to project the motion of this particle onto a diameter of the circle, the resulting motion would be simple harmonic. This is because the projection of the circular motion onto any diameter of the circle will always result in an oscillation that follows the rules of SHM.
The mathematical relationship between circular motion and SHM is also quite clear. The displacement of an object undergoing SHM can be represented as x = A cos(wt + φ), where A is the amplitude, w is the angular frequency, t is the time and φ is the phase constant. This equation is very similar to the one that describes circular motion, r = R cos(θ), where R is the radius and θ is the angle.
In both cases, the object's position is described by a cosine function, indicating a periodic motion. The frequency of the motion in both cases is determined by the angular frequency, w. In circular motion, this is the rate at which the object moves around the circle, while in SHM, it is the rate at which the object oscillates back and forth.
In conclusion, while circular motion and SHM may seem different at first glance, they are closely related. Both involve periodic motion, and the motion of an object in SHM can be thought of as the projection of circular motion onto a line. This relationship is not only conceptual but also mathematical, as the equations describing the two types of motion are very similar.
The net force (Fnet) acting upon an object moving in circular motion is directed inwards. While there may by more than one force acting upon the object, the vector sum of all of them should add up to the net force. In general, the inward force is larger than the outward force (if any) such that the outward force cancels and the unbalanced force is in the direction of the center of the circle. The net force is related to the acceleration of the object (as is always the case) and is thus given by the following three equations:
An equation expresses a mathematical relationship between the quantities present in that equation. For instance, the equation for Newton's second law identifies how acceleration is related to the net force and the mass of an object.
The relationship expressed by the equation is that the acceleration of an object is directly proportional to the net force acting upon it. In other words, the bigger the net force value is, the bigger that the acceleration value will be. As net force increases, the acceleration increases. In fact, if the net force were increased by a factor of 2, the equation would predict that the acceleration would increase by a factor of 2. Similarly, if the net force were decreased by a factor of 2, the equation would predict that the acceleration would decrease by a factor of 2.
Newton's second law equation also reveals the relationship between acceleration and mass. According to the equation, the acceleration of an object is inversely proportional to mass of the object. In other words, the bigger the mass value is, the smaller that the acceleration value will be. As mass increases, the acceleration decreases. In fact, if the mass were increased by a factor of 2, the equation would predict that the acceleration would decrease by a factor of 2. Similarly, if the mass were decreased by a factor of 2, the equation would predict that the acceleration would increase by a factor of 2.
As mentioned previously, equations allow for predictions to be made about the affect of an alteration of one quantity on a second quantity. Since the Newton's second law equation shows three quantities, each raised to the first power, the predictive ability of the equation is rather straightforward. The predictive ability of an equation becomes more complicated when one of the quantities included in the equation is raised to a power. For instance, consider the following equation relating the net force (Fnet) to the speed (v) of an object moving in uniform circular motion.
This equation shows that the net force required for an object to move in a circle is directly proportional to the square of the speed of the object. For a constant mass and radius, the Fnet is proportional to the speed2.
The factor by which the net force is altered is the square of the factor by which the speed is altered. Subsequently, if the speed of the object is doubled, the net force required for that object's circular motion is quadrupled. And if the speed of the object is halved (decreased by a factor of 2), the net force required is decreased by a factor of 4.
The mathematical equations presented above for the motion of objects in circles can be used to solve circular motion problems in which an unknown quantity must be determined. The process of solving a circular motion problem is much like any other problem in physics class. The process involves a careful reading of the problem, the identification of the known and required information in variable form, the selection of the relevant equation(s), substitution of known values into the equation, and finally algebraic manipulation of the equation to determine the answer. Consider the application of this process to the following two circular motion problems.
A 95-kg halfback makes a turn on the football field. The halfback sweeps out a path that is a portion of a circle with a radius of 12-meters. The halfback makes a quarter of a turn around the circle in 2.1 seconds. Determine the speed, acceleration and net force acting upon the halfback.
In Lesson 2 of this unit, circular motion principles and the above mathematical equations will be combined to explain and analyze a variety of real-world motion scenarios including amusement park rides and circular-type motions in athletics.
1. Anna Litical is practicing a centripetal force demonstration at home. She fills a bucket with water, ties it to a strong rope, and spins it in a circle. Anna spins the bucket when it is half-full of water and when it is quarter-full of water. In which case is more force required to spin the bucket in a circle? Explain using an equation as a "guide to thinking."
2. A Lincoln Continental and a Yugo are making a turn. The Lincoln is four times more massive than the Yugo. If they make the turn at the same speed, then how do the centripetal forces acting upon the two cars compare. Explain.
3. The Cajun Cliffhanger at Great America is a ride in which occupants line the perimeter of a cylinder and spin in a circle at a high rate of turning. When the cylinder begins spinning very rapidly, the floor is removed from under the riders' feet. What affect does a doubling in speed have upon the centripetal force? Explain.
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An object is said to be in motion if it changes its position with time. A body which does not move is said to be at rest, motionless, or stationary. Circular motion is a movement of an object along a circular path or a circular orbit. The object in circular motion moves along the circumference of a circle. Motion along a circular with constant speed is called uniform circular motion.
There is a continuous change in direction in uniform circular motion due to which there is acceleration that is perpendicular to the path because of uniform speed. This acceleration is directed towards the center. This acceleration is called the centripetal acceleration or radial acceleration. The force which causes this acceleration is called the centripetal force. Therefore, centripetal force acting on a body in circular motion may be defined as the radial force directed towards the center. The direction of the force changes continuously.
Centrifugal force is a fictitious force due to inertia of rotational motion. There is a misconception that an object moving in a circle has an outward force acting on it called centrifugal force. If a centrifugal force were acting the stone would fly away while whirling. The centrifugal force cannot be created. It is only experience in the context of circular motion.
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Force is a fundamental concept in physics that describes the push or pull acting on an object. In the realm of mathematics, force takes on a different meaning but retains the essence of its physical counterpart. This article explores the concept of force in mathematics, providing definitions, examples, an FAQ section, and a quiz to enhance your understanding.
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