Circular Motion Physics Ib
Odon Irving
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.
In physics, circular motion is a movement of an object along the circumference of a circle or rotation along a circular arc. It can be uniform, with a constant rate of rotation and constant tangential speed, or non-uniform with a changing rate of rotation. The rotation around a fixed axis of a three-dimensional body involves the circular motion of its parts. The equations of motion describe the movement of the center of mass of a body, which remains at a constant distance from the axis of rotation. In circular motion, the distance between the body and a fixed point on its surface remains the same, i.e., the body is assumed rigid.
Examples of circular motion include: special satellite orbits around the Earth (circular orbits), a ceiling fan's blades rotating around a hub, a stone that is tied to a rope and is being swung in circles, a car turning through a curve in a race track, an electron moving perpendicular to a uniform magnetic field, and a gear turning inside a mechanism.
Since the object's velocity vector is constantly changing direction, the moving object is undergoing acceleration by a centripetal force in the direction of the center of rotation. Without this acceleration, the object would move in a straight line, according to Newton's laws of motion.
In physics, uniform circular motion describes the motion of a body traversing a circular path at a constant speed. Since the body describes circular motion, its distance from the axis of rotation remains constant at all times. Though the body's speed is constant, its velocity is not constant: velocity, a vector quantity, depends on both the body's speed and its direction of travel. This changing velocity indicates the presence of an acceleration; this centripetal acceleration is of constant magnitude and directed at all times toward the axis of rotation. This acceleration is, in turn, produced by a centripetal force which is also constant in magnitude and directed toward the axis of rotation.
In the case of rotation around a fixed axis of a rigid body that is not negligibly small compared to the radius of the path, each particle of the body describes a uniform circular motion with the same angular velocity, but with velocity and acceleration varying with the position with respect to the axis.
During circular motion, the body moves on a curve that can be described in the polar coordinate system as a fixed distance R from the center of the orbit taken as the origin, oriented at an angle θ(t) from some reference direction. See Figure 4. The displacement vector r \displaystyle \mathbf r is the radial vector from the origin to the particle location: r ( t ) = R u ^ R ( t ) , \displaystyle \mathbf r (t)=R\hat \mathbf u _R(t)\,, where u ^ R ( t ) \displaystyle \hat \mathbf u _R(t) is the unit vector parallel to the radius vector at time t and pointing away from the origin. It is convenient to introduce the unit vector orthogonal to u ^ R ( t ) \displaystyle \hat \mathbf u _R(t) as well, namely u ^ θ ( t ) \displaystyle \hat \mathbf u _\theta (t) . It is customary to orient u ^ θ ( t ) \displaystyle \hat \mathbf u _\theta (t) to point in the direction of travel along the orbit.
Because the radius of the circle is constant, the radial component of the velocity is zero. The unit vector u ^ R ( t ) \displaystyle \hat \mathbf u _R(t) has a time-invariant magnitude of unity, so as time varies its tip always lies on a circle of unit radius, with an angle θ the same as the angle of r ( t ) \displaystyle \mathbf r (t) . If the particle displacement rotates through an angle dθ in time dt, so does u ^ R ( t ) \displaystyle \hat \mathbf u _R(t) , describing an arc on the unit circle of magnitude dθ. See the unit circle at the left of Figure 4. Hence: d u ^ R d t = d θ d t u ^ θ ( t ) , \displaystyle \frac d\hat \mathbf u _Rdt=\frac d\theta dt\hat \mathbf u _\theta (t)\,, where the direction of the change must be perpendicular to u ^ R ( t ) \displaystyle \hat \mathbf u _R(t) (or, in other words, along u ^ θ ( t ) \displaystyle \hat \mathbf u _\theta (t) ) because any change d u ^ R ( t ) \displaystyle d\hat \mathbf u _R(t) in the direction of u ^ R ( t ) \displaystyle \hat \mathbf u _R(t) would change the size of u ^ R ( t ) \displaystyle \hat \mathbf u _R(t) . The sign is positive because an increase in dθ implies the object and u ^ R ( t ) \displaystyle \hat \mathbf u _R(t) have moved in the direction of u ^ θ ( t ) \displaystyle \hat \mathbf u _\theta (t) .Hence the velocity becomes: v ( t ) = d d t r ( t ) = R d u ^ R d t = R d θ d t u ^ θ ( t ) = R ω u ^ θ ( t ) . \displaystyle \mathbf v (t)=\frac ddt\mathbf r (t)=R\frac d\hat \mathbf u _Rdt=R\frac d\theta dt\hat \mathbf u _\theta (t)=R\omega \hat \mathbf u _\theta (t)\,.
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