Uniform Circular Motion Pdf
Ashlie Mealey
Uniform circular motion can be described as the motion of an object in a circle at a constant speed. As an object moves in a circle, it is constantly changing its direction. At all instances, the object is moving tangent to the circle. Since the direction of the velocity vector is the same as the direction of the object's motion, the velocity vector is directed tangent to the circle as well. The animation at the right depicts this by means of a vector arrow.
An object moving in a circle is accelerating. Accelerating objects are objects which are changing their velocity - either the speed (i.e., magnitude of the velocity vector) or the direction. An object undergoing uniform circular motion is moving with a constant speed. Nonetheless, it is accelerating due to its change in direction. The direction of the acceleration is inwards. The animation at the right depicts this by means of a vector arrow.
The final motion characteristic for an object undergoing uniform circular motion is the net force. The net force acting upon such an object is directed towards the center of the circle. The net force is said to be an inward or centripetal force. Without such an inward force, an object would continue in a straight line, never deviating from its direction. Yet, with the inward net force directed perpendicular to the velocity vector, the object is always changing its direction and undergoing an inward acceleration.
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)\,.
Figure 1 illustrates velocity and acceleration vectors for uniform motion at four different points in the orbit. Because the velocity v is tangent to the circular path, no two velocities point in the same direction. Although the object has a constant speed, its direction is always changing. This change in velocity is caused by an acceleration a, whose magnitude is (like that of the velocity) held constant, but whose direction also is always changing. The acceleration points radially inwards (centripetally) and is perpendicular to the velocity. This acceleration is known as centripetal acceleration.
In a non-uniform circular motion, the net acceleration (a) is along the direction of Δv, which is directed inside the circle but does not pass through its center (see figure). The net acceleration may be resolved into two components: tangential acceleration and normal acceleration also known as the centripetal or radial acceleration. Unlike tangential acceleration, centripetal acceleration is present in both uniform and non-uniform circular motion.
In a non-uniform circular motion, normal force does not always point in the opposite direction of weight. Here is an example with an object traveling in a straight path then looping a loop back into a straight path again.
This diagram shows the normal force pointing in other directions rather than opposite to the weight force. The normal force is actually the sum of the radial and tangential forces. The component of weight force is responsible for the tangential force here (We have neglected frictional force). The radial force (centripetal force) is due to the change in the direction of velocity as discussed earlier.
In a non-uniform circular motion, normal force and weight may point in the same direction. Both forces can point down, yet the object will remain in a circular path without falling straight down. First, let's see why normal force can point down in the first place. In the first diagram, let's say the object is a person sitting inside a plane, the two forces point down only when it reaches the top of the circle. The reason for this is that the normal force is the sum of the tangential force and centripetal force. The tangential force is zero at the top (as no work is performed when the motion is perpendicular to the direction of force applied. Here weight force is perpendicular to the direction of motion of the object at the top of the circle) and centripetal force points down, thus normal force will point down as well. From a logical standpoint, a person who is travelling in the plane will be upside down at the top of the circle. At that moment, the person's seat is actually pushing down on the person, which is the normal force.
The reason why the object does not fall down when subjected to only downward forces is a simple one. Think about what keeps an object up after it is thrown. Once an object is thrown into the air, there is only the downward force of Earth's gravity that acts on the object. That does not mean that once an object is thrown in the air, it will fall instantly. What keeps that object up in the air is its velocity. The first of Newton's laws of motion states that an object's inertia keeps it in motion, and since the object in the air has a velocity, it will tend to keep moving in that direction.
A varying angular speed for an object moving in a circular path can also be achieved if the rotating body does not have a homogeneous mass distribution. For inhomogeneous objects, it is necessary to approach the problem as in.[2]
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