4.1 Circular Motion

In this section we will discuss uniform motion, the motion of a particle in a circular path at a constant speed. The magnitude of the velocity vector (or the speed) is constant, however the direction changes constantly (see Figure 4.2). Thus, there is acceleration , in this case toward the center of the circular path. Now, take for example, the circular path of an object as represented by the circle in Figure 4.1.

At time t1, the radius from the center of the circle to the position of the object is represented by vector r1. At some later time, t2, the position of the object has changed. The new radius is represented by vector r2. The displacement of the object, represented by D r, is equal to the difference between r2 and r1:

The magnitude of r2 and r1 are the same. Both are equal to the radius of the circular path, r. The arc length traveled by the particle over time is represented by D s, and the angular displacement is D q . The angular displacement is equal to the quotient of the arc length, D s, and the radius, r, which is approximately equal to the quotient of D r and r:

Now we return to velocity and Figure 4.2, which shows the same path of travel that is illustrated in Figure 4.1. At time t1, the velocity vector v1 represents the particle’s velocity. Vector v2 represents the particle’s velocity at time t2. Vectors v1 and v2 have the same magnitude; both are equal to the constant speed of the particle.

We can use the end-to-end method to study vectors v1 and v2 in more detail. The change in velocity is represented by D v. As can be seen in Figure 4.2, vector D v points toward the center of the circular path. This type of acceleration, acceleration toward the center of a circular path, is called centripetal acceleration, from the Latin words "centri", meaning center, and "petre", meaning to seek. Centripetal acceleration can also be called radial acceleration, and is represented by ac. We will now represent the change in velocity in mathematical terms:

We can now revise Equation 4.1 to represent a change in velocity. D q , the angular displacement, is equal to the quotient of D v and v (the constant magnitude of the particle’s velocity velocity):

We can equate the right sides of Equations 4.1 and 4.2, as they are both equal to D q .

D r is equal to vD t, thus:

The centripetal acceleration, ac, is equal to the quotient of D v and D t, which is equal to the quotient of the square of velocity and the radius, r:

Using Equations 2.1 and 4.3, we can calculate the centripetal force that causes this centripetal acceleration.

The centripetal force is expressed in Newtons, mass in kilograms, velocity in meters per second, and r in meters.

Centripetal forces affect objects in many different circumstances. Perhaps the most common example of a "center seeking" object is a ball attached to a string that is swung in a circular path. Centripetal force also affects a car traveling in a curved path. As a car travels around a corner, friction between its tires and the pavement makes it travel around the curve. When it is icy or wet, the frictional force is reduced (m k is reduced). Thus, the car may skid off the road because there is not enough centripetal force to keep it moving in a curved path.

The motion of a particle in a circular path with a constant speed is periodic; it repeats over time. The time elapsed to perform one cycle of motion (the time elapsed during one revolution along the circular path) is called the period of the motion, T. The velocity of the particle can also be represented by the quotient of the distance around the circle, 2p r, and the period, T:

The frequency, f, is the number of complete revolutions per unit time. For example, if the frequency of a particle’s circular motion was 5 revolutions per second (5 rev/s), the period of the motion would be 0.2 seconds. Notice that if a particle travels n complete turns in some amount of time, the average velocity is 0 m/s because all velocity vectors are countered by equal but opposite vectors. Similarly, the acceleration is 0 m/s2.