4.4 Uniform Circular Motion
A car moving in a circular path with constant speed v. Such motion is called uniform circular motion, and occurs in many situations. It is often surprising to students to find that even though an object moves at a constant speed in a circular path, it still has an acceleration.
In Figure 4.17b is the same as the angle between the velocity vectors in Figure 4.17c, because the velocity vector v is always perpendicular to the position vector r. Thus, the two triangles are similar. (Two triangles are similar if the angle between any two sides is the same for both triangles and if the ratio of the lengths of these sides is the same.) This enables us to write a relationship between the lengths of the sides for the two triangles:
Thus, in uniform circular motion the acceleration is directed inward toward the center of the circle and has magnitude. In many situations it is convenient to describe the motion of a particle moving with constant speed in a circle of radius r in terms of the period T, which is defined as the time required for one complete revolution. In the time interval T the particle moves a distance of 2,r, which is equal to the circumference of the particle’s circular path. Therefore, because its speed is equal to the circumference of the circular path divided by the period, or ν = 2∏r/T .
4.5 Tangential and Radial Acceleration
The total acceleration vector a can be written as the vector sum of the component vectors:
The tangential acceleration component causes the change in the speed of the particle. This component is parallel to the instantaneous velocity, and is given by
The radial acceleration component arises from the change in direction of the velocity vector and is given by
The motion of a particle along an arbitrary curved path lying in
the xy plane. If the velocity vector v (always tangent to the path) changes in direction
and magnitude, the components of the acceleration a are a tangential component at
and a radial component ar .
4.6 Relative Velocity and Relative Acceleration
In this section, we describe how observations made by different observers in different frames of reference are related to each other. We find that observers in different frames of reference may measure different positions, velocities, and accelerations for a given particle.
Galilean Velocity Transformation
Although observers in two frames measure different velocities for the particle, they measure the same acceleration when is constant. We can verify this by taking the time derivative of Equation 4.22:
That is, the acceleration of the particle measured byan observer in one frame of reference is the same as that measured by any other observer moving with constant velocity relative to the first frame.