A particle is in uniform circular motion if it travels around a circle or a circular arc at constant (uniform) speed. Although the speed does not vary, the particle is accelerating. That fact may be surprising because we often think of acceleration (a change in velocity) as an increase or decrease in speed. However, actually velocity is a vector, not a scalar. Thus, even if a velocity changes only in direction, there is still an acceleration, and that is what happens in uniform circular motion.
Figure 4-18 shows the relation between the velocity and acceleration vectors at various stages during uniform circular motion. Both vectors have constant magnitude as the motion progresses, but their directions change continuously. The velocity is always directed tangent to the circle in the direction of motion. The acceleration is always directed radially inward. Because of this, the acceleration associated with uniform circular motion is called a centripetal (meaning "center seeking") acceleration. As we prove next, the magnitude of this acceleration is
(centripetal acceleration) (4-32)
where is the radius of the circle and is the speed of the particle.
In addition, during this acceleration at constant speed, the particle travels the circumference of the circle (a distance of ) in time
is called the period of revolution, or simply the period, of the motion. It is, in general, the time for a particle to go around a closed path exactly once.
Proof of Eq. 4-32
To find the magnitude and direction of the acceleration for uniform circular motion, we consider Fig. 4-19. In Fig. 4-l9a, particle p moves at constant speed around a circle of radius . At the instant shown, p has coordinates and .
Recall from Section 4-3 that the velocity of a moving particle is always tangent to the particle's path at the particle's position. In Fig. 4-19a, that means is perpendicular to a radius drawn to the particle's position. Then the angle that makes with a vertical at p equals the angle that radius makes with the axis.
The scalar components of are shown in Fig. 4-19b. With them, we can write the velocity as
Now, using the right triangle in Fig. 4-19a, we can replace with and with to write
To find the acceleration of particle p, we must take the time derivative of this equation. Noting that speed and radius do not change with time, we obtain
Now note that the rate at which changes is equal to the velocity component . Similarly, , and, again from Fig. 4-19b, we see that and . Making these substitutions in Eq. 4-36, we find
This vector and its components are shown in Fig. 4-19c. Following Eq. 3-6, we find that the magnitude of is
as we wanted to prove. To orient , we can find the angle shown in Fig. 4-19c:
Thus, , which means that is directed along the radius of Fig. 4-19a toward the circle's center, as we wanted to prove.
Sample Problem 4-9
"Top gun" pilots have long worried about taking a turn too tightly. As a pilot's body undergoes centripetal acceleration, with the head toward the center of curvature, the blood pressure in the brain decreases, leading to loss of brain function.
There are several warning signs to signal a pilot to ease up: when the centripetal acceleration is or , the pilot feels heavy. At about , the pilot's vision switches to black and white and narrows to "tunnel vision." If that acceleration is sustained or increased, vision ceases and, soon after, the pilot is unconscious - a condition known as -LOC for " -induced loss of consciousness." What is the centripetal acceleration, in units, of a pilot flying an F-22 at speed = 2500 km/h (694 m/s) through a circular arc with radius of curvature = 5.80 km?
SOLUTION: The Key Idea here is that although the pilot's speed is constant, the circular path requires a (centripetal) acceleration, with magnitude given by Eq. 4-32:
If an unwary pilot caught in a dogfight puts the aircraft into such a tight turn, the pilot goes into -LOC almost immediately, with no warning signs to signal the danger.