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Uniform Circular MotionA 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 where In addition, during this acceleration at constant speed, the particle travels the circumference of the circle (a distance of
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
Recall from Section 4-3 that the velocity The scalar components of
Now, using the right triangle in Fig. 4-19a, we can replace
To find the acceleration
This vector and its components are shown in Fig. 4-19c. Following Eq. 3-6, we find that the magnitude of
Thus,
"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 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 Date: 2015-01-12; view: 1865
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