In the chapter “ Motion” , you have studied the basics of motion of a body and
equations supporting the validation of concepts , with your understanding draw a
flowchart explaining the correlation between the concepts starting from “ distance”
to
“ uniform circular motion”.
Answers
Explanation:
Centripetal Acceleration
In the previous section, we defined circular motion. The simplest case of circular motion is uniform circular motion, where an object travels a circular path at a constant speed. Note that, unlike speed, the linear velocity of an object in circular motion is constantly changing because it is always changing direction. We know from kinematics that acceleration is a change in velocity, either in magnitude or in direction or both. Therefore, an object undergoing uniform circular motion is always accelerating, even though the magnitude of its velocity is constant.You experience this acceleration yourself every time you ride in a car while it turns a corner. If you hold the steering wheel steady during the turn and move at a constant speed, you are executing uniform circular motion. What you notice is a feeling of sliding (or being flung, depending on the speed) away from the center of the turn. This isn’t an actual force that is acting on you—it only happens because your body wants to continue moving in a straight line (as per Newton’s first law) whereas the car is turning off this straight-line path. Inside the car it appears as if you are forced away from the center of the turn. This fictitious force is known as the centrifugal force. The sharper the curve and the greater your speed, the more noticeable this effect becomes.
Figure 6.7 shows an object moving in a circular path at constant speed. The direction of the instantaneous tangential velocity is shown at two points along the path. Acceleration is in the direction of the change in velocity; in this case it points roughly toward the center of rotation. (The center of rotation is at the center of the circular path). If we imagine Δs becoming smaller and smaller, then the acceleration would point exactly toward the center of rotation, but this case is hard to draw. We call the acceleration of an object moving in uniform circular motion the centripetal acceleration ac because centripetal means center seeking.Figure 6.7 The directions of the velocity of an object at two different points are shown, and the change in velocity Δv is seen to point approximately toward the center of curvature (see small inset). For an extremely small value of Δs, Δv points exactly toward the center of the circle (but this is hard to draw). Because ac=Δv/Δt, the acceleration is also toward the center, so ac is called centripetal acceleration.
Now that we know that the direction of centripetal acceleration is toward the center of rotation, let’s discuss the magnitude of centripetal acceleration. For an object traveling at speed v in a circular path with radius r, the magnitude of centripetal acceleration is
ac=v2r.
Centripetal acceleration is greater at high speeds and in sharp curves (smaller radius), as you may have noticed when driving a car, because the car actually pushes you toward the center of the turn. But it is a bit surprising that ac is proportional to the speed squared. This means, for example, that the acceleration is four times greater when you take a curve at 100 km/h than at 50 km/h.
We can also express ac in terms of the magnitude of angular velocity. Substituting v=rω into the equation above, we get ac=(rω)2r=rω2 . Therefore, the magnitude of centripetal acceleration in terms of the magnitude of agular velocity is
ac=rω2.
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