What is Centripetal Acceleration?

When a particle moves in a curve, the acceleration that points to the center of the circle (center of curvature) is perpendicular to the direction of the tangent to the curve, which is also called the normal acceleration. Centripetal acceleration is a physical quantity that reflects how quickly the direction of circular motion changes. Centripetal acceleration only changes the direction of the speed, it does not change the magnitude of the speed. ^[1]

In the above formula,

Represents centripetal acceleration,

Indicates centripetal force,

Represents the mass of the object,

The linear velocity (tangential velocity) of the circular motion of the object,

Represents the angular velocity of the object's circular motion,

Represents the period of the object's circular motion,

The frequency of circular motion of the object,

Represents the radius of the object's circular motion. (

= 2 / T)
by

The direction is always perpendicular to the direction of movement, the direction changes at all times and points to the center of the circle (center of curvature), regardless of acceleration

Whether the size of

The direction changes from moment to moment, so

mistakenly believe

The discussion and analysis of the following two issues can further consolidate and deepen the students' knowledge and understanding of uniform circular motion.

1. What is the meaning of centripetal acceleration?

To understand this problem, we must first clarify the physical meaning of v in the vector triangle (Figure 4).

It only indicates the change of the speed direction, but not the change of the speed. Therefore, the centripetal acceleration only characterizes the change of the speed direction.

2. Does an object doing a uniform circular motion "fall" toward the center of the circle?

This problem lies in knowledge, and it is worth asking to discuss with students. As shown in Figure 5, if the object no longer has acceleration aa at point a, the object will definitely fly out in the direction of ae and arrive in t seconds. Point e, but now the object "falls" to point b, that is, a distance away from ae eb. When the time t is sufficiently short, point b and point a are very close, and with point a as the limit, it can be considered that ab arc and ab string coincide with each other, eb and ad coincide with each other, and there is ab string = vt, eb = ad Because rt abcrt adb, then ad / ab = ab / ac, that is,

It can be seen that the object does "fall" to the center of the circle from time to time, but it cannot really reach the center of the circle. Obviously, this is the result of centripetal acceleration.

Proof method of centripetal acceleration formula (2) and attached drawings

Picture of proof method (2)

As shown in Figure A, a particle makes a uniform circular motion around point O, and the tangents from point A to point B, that is, the linear velocities V _a and V _b , are equal in magnitude. The centripetal acceleration a is the unit change vector of the linear velocity from V _b to V _a . Method: As shown in Figure B, translate the vector V _a so that its starting point coincides with point B. Then the vector V = vector V _b vector V _a (that is, the amount of change in linear velocity when turning over a certain radian). Set the vector V _a The angle with V _b is the angle (in radians) that the particle has rotated in a uniform circular motion.

As shown in Figure D (a part of circle O, that is, a sector, OQ = OP = r, and a chord PQ and arc PQ), let be the number of radians between the angle of OQ and OP (actually, the arc length corresponding to this angle in mathematics The ratio to the radius of the circle, namely the arc PQ: the value of the radius r, such as one radian 57.3 °) then we know that X · Y / X = Y, then the length of the arc PQ can be expressed as "radius r · arc PQ / radius r" That is, arc length = radius × corresponding radian . When the included angle is very small, the arc PQ = chord PQ can be approximated, that is, the curved arc length is almost the same as the straight line segment length, which provides a basis for the subsequent determination of V.

Return to Figure B. As shown in the figure, when the angle between OB and OA (equal to the angle between V _b and Va) is small and very small, then the corresponding V is very small and very small, with B as the vertex. The arc V swept from point A to point B in a sector with _a bus length of V _a (or V _b ) can be approximately equal to the chord V. According to the introduction of Tuding, if the radius r in Tuding is Considered as linear velocity V _a (or V _b ), arc length = radius × corresponding radian (that is, the previous V = · r ) used in Figure B is the arc V = V = linear velocity (considered as radius r ) × radian (the ratio of the arc V to the linear velocity Va or Vb which can be regarded as the circle radius r)

And when the amount of V is small to the unit (that is, the amount of V in one second), then this V is what we call the centripetal acceleration a, the centripetal acceleration a = V / t , and the arc V = chord V, so the centripetal acceleration a = arc V / t .

First, radian is the number of radians of the angle that the particle has rotated in a circular motion after a certain time ( t), then the angular velocity = / t is the number of radians rotated in one second, that is, radian = t , and V = arc V = centripetal acceleration a × t .

Then according to arc length = radius × corresponding radian, arc V = V = linear velocity V × radian (as shown in Figure C, when is small to a certain degree, arc V = V exists as small as unit radian Such a relationship ) According to and , we obtain the centripetal acceleration a × t = linear velocity V (the magnitude of this vector is always the same) × angular velocity · t , and at the same time remove t around the equation. Simplified to:

Centripetal acceleration a = angular velocity × linear velocity V , that is, a (n) = · V , and a ( n) = 2 · r, a (n) = V ^ 2 / r, etc. are all based on this formula And V = · r is inferred.