Centripetal Acceleration In one-dimensional kinematics, objects with a constant speed have zero acceleration. However, in two- and three-dimensional kinematics, even if the speed is a constant, a particle can have acceleration if it moves along a curved trajectory such as a circle. In this case the velocity vector is changing, or dv/dt / 0. This is shown in the figure below. As the particle moves counterclockwise in time At on the circular path, its position vector moves from if (t) to F (t+At) . The velocity vector has constant magnitude and is tangent to the path as it changes from v (t) to v (t+At), changing its direction only. Since the velocity vector v (t) is perpendicular to the position vector Ff (t) , the triangles formed by the position vectors and AF, and the velocity vectors and Av are similar. Furthermore, since |F(t) = |F(++At)| and |v (t)] =|v(+At)], the two triangles are isosceles. From these facts we can make the assertion A = AF or Av = Ar. v(t + At) F(t + At) C ?? AF Av v(t) V(1 + Ar) v(!) (a) (b) (a) A particle is moving in a circle at a constant speed, with position and velocity vectors at times t and t+At. (b) Velocity vectors forming a triangle. The two triangles in the figure are similar. The vector Av points toward the center of the circle in the limit At -+ 0.
We can find the magnitude of the acceleration from a = lim At-+0 =(im Ar) = ;. The direction of the acceleration can also be found by noting that as At and therefore 40 approach zero, the vector Av approaches a direction perpendicular to v. In the limit At -> 0,Av is perpendicular to v. Since V is tangent to the circle, the acceleration dv/dt points toward the center of the circle. Summarizing, a particle moving in a circle at a constant speed has an acceleration with magnitude The direction of the acceleration vector is toward the center of the circle. This is a radial acceleration and is called the centripetal acceleration, which is why we give it the subscript c. The word centripetal comes from the Latin words centrum (meaning "center") and petere (meaning "to seek"), and thus takes the meaning "center seeking." V The centripetal acceleration vector points toward the center of the circular path of motion and is an acceleration in the radial direction. The velocity vector is also shown and is tangent to the circle. Let's investigate some examples that illustrate the relative magnitudes of the velocity, radius, and centripetal acceleration. Creating an Acceleration of 1 g A jet is flying at 134.1 m/s along a straight line and makes a turn along a circular path level with the ground. What does the radius of the circle have to be to produce a centripetal acceleration of 1 g on the pilot and jet toward the center of the circular trajectory?
Strategy Given the speed of the jet,