The most common problem in the determination of the radius of curvature of trajectory of the thrown body in a given period of time. The trajectory in this case is described by the equations on the coordinate axes: x = f(t), y = f(t), where t is the time and date at which you want to find the radius. Evaluation will be based on the formula AP = V2/R. Here the radius R is detected from the ratio of the normal acceleration AP and the instantaneous speed V of the movement of the body. Knowing these values, you can easily find the desired component R.
Calculate the projection velocity of the body on the axes (OX, OY). The mathematical sense of speed is the first derivative of the equations of motion. So they are easily found by taking the derivative of the set of equations: Vx = x, Vy = y'. When considering geometric data display projection in a coordinate system shows that they are legs of a right triangle. And the hypotenuse in it – the desired velocity. Based on this, calculate the magnitude of the instantaneous velocity V by the Pythagorean theorem: V = √( Vx2 + Vy2). Substituting in the expression is a known value of time, find the index of V.
Module normal acceleration is also easy to identify, having another right triangle formed by the module full acceleration and tangential acceleration AK. And here the normal acceleration is the leg and calculated as: AP = √( A2 - ak2). To find the tangential acceleration differentiate in time equation instantaneous speed of motion: AK = |dV/dt|. Full acceleration calculate from its projections on the axis, similar to finding the instantaneous velocity. Just for this take from the set of equations of motion of second order derivatives: Ah = x," y = y". The module of the acceleration a = √( ах2 + ау2). Substituting all the found values, determine a numeric value for the normal acceleration AP = √( A2 - ak2).
Express from the formula AP = V2/R the desired variable radius of curvature of the trajectory: R = V2/ AP. Substitute the numerical values of speed and acceleration, and calculate the radius.
Advice 2 : How to determine the radius of curvature
To study the motion of a physical object (car, cyclist, ball in roulette, it is sufficient to study the motion of some of its points. In the study of motion is that all points describe some of the curves.
Be aware that curves can describe the motion of a fluid, gas, light rays, lines current. The radius of curvature for a plane curve at a point is the radius of the tangent circle at that point. In some cases the curve is given by equations, and the radius of curvature is calculated according to the formulas. Accordingly, in order to know the radius of curvature, it is necessary to know the radius of the circle relating to a certain point.
Define a plane curve point And, near it, take another point B. Construct a tangent to an existing curve that pass through points A and B.
Swipe through point A and the line perpendicular to the constructed tangent, extend them to the intersection. Label the point of intersection as O. the Point O is the center of the tangent circle at a given point. So OA is the radius of the circle, i.e. the curvature in a given point A.
Note that when a point moves along any curved path at any point in the motion it moves in some circles, which varies from point to point.
If the point in space to define curvature in two mutually perpendicular directions, these curvature will be referred to the principal. The direction of the main curvatures must be 900. For the calculations often use the average curvature is equal to the sum of the main curvatures, and Gaussian curvature, equal to their work. There is also a notion of curvature of a curve. It is the reciprocal of the radius of curvature.
Acceleration is an important factor in the motion of a point. The curvature of the trajectory directly affects the acceleration. Acceleration occurs when a point with a constant speed begins to move along the curve. Changes not only the absolute value of the velocity, but its direction, there is a centripetal acceleration. I.e. in reality, the point begins to move along the circumference, which at this point in time.