Tip 1: How to find the radius of curvature of the trajectory

In considering the motion of bodies used a number of characterizing quantities such as tangential and normal (centripetal) acceleration, speed and trajectory curvature.The radius of curvature - geometric concept that refers to the circle radius R, on which the body moves.This option can be found on the relevant formulas via a predetermined path of movement.
most common tasks for the determination of the radius of curvature of the trajectory of a body thrown in a specified period of time.The motion path in this case is described by the equations on the coordinate axes: x = f (t), y = f (t), where t - the time at which you want to find the radius.Its calculation is based on the application of the formula AN = V² / R.Here, the radius R is detected from the ratio of AN and normal acceleration instantaneous velocity V of the body.Having these values, you can easily find the desired component R.
Calculate the velocity of the body projections on the axes (OX, OY).The mathematical sen
se of speed - this is the first derivative of the equation of motion.Therefore, they are easily found by taking the derivative of the given equations: Vx = x ', Vy = y'.When considering the geometrical projection display data in the coordinate system it shows that they are the legs of a right triangle.And the hypotenuse in it - the desired instantaneous speed.Based on this, calculate the magnitude of the instantaneous velocity V of the Pythagorean theorem: V = √ (Vx² + Vy²).Substituting in the expression of the known value of time, get a numerical indicator V.
normal acceleration unit is also easy to identify, consider another right-angled triangle formed by the module and the complete acceleration and tangential acceleration ak body.And here, the acceleration is the leg and is calculated as follows: AN = √ (a² - ak²).To find the tangential acceleration Differentiate the time equation of instantaneous speed: ak = | dV / dt |.Full acceleration is calculated by its projections on the axis, analogous to finding the instantaneous velocity.Only take this set of equations of motion of second-order derivatives: dx = x ', AY = y' '.Module acceleration a = √ (ax2 + ay2).Substituting the obtained values, determine the numerical value of the normal acceleration AN = √ (a² - ak²).
Express from the formula AN = V² / R desired variable radius of curvature of the trajectory: R = V² / AN.Substitute numerical values ​​of velocity and acceleration, calculate the radius.

Tip 2: How to find the radius of curvature

curvature - a concept borrowed from differential geometry.It is a collective name for a number of quantitative characteristics (vector, scalar, tensor).The curvature indicates the deviation of the geometrical "object", which can be surface and curve, and Riemann space, from other known "flat" objects (flat, straight, Euclidean space, and so on. D.).
How to find the radius of curvature
Typically the curvature is determined individually for each desired point on a given "object" and designate it as the second-order differential expression.For objects having a lower smoothness and curvature can be determined in an integral way.As a general rule, if at all points curvature identical treatment is zero, then this should be a local match of the studied "object" with a "flat" object.
Suppose that you need to make a plano-convex lens.You only know that the optical power is equal to 5 diopters.Find radius of curvature convex surface of the linzy.Vspomnite equality:
D = 1 / f
D - is the optical power (lens), f - is the focal rasstoyanieZapishite equality:
1 / f = (n-1) *(1 / r1 + 1 / r2)
n - is the refractive index (this type of material)
r1 - radius lens with one side
r2 - on the other hand
Simplify expression soas a lens plano, it radius of one of the parties will seek to infinity means 1 divided by infinity, will tend to zero.You should receive a simplified expression: 1 / f = (n-1) * 1 / r2
Since you know the optical power of the lens, then find the focal length:
D = 1 / f
1 /f = 5 diopters
f = 1/5 diopters
f = 0,2 m
Given the task lenses are made of glass.Remember that glass is the refractive index of 1.5, thus the expression you should look like this:
(1,5 - 1) * 1 / r2 = 0,2 m
0,5 * 1 / r2 = 0,2m
Divide all parts of the expression 0.5.You should get:
1 / r2 = 0,4 m
r2 = 1 / 0.4 m
r2 = 2,5 mZapishite result: D.You will receive from the plano-convex lens radius of curvature 2,5 meters.
determine the radius of curvature of the lens you need, can Ophthalmologist, making the necessary measurements.It is necessary to select appropriate contact lenses.There are contact lenses, the radius of curvature of which increases continuously from the center to the edge - lens, the central portion of the rear surface of which has a non-spherical shape.
Helpful Hint
base curvature - a curvature of the central part of the rear surface of the contact lens.Most part of the contact lens has a spherical shape, which is characterized by a radius of curvature of the base.The medical center or clinic Ophthalmologist after the special measurements (eg using avtorefkeratometra), determine the base curve of your eyes and write you a prescription.
  • radius of curvature of the lens surface