Instruction

1

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.

2

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.

3

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).

4

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 find the radius of the circle

The definition

**of the radius****of the circle**is one of the main tasks of mathematics. There are many formulas to account for the**radius**, you just know some standard options. Graphically, the radius is denoted by letter R in the Latin alphabet.Instruction

1

A circle is a closed curve. The points that lie in its plane, equidistant from the centre, which lies in one plane together with a curve. Radius - a segment

**of a circle**connecting its center with any point. With its help, you can learn many other parameters of the figure, so it is a key parameter. The numerical value**of the radius**will be the length of this segment.2

You should also distinguish the radius of the shape from its diameter (diameter connects the two most remote from each other point). To use a mathematical method of finding

**the radius**you need to know the length or diameter**of the circle**. In the first case the formula will look like R = L/2?", where L is the known length**of the circumference**, and the number ? equal to 3.14 and is used to denote a particular irrational number.3

In the case that the only known diameter, the formula will look like R = D/2".

4

If the length

**of the circle**is unknown, but there are data on the length and height of a segment, the formula would be R = (h^2*4 + L^2)/8*h where h is the height of the segment is the distance from mid-chord to the most protruding part of the said arc) and L is the length of the segment (which is not the chord length).Chord – a line segment that connects two points**of a circle**.Note

It is necessary to distinguish the concept of "circumference" and "circle". The circle is part of the plane, which, in turn, is limited by the circumference of a certain radius. To find the radius, you must know the area of a circle. In this case, the equation would be R = (S/π)^1/2, where S is a square. To calculate the area, in turn, should know the radius (S = NR^2").

# Advice 3: 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.

Instruction

1

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.

2

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.

3

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.

4

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.

5

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.

6

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.

# Advice 4: How to find the centripetal acceleration

Centripetal acceleration appears when a body moves in a circle. It is directed to its center, measured in m/S2. A feature of this type of acceleration is that it is, even when the body is moving with constant speed. It depends on the radius of the circle and the linear velocity of the body.

You will need

- speedometer;
- - instrument for measuring distances;
- - stopwatch.

Instruction

1

To find the centripetal acceleration, measure the velocity of a body moving along a circular trajectory. You can do this using a speedometer. If you install this device is not possible, calculate the linear speed. To do this, note the time that it took for a full rotation on a circular path.

2

It is the period of rotation. Let's Express it in seconds. Measure the radius of the circle on which the moving body with a ruler, measuring tape or a laser rangefinder in meters. To find the speed find the product of the number 2 to the number π≈3.14, and the radius R of the circle and divide the result by the period T. This will be linear velocity of the body v=2∙π∙R/T.

3

Find the centripetal acceleration AC by dividing the square of the linear velocity v on the radius of the circle on which the moving body is R (AC=v2/R). Using formulas to determine the angular velocity, frequency and period of rotation, find that value in other formulas.

4

If you know the angular velocity ω and the radius of the trajectory (the circle in which the body moves) R the centripetal acceleration will be equal to AC= ω2∙R. When you know the period of rotation of the body T, and the radius of the trajectory R, then AC= 4∙π2∙R/T2. If we know the speed ν (the number of complete rotations per second), determine the centripetal acceleration by the formula AC= 4∙π2∙R∙ν2.

5

Example: the car, the radius of the wheels of which 20 cm is moving down the road at a speed of 72 km/h. Determine the centripetal acceleration of the extreme points of the wheels.

Solution: the linear velocity of points of any wheel is 72 km/h=20 m/s. wheel Radius in meters R=0.2 m. Calculate the centripetal acceleration substituting the resulting numbers into the formula AC=v2/R. Get ATS=202/0,2=2000 m/S2. This centripetal acceleration under uniform rectilinear motion will be at the extreme points of all four wheels of the car.

Solution: the linear velocity of points of any wheel is 72 km/h=20 m/s. wheel Radius in meters R=0.2 m. Calculate the centripetal acceleration substituting the resulting numbers into the formula AC=v2/R. Get ATS=202/0,2=2000 m/S2. This centripetal acceleration under uniform rectilinear motion will be at the extreme points of all four wheels of the car.

# Advice 5: How to find tangential acceleration

Tangential

**acceleration**is the bodies moving along a curved path. It is aimed in the direction of change of the velocity of the body at a tangent to the trajectory. Tangential acceleration happens to bodies moving uniformly in a circle, they have only centripetal acceleration.You will need

- speedometer or the radar;
- - ruler or measuring tape;
- - stopwatch.

Instruction

1

Find the tangential acceleration aτ, if you know the full acceleration of a particle moving along a curved trajectory a and its centripetal acceleration an. To do this, from the square full acceleration, subtract the square of the centripetal acceleration, and from this value, extract the square root aτ=√(a2-an2). If you do not know the centripetal acceleration, but there is a value of instantaneous speed, measure with a ruler or tape measure the radius of curvature of the trajectory and find its value by dividing the square of the instantaneous velocity v, which measure the speedometer or the radar on the radius of curvature of the trajectory R, an=v2/R.

2

Example. The body moves in a circle with a radius of 0.12 m. Its full acceleration is 5 m/S2, determine its tangential acceleration, at a time when its velocity is 0.6 m/s. First, find the centripetal acceleration of a body at a specified speed, it it square divide by the radius of the trajectory of an= v2/R= 0,62/0,12=3 m/S2. Find the tangential acceleration by the formula aτ=√(a2-an2)=√(52-32)=√(25-9)= √16=4 m/S2.

3

Find the magnitude of tangential acceleration through a change in the speed module. To do this using a speedometer or radar, determine the initial and final velocity of the body for a certain period of time, which you measure with a stopwatch. Find the tangential acceleration, v the ultimate subtracting from the initial value of speed v0 and dividing the time interval t during which that change occurred: aτ= (v-v0)/t. If the value of the tangential acceleration turned negative, so the body slows down, if positive accelerating.

4

Example. 4 with the velocity of a body moving along the circumference, decreased from 6 to 4 m/s. Determine the tangential acceleration. Calculated by applying the formula, obtain aτ= (v-v0)/t=(4-6)/4=-0,5 m/S2. This means that the body slows down with an acceleration of the absolute value of which is equal to 0.5 m/S2.