Determine the radius of curvature - motion of a particle, Mechanical Engineering

Determine the radius of curvature - motion of a particle:

The motion of a particle in XOY plane is defined by the equation

r (t ) = 3t i^ + (4t - 3t 2 ) j^

The distances are in metres. Determine its radius of curvature and its acceleration while it crosses the x axis again.

Solution

We have  x = 3t,           y = (4t - 3t 2 )

∴ t = x/3 ,                   ∴ y = 4 x/3 - x 2/3

∴          The equation is a second degree curve and if we equate it to zero, we shall get two values of x.

The path crosses the x axis at x = 0, y = 0 and t = 0 second, x = 4, y = 0, t = 4 /3 second as shown in Figure.

The radius of curvature is attained as below.

          1/ ρ = ±  (d 2 y/ d x 2) / [1+ (dy/dx)2] (3/2)

           y = 4 /3 x - x 2 /3

 dy / dx = 4/3  - (2/3) x

and

d2  y /dx2= - 2/3

∴ 1/ ρ =  ± (2/3) / [ 1+ ((4/3)-(2/3))2](3/2)

at         x = 0    or         at         x = 4 m

 ∴ 1/ ρ =  ± (2/3) / [ 1+ ((4/3) 2](3/2)                ∴ 1/ ρ =  ± (2/3) / [ 1+ ((-4/3) 2](3/2)

            ±( 2 /3) /(25/9)(3/2);                                ±( 2 /3) /(25/9)(3/2)

               =  18/125 ;                                                    =  18/ 125

               ρ = 6.94 m                                                       ρ = 6.94 m

We have, x = 3t  y = 4t - 3t 2

∴ vx  = 3 m / sec                                    ∴ v y  = 4 - 6t m/ sec.

∴ for  t = 0,       vx  = 3 m/sec.,              vy  = 4 m/sec.

1472_Determine the radius of curvature - motion of a particle.png

Differentiating further, we obtain

d 2 x/dt2  = ax  = 0,                                             d 2 y /dt 2  = a y  = - 6

460_Determine the radius of curvature - motion of a particle1.png

The total acceleration is constant and equal in magnitude to 6 m/sec2.

At both of instants t = 0 and t = 4/3 seconds. The normal acceleration may be found as

a n   = v2 / ρ =   25 /6.94  = 3.6 m / sec2

and tangential acceleration

1558_Determine the radius of curvature - motion of a particle2.png

Posted Date: 1/29/2013 1:54:58 AM | Location : United States







Related Discussions:- Determine the radius of curvature - motion of a particle, Assignment Help, Ask Question on Determine the radius of curvature - motion of a particle, Get Answer, Expert's Help, Determine the radius of curvature - motion of a particle Discussions

Write discussion on Determine the radius of curvature - motion of a particle
Your posts are moderated
Related Questions
The crank shaft of a high speed pump has a parallel shaft of 2.5 cm diameter and 48 cm in length. It carries three eccentrics each of 6 cm in diameter and uniform thickness of 1.8

Find the speed of shaft : Find the speed of shaft driven with the belt by an engine running at 600 rotation per minute. The thickness of belt is 2cm, diameter of engine pulle


have assignment in FEM calculation and ANSYS are you able to solve this kind of assignment

Q. Measurement of river water levels? In designing bridges and other engineering structures on rivers, maximum discharge and highest flood level form essential data for design

If Low Resistance or Very High Current in Primary Coil Circuit Causes of Problem Remedy Primary ignition coil short circuited

1. A continuous steady-state process involving components A, B, and C is shown below, along with the known process information a) On the flowchart, assign variables for the unkn

Q. Show Corrosion of BFW Pump? BFW pumps are in high-pressure high-temperature service and may be subject to cavitation and erosion-corrosion. Also overheating of these pumps i

(a) Illustrate reversible, irreversible and Quasi-static process with examples. (b) What do you mean by terms change of state, path, process, property, cycle. (c) Distinguish

how will you detect the depth of sea and application?