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
Compute the outside diameter of the shaft: A hollow shaft of diameter ratio is 0.4 is needed to transit 600 kW at 120 rpm. The maximum torque being 20% greater than the mean t

Single piece pattern These are inexpensive and the simplest type of patterns. As the name indicates, they are made of a single piece as shown in Figure.2.2 this type of pattern

A light cantilever of rectangular section (5cm deep and 2.5 cm wide) has a mass fixed at its free end. Find the ratio of frequency of free lateral vibration in vertical plane to th

The Linear Dimension Command                                    You can use this command to generate horizontal and vertical dimensions.Creating a linear dimension is eas

steam coming out of the whistle of a pressure cooker is ?

Calculate the value of ∫ Π/2 sinx dx by Simpson's 1/3 rule using 11 ordinates. Determine the root of equation xe x = cosx using secant method correct to four decimal places.

brief description of resonant cavities

Q. Explain working of plate heat exchangers? For gasket plate heat exchangers handling flammable materials, the same spacing requirements for air coolers should be applied. If

Necessity of Lubrication To lubricate the various parts of engine. The supplementary function of lubrication is to cool and clean the engine parts. To prevent en

what is dry ness fraction?