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
Materials requirements planning: Materials requirements planning (MRP) is a set of process for converting forecasted demand for a manufactured product into a requirement sched

Determine the width and depth of the beam: A timber beam of rectangular section is only supported over a span of 5 m. It carries an uniformly distributed load of 15 kN/m over

Types of stresses: In general stress is divided in to three group as: various types of stresses may be classified as: 1. Simple or direct stress (Tension, Compressi

Resulting structure of steel: Heat treatments as mentioned below are given to thin steel strips for which TTT diagram is as shown in Figure 19. What will be the resulting stru

ACSR (Aluminium Conductor Steel Reinforced) are used as  (A)  over head transmission lines.  (B)  super conductors. (C)  fuse  (D)  underground cables.

Q. Requirements for vessels with thickness? The following additional requirements are applicable, as specified,  to Carbon Steel and Low Alloy steel pressure vessels with main

Explain concepty of Worktable in radial drilling machine The worktable is supported on an arm mounted to the column. The worktable can be adjusted vertically to accommodate dis

unstable governors

Impact Factor on Vessel rigging analysis? Unless otherwise specified by the user, a minimum impact factor of 1.5 shall be applied to the lift weight for designing the lifting d

Q. What is Fusion Bonding? Fusion Bonding shall refer to the fusion bond coating process which utilizes heat to melt adhere the coating material to the metal surface. Fusion Bo