Motion of this particle, Mechanical Engineering

Assignment Help:

Motion of this particle:

A particle of mass m is resting on a smooth horizontal plane as illustrated in Figure (a). It is associated to a spring which has a constant of k in force units per unit deformation. The mass is displaced to a distance xo from its equilibrium position and released with zero velocity. Learn the motion of this particle.

507_Motion of this particle6.jpg

Solution

A free-body diagram is drawn showing the particle in a position at a distance x from equilibrium position. Referring Figure (b). A force T is acting on this body in a horizontal direction which is developed due to the stretching of the spring through the distance x.

Within elastic limit, we may assume that the tension in the spring is proportional to its modified in length from the equilibrium position. Then T = k x, where T is the spring force, k is constant and x is the change of length.

Let ax be the acceleration with which the spring shall come back to its equilibrium position. Adding up forces in horizontal direction, we have - T = max.

Please note that x is to the right, thus ax is written as positive and T is to the left, hence it is written as negative.

∴ - k x = m (d 2 x   /dt2)            - a Simple Harmonic Motion.

The solution of this second order differential equation may be written as following

1449_Motion of this particle.png

The values of A and B may be calculated as below. Value of x is x0, when t = 0.

1046_Motion of this particle1.png

 = 0 + B

∴ B = x0

and

1702_Motion of this particle2.png

To evaluate A, we differentiate x w. r. t. time as we know v = 0, when t = 0.

1081_Motion of this particle3.png

when t = 0, v = 0 and

940_Motion of this particle4.png

∴ we get                  A = 0.

∴ The equation of motion for this problem is

1571_Motion of this particle5.png


Related Discussions:- Motion of this particle

What is the eutectoid composition, Question 1 What is the eutectoid com...

Question 1 What is the eutectoid composition in the phase diagram? Question 2 For an alloy of 40%C/60%D at 400°C, indicate the phase/s that are present. It is not neces

Moment of resistance, Explain the Moment of resistance? Sol.: The two...

Explain the Moment of resistance? Sol.: The two equal and unlike parallel forces, the lines of action of which are not same, form couple. The resultant compressive force ( P

Calculate the total radiation to floor, An enclousure measures 1.5 x 1.75m ...

An enclousure measures 1.5 x 1.75m with a height of 2m.Under steady state equilibrium conditions , the ceiling and walls are maintained at 525K and floor at 400K. Calculate the

Flux cored arc welding (fcaw), Flux Cored Arc Welding (FCAW) This is a ...

Flux Cored Arc Welding (FCAW) This is a variation of GMAW where solid wire is replaced by flux cored wire. The equipments and accessories are the same. Generally, flux cored wi

Matlab program of location a point with respect a line, matlab program of l...

matlab program of location a point with respect a line

Spring, strain energy = m2l/2Ei

strain energy = m2l/2Ei

Determine concept of hydrostatic sensor paver machine, Determine concept of...

Determine concept of Hydrostatic sensor paver machine A sensor asphalt paver is a machine used to deal out, shape, and partially compact a layer of asphalt on the surface of a

Computer aided synthesis, The basic aim of synthesis in design process is t...

The basic aim of synthesis in design process is to provide information which can tell you how to improve the design. In general the synthesis process gives many alternatives of

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd