Question

Suppose a spring with spring constant is horizontal and has one end attached to a wall and the other end attached to a mass. Suppose that the friction of the mass with the floor (i.e., the damping constant) is . Set up a differential equation that describes this system. Let to denote the displacement, in meters, of the mass from its equilibrium position, and give your answer in terms of . Assume that positive displacement means the mass is farther from the wall than when the system is at equilibrium. x'' x'/2 7/2x

Answer 1

x'' + (2/3)x' + (7/3)x=0 would be the differential equation describing the system.

Given:

spring constant, k= 7 N/m ; mass = 3kg  ;and damping constant, c=2 Ns/m

For Differential Equation:

m(d²x/dt²) = -kx -cdx/dt   ........... (1)

Substitute the values of m,k, and c in (1).

3(d²x/dt²) = -7x -2dx/dt  

3(d²x/dt²) -7x -2dx/dt = 0

i.e , 3x'' +2x' + 7x =0

By dividing the equation with ;

  x'' + (2/3)x' + (7/3)x =0

The differential equation for the system is

x'' + (2/3)x' + (7/3)x =0

What is the importance of differencial equation ?

Differential equations are a useful tool for scientists and engineers because they can be used to simulate complicated situations. They may better comprehend a system's behavior and how it could be changed to get a desired result by resolving a differential equation.

Because they may be used to simulate a wide range of real-world scenarios, differential equations are significant in both mathematics and the sciences. Differential equations, for instance, can be used to simulate the motion of particles in a fluid or the trajectory of a projectile in physics. Differential equations may be used in biology to simulate population increase and the transmission of diseases.

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