Question

Differntial Equations Forced Spring Motion

1. A 1 kg mass is attached to a spring of spring constant k = 4kg/82, The spring-mass system is attached to a machine that supplies an external driving force of f(t) = 4 cos(wt). The systern is started from equilibrium i.e. 2(0) = 0 and z'(0) = 0. There is no damping. (a) Find the position x(t) of the mass as a function of time (b) write your answer in the form r(t)-1 sin(6t) sin(wt), where δ is the half difference and is the mean frequency. (c) Now suppose w = 1.5. Sketch a graph that shows the "beats" of the system and include the envelope of the beating motion in your plot like we did in class. 2. An undamped spring-mass system with external force is modeled by the differential equation: z', + 25x = 4 cos(5t). Suppose (01 and ' (00. Find the position of mass as a function of time. What part of this solution guarantees that this system resonates rather than demonstrating the "beating" of the previous question

Answer 1

Here total force on 1kg mass is given by

f (t)- kr
and acceleration is written as then

or
$m{x}''=f(t)-kx$
or

kr 4 r" =--+-cos(wt) ?7m ?7m

Now by integrating

here C₁=0 as x'(0) =0

sunlu ?7m

hence

Now by farther integration

sunlu ?7m

4cos(ut

here
as x(0) =0

mu2

now equation become

$x(t)=-\frac{kx}{2m}\times t^{2}-\frac{4}{m\omega^{2} }\times cos(\omega t)+\frac{4}{m\omega^{2} }$

by substituting value of m=1kg k=4  and

mmu

- COsl