A mass of 1 g is set in motion from its equilibrium position with an initial velocity of 6in/sec, with no damping and a spring constant of k = 9. Find (a) the solution y for the displacement of the spring, (b) the time when the mass first returns to equilibrium (i. e. displacement y = 0), (c) it's velocity at the time you found in (b). slader

a) [m]

b) 0.033 s

c) -0.152 m/s

Step-by-step explanation:

a)

The force acting on the mass-spring system is (restoring force)

where

k = 9 is the spring constant

y is the displacement

Also, from Newton's second law of motion, we know that

where

m = 1 g = 0.001 kg is the mass

y'' is the acceleration

Combining the two equations,

This is a second order differential equation; the solution for y(t) is

where

A is the amplitude of motion

is the angular frequency

The spring starts its motion from its equilibrium position, this means that y=0 when t=0; therefore, the phase shift must be

So the displacement is

The velocity of the spring is equal to the derivative of the displacement:

We know that at t = 0, the initial velocity is 6 in/s; since 1 in = 2.54 cm = 0.0254 m,

And since at t = 0,

Then we have:

From which we find the amplitude:

So the solution for the displacement is

[m]

b)

Here we want to find the time t at which the mass returns to equilibrium, so the time t at which

This means that

We know already that the first time at which this occurs is

t = 0

Which is the beginning of the motion.

The next occurence of y = 0 is instead when

which means:

c)

As said in part a), the velocity of the mass-spring system at time t is given by the derivative of the displacement, so

where we have

is the angular frequency

is the amplitude of motion

t is the time

Here we want to find the velocity of the mass when the time is that calculated in part b):

t = 0.033 s

Substituting into the equation, we find:

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step-by-step explanation:

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