Solve Modeling Higher-Order Diff Eqn: Free Undamped Motion w/ Spring

[Matthewmccoy6]

A mass of 1 slug is suspended from a spring whose spring constant is 9lb/ft. The mass is initially released from a point 1 foot above the equilibrium position with an upward velocity of square root 3 ft/s. Find the times at which the mass is heading downward at a velocity of 3 ft/s.

I found the differential equation to be x(t)=Acos(3t) + Bsin(3t).

What I did was I took downward as positive while upward as negative. I'm having trouble finding t.

My equation for the motion was x(t)= -cos 3t - (square root 3/ 3) sin 3t

Please help, thanks!

 

[HallsofIvy]

A mass of 1 slug is suspended from a spring whose spring constant is 9lb/ft. The mass is initially released from a point 1 foot above the equilibrium position with an upward velocity of square root 3 ft/s. Find the times at which the mass is heading downward at a velocity of 3 ft/s.

I found the differential equation to be x(t)=Acos(3t) + Bsin(3t).

That is NOT a differential equation. I presume you mean you found that as the general solution to the differential equation.

What I did was I took downward as positive while upward as negative. I'm having trouble finding t.

My equation for the motion was x(t)= -cos 3t - (square root 3/ 3) sin 3t

Please help, thanks!

So you need to solve
$$-cos(3t)- (\sqrt{3}/3)sin(3t)= 3$$.
That is the same as
$$cos(3t)= 3- (\sqrt{3}/3)sin(3t)$$

Squaring both sides of that gives
$$cos^2(3t)= 9- (2\sqrt{3}/3)sin(3t)+ sin^2(3t)$$

Replace $cos^2(3t)$ with $1- sin^2(3t)$ and you have
$$1- sin^2(3t)= 9- (2\sqrt{3}/3)sin(3t)+ sin^2(3t)$$
a quadratic equation you can solve for sin(3t) and then for t.

Be sure to check if an "extraneous" solutions were introduced by the squaring.

 

[Matthewmccoy6]

I know, this is a differential equation problem though and using the equation of simple harmonic motion I found the above equation.
Also, I was hoping there was a simple identity that I could use to find t rather than something really unpleasant. Thanks though.