At t = 0 a block with mass M = 5 kg moves with a velocity v = 2 m/s at position xo = -.33 m from the equilibrium position of the spring. The block is attached to a massless spring of spring constant k = 61.2 N/m and slides on a frictionless surface. At what time will the block next pass x = 0, the place where the spring is unstretched?
t1 =

An explanation would be awesome if possible.

Question

At t = 0 a block with mass M = 5 kg moves with a velocity v = 2 m/s at position xo = -.33 m from the equilibrium position of the spring. The block is attached to a massless spring of spring constant k = 61.2 N/m and slides on a frictionless surface. At what time will the block next pass x = 0, the place where the spring is unstretched?
t1 =

An explanation would be awesome if possible.

anonymous · Answer

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anonymous · Answer

The mass-spring system is performing a simple harmonic oscillatory motion.
as given, at t = 0, the mass-spring system is not at its relaxed position. 
hence, in the S.H.M equations we will have to introduce some phase lag.

Now the equations for S.H.M motion for the mass-spring system is given by
$$x = Asin( wt - \delta)$$

$$\delta$$ marks the phase lag because at t = 0, the spring is unstretched. 
A is the amplitude of the motion.
w is the angular frequency = $$\sqrt{k/m}$$ = 3.5

and the corresponding equation for velocity can be obtained by differentiating the above eq. w.r.t. time.
$$v = Awcos(wt - \delta)$$

Now we can play with our question.
at t = 0, x = -0.33, v = 2
$$-0.33 = -Asin \delta$$     
$$2 = Awcos \delta$$          

dividing top equation by bottom equation we get,

$$0.33*w/2 = \tan \delta$$ = 0.523 radian

$$\delta = \tan^{-1} (0.33 * w/2)$$

this way we get our delta 

now everything is easy-peesy,

we need to find out when will we next reach x = 0, 
just look at our x equation again

$$x = Asin (wt - \delta)$$

the first instance where x = 0 will be -
$$wt - \delta = 0$$

$$t = \delta/w$$
  = 0.523/3.5
  = 0.15 seconds

anonymous · Answer

@zaphodplaysitsafe you're a physics beast.

anonymous · Answer

Thanks @kobesaurus, I really love physics. talking of beasts, your name suggests, you are quite a beast yourself :)

anonymous · Answer

@zaphodplaysitsafe Thats dope! 200 million years and I still struggle with Physics unfortunately.

anonymous · Answer

we all do mate, there is so much to understand. I blame our short lifespan :)

anonymous · Answer

@zaphodplaysitsafe in time I will acquire your Jedi like skills on the Physics front. Thanks again for all the help I really appreciate it!

anonymous · Answer

@kobesaurus may the force be with you.  force sounds so appropriate right now :P