Is it possible to solve spring questions with Calculus?

[docholliday]

I found this on a previous thread and someone mentions that you can use calculus to solve it. I already solved it using the law of conservation and got .027816 m or 2.7 cm, but it's driving me nuts figuring out how to use calculus to solve.

How do I go about integrating acceleration since it's dependent on distance? Please give any insight.

1. A block of mass 0.3 kg and spring constant 24 N/m is on a frictionless surface. If the block is set into motion when compressed 3.5 cm, what is the maximum velocity of the block? How much is the spring compressed when the block has a velocity of 0.19 m/s?

 

[Nugatory]

I found this on a previous thread and someone mentions that you can use calculus to solve it. I already solved it using the law of conservation and got .027816 m or 2.7 cm, but it's driving me nuts figuring out how to use calculus to solve.

How do I go about integrating acceleration since it's dependent on distance? Please give any insight.

1. A block of mass 0.3 kg and spring constant 24 N/m is on a frictionless surface. If the block is set into motion when compressed 3.5 cm, what is the maximum velocity of the block? How much is the spring compressed when the block has a velocity of 0.19 m/s?

Google "simple harmonic oscillator" to find the standard calculus-based solution.

And for a quick preview... You'll start with F=ma, where F is just the spring constant k times the displacement x, and the acceleration a is the second derivative (this is where the calculus comes in) of the displacement with respect to time:
$$kx=m\frac{d^2x}{dt^2}$$

Now try plugging in $x=sin(At+B)$, use the initial conditions, solve for A and B, and you'll have the displacement as a function of time. And of course the velocity as a function of time will be the first derivative of that.