# Accelerated mass on a spring

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1. Aug 26, 2015

### einaap

Hi fellow physicists,
Suppose a spring with a stiffness k, is attached to wall and with the other side a block with a mass, m, a force F, then pulls the block away from the wall. How do you derive a function for acceleration of the block as a function of time, a(t)?
When trying to solve this I derived the equation: (F-K*s)/m=a but I dont know how to derive a function of s (displacement) as a function of t (to substitute for s in previous equation). The problem seems to be that s(t) is in its turn again a function acceleration which is a function of how far the spring is stretched which is a function of displacement, s :)
Does anyone know how to do this? Thanks

2. Aug 26, 2015

### paisiello2

Yes, you are right, acceleration is a function of displacement but you should know what this function is from kinematics.

3. Aug 26, 2015

### Chandra Prayaga

There should be no F in the equation that you wrote. The only unbalanced force is -ks and this should be equal to ma, So your equation should read -ks/m = a. As you can see, and as paisiello2 pointed out, this shows that the acceleration depends on s. This is a second order differential equation whose solution gives you s as a function of t.

4. Aug 26, 2015

### paisiello2

No, I think F is an externally applied force so there is an F in the equation.

5. Aug 26, 2015

### Chandra Prayaga

Agreed. But if there is an extra applied force in addition to -kx, then it may no longer be simple harmonic motion. The solution of s as a function of t will depend on what F is. for example, if the spring and object are hanging vertically from the ceiling, F would be gravity, and you would still have simple harmonic motion. If F is friction, it would damped harmonic motion, and if F is itself an oscillatory motion, then you would have a driven harmonic oscillator.

6. Aug 27, 2015

### Korak Biswas

Replace acceleration 'a' with $\frac{d ^2s}{dt^2}$ in your equation and solve this differential equation. Then you will find s as a function of t. Then differentiate it twice. You will get the answer. But at first you have to know about the nature of external force F.