# Find the Maximum speed

1. Dec 6, 2009

### Dark Visitor

The motion of a piston in an automobile engine is nearly simple harmonic. If the 1 kg piston travels back and forth over a total distance of 10 cm, what is the maximum speed (magnitude of velocity) when the engine is running at 3000 rpm? (Hint: the 10 cm distance has something to do with amplitude; 3000 rpm needs to be converted to rad/sec)

I did convert the 3000 rpm, and I got 314.159 rad/sec.

I need some help with this problem a.s.a.p. I don't know where to go with it. Thanks for any help you can give me.

2. Dec 6, 2009

### rock.freak667

What formula do you think you will need here?

3. Dec 6, 2009

### Dark Visitor

I don't know, and I don't know any way to find A. I keep thinking that A has something to do with the 3000 rpm, but I am not sure.

4. Dec 6, 2009

### diazona

What formula(s) do you know that would allow you to find the maximum velocity?

5. Dec 6, 2009

### rock.freak667

A is the amplitude. If the piston moves, from minimum to maximum, a distance of 10cm, what distance does it move when it goes from 0 to maximum?

6. Dec 6, 2009

### Dark Visitor

5 cm? Cause 10 cm would be the total length, while +A is half, and -A is the other half.

7. Dec 6, 2009

### rock.freak667

Right good. Now can you formulate an equation for speed given kinetic energy is 1/2mv2 and potential energy is 1/2kx2. What is the formula for the total energy?

8. Dec 6, 2009

### Dark Visitor

E = K + U

(K is kinetic energy, u is potential energy)

But using the equations you gave, I am missing most of the quantities. How can I use them?

9. Dec 6, 2009

### rock.freak667

I told you K and U, if you get an expression for E, you will get what you need with the quantities you know.

10. Dec 6, 2009

### Dark Visitor

But for K, I need a velocity, which I don't have.

And for U, I need a spring constant value for k, but we have x (10 cm).

11. Dec 6, 2009

### rock.freak667

you don't need to plug in all the values at one time. Just formulate and then see if you can get the terms needed.

what is E in terms of k and A?

12. Dec 6, 2009

### Dark Visitor

Well since I don't have all the values I need, I get:

K = 1/2(1 kg)(v)2

and

U = 1/2(k)(.01 m)2

and then we still have E = K + U

13. Dec 6, 2009

### rock.freak667

do not put in the values as yet.

$$E=\frac{1}{2}mv^2+\frac{1}{2}kx^2$$

Do you know another way to re-write E in terms of k and A?

14. Dec 6, 2009

### denverdoc

I think what is being asked for is an equation that resembles the other oscillators.

x(t)=A Sin(argument in t) where your argument has you hitting both extremes at a rate of 50 times a second, say by multiplying t by something like 100 * pi (314.15)
Now remember to get the max velocity, you need to know when acceleration is zero.
The velocity is -wA sin(wt). sin(wt) is at most 1.0. Hope this helps, I've been at this most of the day and am burnt to a crisp.

15. Dec 6, 2009

### Dark Visitor

So do I use the equation:

vmax = wA ? Or what? I am really confused on this one...

16. Dec 6, 2009

### denverdoc

YES! Vmax=wA. Good job. Sorry if I'm a little fried and testy.

17. Dec 6, 2009

### rock.freak667

yes, had you derived it, you would get $v_{max}=\omega A$

18. Dec 6, 2009

### Dark Visitor

No, don't be. I owe you big time for all of your help today, and I would be the same way if I was helping someone like me.

Now that I got that right, where do I get my w and A values?

19. Dec 6, 2009

### rock.freak667

A is the amplitude that you found, and w is the angular velocity you converted in the first post.

20. Dec 6, 2009

### Dark Visitor

Okay, and I got: