An object of mass *m* is hung on the end of a spring of relaxed
length *l* and spring constant *k. *The object is pulled down
until the stretch of the spring is *s*_{o}*. *What is
the length of the spring when the object next comes to rest? (i.e. how
high does the object rise before it comes to rest? What is the maximum
speed that the object reaches, and where does it occur?

**Diagrams.**

**Initial state ("before")
Object at max height**
**Object at max speed**

Mark the reference level you choose for gravitational PE *clearly
*on
the diagrams.

**Set-up of solution**

The objects** **in the system** **are:

The forms of mechanical energy that this system possesses are :

The kinetic energy is greatest when the ________________________________is least.

Fill in the first two columns of the table below. If you don't know the value of an entry in the top 3 rows, give it a name (algebraic symbol).

Quantity |
Before(initial state) |
Object at max height |
Object at max speed |

stretch of spring | |||

height of object above reference level | |||

speed of object | |||

___________________ | _____________________ | _________________ | _________________ |

Spring PE | |||

Gravitational PE | |||

Kinetic energy | |||

_____________________ | _________________ | _________________ | |

Total mechanical energy |

**Solution to first problem.**

Set the total energy in the before state equal to the total energy in the "max height" state. You will get one equation so you can solve for only one unknown. If you have more than one unknown in your equation, go back and see if you can fix something in your table. Your answer should contain the given quantities:

*m*, *k , g, l , s*_{o}

**Solution to second problem**

The second part of the problem is harder. We need to fmd where the PE
is least. So first write an expression for the PE in terms of the stretch
of the spring *s*. Note this is not *s*_{o} or *s*_{f
}but
just some arbitrary, so far unknown, value of *s*. Then find the derivative
d*U*/d*s* and set it equal to zero to find the value of *s*
that gives the minimum PE. Check that it really is a minimum (not a maximum,
for example). Look at the answer -does it make sense? Once you have the
answer, go back to your table, and fill in the final column. Set the totals
in the bottom row equal, and solve for *v*_{max}.