# Spring - 3 Differential Equations

#### joshuaa

The book has already answered those three problems. The book took stretching the spring downward as a positive displacement.
When the problems were solved, I was confused why it took velocity negative in all of them

at time = 0, velocity was negative in problem 1, 2, and 3. Why?

1. A spring is stretched 20 cm by a 4-kg mass. The weight is released with a downward velocity of 2 m/s. Neglect damping. Find an equation for the position of the spring at any time t and graph the position function.

2. A spring is stretched 10 cm by a 4-kg mass. The weight is pulled down an additional 20 cm and released with an upward velocity of 4 m/s. Neglect damping. Find an equation for the position of the spring at any time t and graph the position function. Find the amplitude and phase shift of the motion.

3. A spring is stretched 2 inches by a 6-pound weight. The weight is then pulled down an additional 4 inches and released with a downward velocity of 4 ft/s. Neglect damping. Find an equation for the position of the spring at any time t and graph the position function. Find the amplitude and phase shift of the motion.

#### topsquark

Forum Staff
Without knowing what the book is doing, I can only guess. For a problem with both a spring and a weight we have to set two origins and positive directions. It would seem that we have an origin for the spring at it's equilibrium point (the usual choice) and positive direction in the direction of a compression (also the usual choice.) So when we stretch the spring we say it's a negative displacement.

The choice of positive direction for the vertical axis is not part of your question. Presumably the choice will be for the origin at the spring's equilibrium point and postive upward.

-Dan

• joshuaa

#### joshuaa

Hey Dan,

If you would solve the three problems, what would your choice be for the three velocities?

1. Negative?
2. Positive ?
3. Negative?

#### topsquark

Forum Staff
Hey Dan,

If you would solve the three problems, what would your choice be for the three velocities?

1. Negative?
2. Positive ?
3. Negative?
Using the coordinate system I mentioned in post #2, yes to all three. It doesn't really matter what you choose, just as long as you sketch in (or write in) your choice of positive direction and be consistent. Check with your instructor, though. I've run into Professors that insist that the positive direction for the spring is always a compression.

-Dan

• joshuaa

#### joshuaa

that makes sense. thanks Dan.