# Degrees of freedom

1. Mar 23, 2014

### rcummings89

Hello,

I just want to make sure I am understanding how to define degrees of freedom of an object.

If you have a rolling ball moving strictly in the x-direction, it has two degrees of freedom: one from its rotation, and one from its translation? Or is it just one DoF because its translation is proportional to the angle of rotation (x = RΘ)?

Also a hanging mass on a spring only has one degree of freedom because it can only translate in the y-direction?

Thanks!

2. Mar 23, 2014

### euquila

Assuming the ball does not slip, then there is only one degree of freedom because a change in angle of the ball corresponds exactly to how much the ball displaced along the x axis according to the relationship x = R * theta. The rotation of the ball and its displacement along the x-axis are not independent of each other.

A mass/spring system also only has one degree of freedom because it can only displace in one direction.

3. Mar 23, 2014

### Staff: Mentor

if it's rolling without slipping, then there's only one degree of freedom. If it's also slipping, the there are two degrees of freedom as you expect.

in practice it's very difficult to hang a weight in such a way that it doesn't swing sideways like a pendulum, as least a little bit. So although you can idealize the problem down to one degree of freedom along one axis, any realistic setup will either have to constrain the motion (a vertically oriented track, for example) or will pick up two more degrees of freedom from the two sideways directions. When you're learning, it's generally best to focus on the idealized setup - just try to remain aware of the simplifying assumptions that go into the idealization.