# Degrees of Freedom: What's the Definition?

• rcummings89
In summary, when considering a rolling ball, there is only one degree of freedom due to the relationship between rotation and displacement along the x-axis. Similarly, a hanging mass on a spring also has one degree of freedom, although this may not be the case in a realistic setup due to additional sideways motion. It is important to be aware of the idealization and simplifying assumptions when studying these systems.
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!

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.

rcummings89 said:
Hello,
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)?
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.

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

## 1. What is the definition of degrees of freedom?

Degrees of freedom is a statistical concept that refers to the number of independent variables or components that can vary in a system or dataset without affecting the value of a specific outcome or measurement.

## 2. How are degrees of freedom calculated?

The calculation of degrees of freedom depends on the specific statistical test or model being used. In general, it is the number of observations or data points in a sample minus the number of parameters being estimated.

## 3. Why is degrees of freedom important?

Degrees of freedom is important in statistical analysis because it affects the accuracy and reliability of the results. A higher number of degrees of freedom can lead to more precise estimates, while a lower number can result in inflated or biased results.

## 4. How does degrees of freedom relate to sample size?

As the sample size increases, the degrees of freedom also increase. This is because a larger sample size provides more data points, which allows for a more accurate estimation of the parameters being tested.

## 5. What is the significance of degrees of freedom in hypothesis testing?

In hypothesis testing, degrees of freedom are used to determine the critical value for a specific level of significance. This critical value is then compared to the test statistic to determine if the null hypothesis can be rejected.

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