Frame of Reference

Frame of Reference Characteristics

The great thing about a frame of reference is that you can define all the characteristics! Don't get me wrong; you still have to obey all the laws of physics. You can't just say, "I'm going to define Newton's Second Law to be invalid!" You can, however, define the characteristics of your system to benefit you the most and make the most sense for you. For example, maybe you dropped an object above a fan blowing upward, causing the object to rise into the air. Seeing how the object is moving upward, it might not be such a bad idea to define the upward direction as negative to see how much the force of the fan is counteracting gravity!

For example, in Fig. 2 below, we choose to define the up direction as positive. The free-body diagram off to the left shows that the force of the fan is much greater than the gravity of the ball. By defining upwards as positive, we can subtract \(F_g \) from \(F_\text{fan}\) and figure out how much the fan counteracts gravity.

Fig. 1 - In this problem, we define up in the \(y\) direction as positive

The Observer

We cannot talk about reference frames without talking about observers. Physics is all about how objects in the universe interact with each other. However, depending on location, one person might see an interaction differently than another.

For example, a person standing on the ground may look in the sky and see a plane. Then, two minutes later, that person looks up again and sees the same plane in the sky. "It didn't travel that far," he may think to himself because he can still see it. People in the plane, however, would think differently. In those short two minutes, they would pass the airport, maybe some mountains, and probably an entire city. The location from which an interaction is observed can mean everything when describing how it all happened.

Perspective is key. Similar to how people can get into arguments because they see things differently, if a proper frame of reference is not defined, physicists will see a problem differently and come up with diverse answers. Therefore, by choosing a reference frame, one is also choosing to define the direction and magnitude of the parts of the physical system measured by an observer in that frame of reference.

Inertial Reference Frames

Recognize that measurements within a given reference frame may be converted to measurements within another reference frame.

Therefore, using formulas and calculations, different reference frames can be compared and related to each other. Think of unit conversions: to convert from meters to kilometers, you divide by \(1000\). Converting measurements in one frame of reference to another is no different. However, the conversion process is often more complex.

There are two types of reference frames where these conversions can take place: inertial and non-inertial.

Inertial Reference Frames

Inertial reference frames assume that acceleration is constant and Newton's first law is directly applicable. Remember that Newton's first law is that an object in motion will remain in motion unless acted upon by an outside force: sometimes, we refer to this fundamental physics principle as the law of inertia because it has everything to do with an object's resistance to a change in motion.

So, what does it mean for Newton's first law to be directly applicable? The first phrase, "an object in motion will remain in motion," identifies an inertial reference frame. Any frame of reference with no external net force acting on it is an inertial reference frame because its acceleration will remain constant relative to all other frames.

Non-inertial Reference Frames

Non-inertial reference frames address the second part of Newton's first law: "unless acted upon by an outside force." These frames of reference have an acceleration with respect to an inertial frame - or in other words, an accelerator at rest in a non-inertial frame of reference would detect a non-zero acceleration. This results in classical mechanics explaining the motion of bodies using fictitious forces (also known as pseudo-forces), such as centrifugal force.

Frame of Reference and Motion

Frames of reference help us to understand motion better. Although every physicist would like it to be, the real world is not ideal. In a world designed for simple analysis of motion, every object and person would take its turn to move. Can you imagine?

The reality is that everything is moving relative to each other. Frames of reference allow us to "frame" motion with respect to other things that are moving.

I know, pretty confusing, right? There's a reason that physicists did not unlock how relative motion and the speed of light get along until Albert Einstein. Remember how we talked about converting the measuring of quantities from one reference frame to another? Let's say you were moving with some velocity in a car. To calculate your velocity relative to another car's velocity, you simply add the observed car's velocity to your own.

Therefore, an observed object's velocity is found by adding the observed object's velocity and the observer's velocity. This is done through the addition or subtraction of vectors. This addition or subtraction looks something like this:

$${\vec v_a}_b= ({\vec v_a}_d+ {\vec v_d}_b).$$

To translate this formula into laymen's terms, it means, "The velocity of object \(a\) in \(b\)'s frame of reference is equal to the velocity of \(a\) in \(d\)'s frame of reference plus the velocity of \(d\) in \(b\)'s frame of reference."

Frame of Reference Examples

Now that our head is spinning, let's try to straighten it out by actually applying this to the real world.

Example 1

First, we'll begin with adding relative velocities.

Example 2

Now let's try a problem with the subtraction of relative vector velocities.