6. Non-Inertial Frames
We stated, long ago, that inertial frames provide the setting for Newtonian mechanics.
But what if you, one day, find yourself in a frame that is not inertial? For example,
suppose that every 24 hours you happen to spin around an axis which is 2500 miles
away. What would you feel? Or what if every year you spin around an axis 36 million
miles away? Would that have any e↵ect on your everyday life?
In this section we will discuss what Newton’s equations of motion look like in non-
inertial frames. Just as there are many ways that an animal can be not a dog, so
there are many ways in which a reference frame can be non-inertial. Here we will just
consider one type: reference frames that rotate. We’ll start with some basic concepts.
6.1 Rotating Frames
Let’s start with the inertial frame S drawn in the figure z=z
with coordinate axes x, y and z. Our goal is to understand
the motion of particles as seen in a non-inertial frame S 0 ,
with axes x0 , y 0 and z 0 , which is rotating with respect to S. y y
We’ll denote the angle between the x-axis of S and the x0 -
axis of S 0 as ✓. Since S 0 is rotating, we clearly have ✓ = ✓(t) x
and ✓˙ 6= 0. θ
x
Our first task is to find a way to describe the rotation of
Figure 31:
the axes. For this, we can use the angular velocity vector !
that we introduced in the last section to describe the motion of particles. Consider a
particle that is sitting stationary in the S 0 frame. Then, from the perspective of frame
S it will appear to be moving with velocity
ṙ = ! ⇥ r
˙ Recall that in general, |!| = ✓˙ is the angular speed,
where, in the present case, ! = ✓ẑ.
while the direction of ! is the axis of rotation, defined in a right-handed sense.
We can extend this description of the rotation of the axes of S 0 themselves. Let ei0 ,
i = 1, 2, 3 be the unit vectors that point along the x0 , y 0 and z 0 directions of S 0 . Then
these also rotate with velocity
ėi0 = ! ⇥ ei0
This will be the main formula that will allow us to understand motion in rotating
frames.
– 93 –
, 6.1.1 Velocity and Acceleration in a Rotating Frame
Consider now a particle which is no longer stuck in the S 0 frame, but moves on some
trajectory. We can measure the position of the particle in the inertial frame S, where,
using the summation convention, we write
r = ri ei
Here the unit vectors ei , with i = 1, 2, 3 point along the axes of S. Alternatively, we
can measure the position of the particle in frame S 0 , where the position is
r = ri0 ei0
Note that the position vector r is the same in both of these expressions: but the
coordinates ri and ri0 di↵er because they are measured with respect to di↵erent axes.
Now, we can compute an expression for the velocity of the particle. In frame S, it is
simply
ṙ = ṙi ei (6.1)
because the axes ei do not change with time. However, in the rotating frame S 0 , the
velocity of the particle is
ṙ = ṙi0 ei0 + ri0 ėi0
= ṙi0 ei0 + ri0 ! ⇥ ei0
= ṙi0 e0i + ! ⇥ r (6.2)
We’ll introduce a slightly novel notation to help highlight the physics hiding in these
two equations. We write the velocity of the particle as seen by an observer in frame S
as
✓ ◆
dr
= ṙi ei
dt S
Similarly, the velocity as seen by an observer in frame S 0 is just
✓ ◆
dr
= ṙi0 ei0
dt S 0
From equations (6.1) and (6.2), we see that the two observers measure di↵erent veloc-
ities,
✓ ◆ ✓ ◆
dr dr
= +!⇥r (6.3)
dt S dt S 0
This is not completely surprising: the di↵erence is just the relative velocity of the two
frames.
– 94 –
We stated, long ago, that inertial frames provide the setting for Newtonian mechanics.
But what if you, one day, find yourself in a frame that is not inertial? For example,
suppose that every 24 hours you happen to spin around an axis which is 2500 miles
away. What would you feel? Or what if every year you spin around an axis 36 million
miles away? Would that have any e↵ect on your everyday life?
In this section we will discuss what Newton’s equations of motion look like in non-
inertial frames. Just as there are many ways that an animal can be not a dog, so
there are many ways in which a reference frame can be non-inertial. Here we will just
consider one type: reference frames that rotate. We’ll start with some basic concepts.
6.1 Rotating Frames
Let’s start with the inertial frame S drawn in the figure z=z
with coordinate axes x, y and z. Our goal is to understand
the motion of particles as seen in a non-inertial frame S 0 ,
with axes x0 , y 0 and z 0 , which is rotating with respect to S. y y
We’ll denote the angle between the x-axis of S and the x0 -
axis of S 0 as ✓. Since S 0 is rotating, we clearly have ✓ = ✓(t) x
and ✓˙ 6= 0. θ
x
Our first task is to find a way to describe the rotation of
Figure 31:
the axes. For this, we can use the angular velocity vector !
that we introduced in the last section to describe the motion of particles. Consider a
particle that is sitting stationary in the S 0 frame. Then, from the perspective of frame
S it will appear to be moving with velocity
ṙ = ! ⇥ r
˙ Recall that in general, |!| = ✓˙ is the angular speed,
where, in the present case, ! = ✓ẑ.
while the direction of ! is the axis of rotation, defined in a right-handed sense.
We can extend this description of the rotation of the axes of S 0 themselves. Let ei0 ,
i = 1, 2, 3 be the unit vectors that point along the x0 , y 0 and z 0 directions of S 0 . Then
these also rotate with velocity
ėi0 = ! ⇥ ei0
This will be the main formula that will allow us to understand motion in rotating
frames.
– 93 –
, 6.1.1 Velocity and Acceleration in a Rotating Frame
Consider now a particle which is no longer stuck in the S 0 frame, but moves on some
trajectory. We can measure the position of the particle in the inertial frame S, where,
using the summation convention, we write
r = ri ei
Here the unit vectors ei , with i = 1, 2, 3 point along the axes of S. Alternatively, we
can measure the position of the particle in frame S 0 , where the position is
r = ri0 ei0
Note that the position vector r is the same in both of these expressions: but the
coordinates ri and ri0 di↵er because they are measured with respect to di↵erent axes.
Now, we can compute an expression for the velocity of the particle. In frame S, it is
simply
ṙ = ṙi ei (6.1)
because the axes ei do not change with time. However, in the rotating frame S 0 , the
velocity of the particle is
ṙ = ṙi0 ei0 + ri0 ėi0
= ṙi0 ei0 + ri0 ! ⇥ ei0
= ṙi0 e0i + ! ⇥ r (6.2)
We’ll introduce a slightly novel notation to help highlight the physics hiding in these
two equations. We write the velocity of the particle as seen by an observer in frame S
as
✓ ◆
dr
= ṙi ei
dt S
Similarly, the velocity as seen by an observer in frame S 0 is just
✓ ◆
dr
= ṙi0 ei0
dt S 0
From equations (6.1) and (6.2), we see that the two observers measure di↵erent veloc-
ities,
✓ ◆ ✓ ◆
dr dr
= +!⇥r (6.3)
dt S dt S 0
This is not completely surprising: the di↵erence is just the relative velocity of the two
frames.
– 94 –
