A body has non-uniform circular motion when its trajectory is a circumference and its angular acceleration is constant. In this section, we are going to study:

Non-uniform circular motion graphs

Angular position - time (φ-t) graph

φ=φ0+ω·t+12·α·t2

The angular position–time graph (φ–t) of a non-uniform circular motion represents time on the horizontal axis (t-axis) and angular position on the vertical axis (φ-axis). The angular position, φ, measured in radians in the International System (S.I.), increases (or decreases) in a non-uniform manner with the passage of time. We can distinguish two cases, depending on whether the angular velocity is positive or negative:

angular position - time graph in non uniform circular motion.

 

Angular velocity - time (ω-t) graph

ω=ω0+α·t

The angular velocity–time graph (ω–t) of a non-uniform circular motion (u.a.c.m.) represents time on the horizontal axis (t-axis) and angular velocity on the vertical axis (ω-axis). Angular velocity, measured in the S.I. in radians per second (rad/s), increases (or decreases) in a uniform manner with the passage of time. This is due to the action of the acceleration. Again, we can distinguish two cases:

angular velocity - time graph in non uniform circular motion

We can get the acceleration from the angle θ. To do it just remember that in a right triangle the tangent of each of its angles is defined as the opposite cathetus divided by the adjacent one:

tanθ=opposite cathetusadjacent cathetus=ωt=ω-ω0t=α

The value of the slope is the value of the angular acceleration. So, the greater the slope of the straight line, the higher the angular acceleration of the body.

Notice that the area under the curve ω between two instants of time is numerically the same as the distance traveled. Can you tell why?

angular distance traveled in non-uniform circular motion as are under the angular velocity curve

The area under the curve can be calculated as the area of the rectangle S1 that would correspond to a uniform circular motion (u.c.m.) to which we will add the area of the triangle S2:

φ=φ-φ0=S1+S2=1ω0t+ω-ω0t2=2ω0t+12αt2

Where we have applied:

  • S1=ω0tS2=Srectangle2=ω-ω0t2
  • ω-ω0=αt

Angular acceleration - time (α-t) graph

α=constant

The angular acceleration–time graph (α–t) of a non-uniform circular motion shows that the angular acceleration, measured in the international system (S.I.) in radians per second squared (rad/s2), is always constant. We can distinguish two cases:

angular acceleration - time graph in non-uniform circular motion

Notice that the area under the curve α, limited by two instants of time, is numerically the same as the experienced increase in the angular velocity. Can you tell why?

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Formulas worksheet

Here is a full list of formulas for the section Non-Uniform Circular Motion Graphs. By understanding each equation, you will be able to solve any problem that you may encounter at this level.

Click on the icon   to export them to any compatible external program.

Angular position in non-u.c.m.

φ=φ0+ω·t+12·α·t2

Angular velocity in non-u.c.m.

ω=ω0+α·t

Angular acceleration in non-u.c.m.

α=constant

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