Centripetal acceleration in UCM
This activity belongs to the GeoGebra book The Domain of the Time.
In the UCM activity, we saw how the mass m, represented by point M, moved in uniform circular motion (UCM) around point O, that is, at a distance r with a constant angular velocity ω. There was also a tangential velocity v, whose magnitude is the constant ω r.
However, the fact that v has a constant magnitude does not mean that velocity v is constant, since its direction is not. This means that there must be a force (a rigid body, the tension of a rope, gravity, a magnetic force...) that forces mass m to maintain the circular motion. Otherwise, as we've seen, it would follow a uniform rectilinear motion due to inertia.
This force is known as centripetal force, because its direction are toward the center of the circle. This force causes a centripetal acceleration c, represented by the green vector. The magnitude of this acceleration is exactly what is needed to keep the mass in circular motion and prevent it from continuing in a straight line by inertia.
If we place ourselves at point M, we will feel a force that seems to pull us away from the circular path, as if we were "get off on a tangent". This apparent (i.e., fictitious) force is called the "centrifugal force", but it is simply our perception of the resistance inertia offers to the real centripetal force.
To better observe the relationship between acceleration c and velocity v, activate the "View variation of v" box (a diagram known as the hodograph of the motion).
- Note: In the hodograph, point A travels 2π|v| with each lap, that is, every T = 2π/ω segundos. seconds. As A moves at speed c (acceleration is the rate of change of velocity), we have |c| = 2π|v|/T = ω |v| = ω2 r = v2/r.
SCRIPT FOR SLIDER anima
# Calculate the elapsed seconds dt; add one second if t1(1) < tt
SetValue(tt, t1(1))
SetValue(t1, First(GetTime(), 3))
SetValue(dt, (t1(1) < tt) + (t1(1) − tt)/1000)
# Register the lap time and the number of laps completed
SetValue(reg, If(x(v)>0 ∧ x(v + dt c) ≤ 0, Append(t, reg), reg))
SetValue(laps, If(x(v) > 0 ∧ x(v + dt c) ≤ 0, laps + 1, laps))
# Move M
SetValue(v, v + dt c)
SetValue(M, M + dt v)
Author of the activity and GeoGebra construction: Rafael Losada.