The first lecture saw that the shadow of a point going round a circle gives the position in simple harmonic motion. That same point also has a velocity. In uniform circular motion the velocity points along the tangent and keeps a constant size of .
Shine a light straight down on that velocity vector too. The shadow cast below is the velocity of the weight, .
At the ends the tangent points straight up or straight down, so the shadow has length . At the centre the tangent is horizontal, so the shadow is at its longest. The weight is at rest at the ends and fastest at the centre. The picture of the circle says it outright.
The acceleration works the same way. In uniform circular motion it always points at the centre of the circle, with size .
Cast its shadow and it becomes . Since the position was , this may equally be written . We have come straight back to the equation of motion set up in the first lecture.
Look at the direction as well. The shadow of a vector pointing at the centre always points toward the origin , the same way as the arrow of the restoring force. With it could hardly be otherwise.
Stack position, velocity and acceleration one above the other on a common time axis.
The velocity runs a quarter of a period ahead of the position. When the position is at an end the velocity is ; when the position is at the centre the velocity is greatest. The acceleration points exactly opposite to the position, half a period out of step.
Differentiate with respect to time and you get ; differentiate and you get . Reasoning from the shadow of a circle or differentiating gives the same expressions.
Put both the velocity arrow and the acceleration arrow on the same weight. As one grows, the other shrinks.
At the ends () the velocity is . But the acceleration is , its largest, and so is the force. The weight seems to pause for an instant, and it is exactly where it is being pulled back hardest.
At the centre () it is the other way round. The acceleration and the force are , while the speed is , its largest. No force acting, and moving fastest. A great many people trip over this. What sets the speed is the accumulation of the forces received up to that moment, not the force at that instant.