Have you read part 1 of this post?

Without Calculus

- It’s possible to figure out that velocity is cosine and acceleration is negative sine without actually differentiating (although you don’t get the actual equations)
- To help explain, I’ve sketched you some graphs and
~~scanned them in, like a professional~~taken photos of them with my phone - The first shows how the displacement varies. I’ve forgotten to actually label the axes, but I’ve annotated the peaks and some of the x-intercepts to explain the displacement, velocity and acceleration

- At the first peak, displacement is at its maximum positive value
- Here, the pendulum (for example) stops swinging the way it was going and starts to go back the other way, so the velocity at that instant is zero
- The acceleration is at its maximum negative value, because the acceleration is proportional to the displacement but in the opposite direction
- At the x-intercept in the middle of the graph, displacement is zero
- The object is now up to its full speed, in the negative direction. (Pendulums are fastest in the middle)
- The acceleration is zero, because it’s at the point of zero displacement
- At the negative peak, the displacement is at its maximum negative value
- Again, the velocity is zero because the oscillator has stopped and is about to go back the other way
- The acceleration is fully positive, because the displacement is fully negative
- At the x-intercept on the right of the graph, the displacement and acceleration are zero (like at the other x-intercept), but the velocity is positive
- The oscillator is returning from negative displacement to enter positive displacement. (When a pendulum is in the middle, it’s sometimes swinging left to right and sometimes swinging right to left)

[Graph Cancelled]

- I won’t draw the other two graphs (velocity and acceleration) because I don’t have time
- (I have a lot more Physics to revise today)
- I could have done it by drawing some empty axes and putting dots where the velocity would be at its maximum positive value, its maximum negative value and where it would cross the axes
- These would be worked out by examining the peaks and x-intercepts on the displacement graph
- I could then draw a curve through the points, knowing that it would form a phase-shifted sine curve, and I’d end up with a cosine graph of velocity and time
- I could do the same thing for acceleration – work out where it will be at its greatest positive and negative values and where it will be zero, then draw a curve through it

Useful Points

- No matter what’s in the brackets, a sine or a cosine always evaluates to something between -1 and 1 (inclusive)
- If you’re asked for the maximum displacement, velocity or acceleration but not given all the information (such as what time it occurs), you can simplify your equation by assuming that the sine or cosine evaluates to 1 since that will produce a maximum
- The same goes for the maximum displacement, velocity or acceleration in the negative direction – replace the sine or cosine with -1 (but the magnitude will be the same as the maximum in the positive direction anyway)
- The gradient of the displacement graph is the velocity at that point and the gradient of the velocity graph is the displacement at that point
- You can draw a tangent and measure the gradient instead of differentiating and substituting, but it’s less accurate
- The graphs are steepest at the points where they cross the x-axis, so draw tangents there to obtain maximum gradients
- The line from a peak to the next point where the line crosses the x-axis has the average gradient for that interval
- The displacement could be drawn as a cosine curve. In this case, the behaviour is the same, but the timer has been started when displacement is maximum instead of when displacement is zero. The velocity would be a negative sine and the acceleration would be a negative cosine
- The important thing is to get the phase shifts right. Think of the either rules of differentiation or what the gradient is at various points
- Don’t forget radians mode!

Data, Formulae and Relationships Booklet

- No formulae in this part of the post!

There’s also a part 3 of this post…

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