In an assembly operation illustrated in the figure, a robot moves an object first straight upward and then also to the east around an arc forming one-quarter of a circle of radius 4.80 cm that lies in an east-west vertical plane. The robot then moves the object upward and to the north through one-quarter of a circle of radius 3.70 cm that lies in a north- south vertical plane. Find (a) the magnitude of the total displacement of the object and (b) the angle the total displacement makes with the vertical.
When adding vectors in 3D, the steps are basically the same as adding vectors in 2D, except that drawing diagrams for 3D problems is often skipped because it's more complicated to do.
For step 1, we'll usually use equations of this form to find our components when the components aren't given to us:
Steps 2 and 3 can be done using a table.
For step 4, we'll use the same equations from step 1 solved for α, plus the equation for magnitude:
This problem looks tricky at first because it's very wordy! But let's tackle it one piece at a time:
Step 1. Notice that the first displacement, let's call it , is a quarter-circle that starts with a motion straight upwards. The path is along an arc, but the displacement is from the starting point at the edge of the arc to the final point. If we imagine an origin (0,0) at the center of the arc (or sketch it out that way), it's easy to see that the initial position vector is horizontal, the final position vector is vertical, and both of them have magnitudes equal to the radius of the circle (4.80 cm).
Knowing that, we don't have to do any math to find the components: we can just say
We can use the same reasoning for the second displacement, . We know it's a quarter-circle up and to the north with radius 3.70 cm, so