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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 4.30 cm that lies in an east-west vertical plane. The robot then moves the object upward and to the north, through a quarter of a circle of radius 2.80 cm that lies in a north-south vertical plane.

(a) Find the magnitude of the total displacement of the object. (cm)


(b) Find the angle the total displacement makes with the vertical

Explanation / Answer

here,

calculation X and Y components with arc radius 4.3cm

total X = 4.3 cm
total Y = 4.3 + 2.8 = 7.1 cm

The coordinate system being referenced is positive x is to the left ,Positive y is up with respect to the paper ,Positive z is coming out of the paper.
so, total Z = 2.3 cm

calculating resulatant

C = sqrt(4.3^2 + 7.1^2 + 2.3^2)

the magnitude of displacement is 8.61 cm

Now ,

a = 4.3i+7.1j+2.3k
|a| = 8.61

b(0,1,0) unit vector corresponding to vertical

Cos a = a*b / |a||b|
Cos a = 7.1 / |8.61||1|

angle the total displacement makes with the vertical is 34.915 degrees

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