Vector Displacement Calculation for Assembly Operation | Homework Solution

[blob84]

Homework Statement

In an assembly operation illustrated in Figure P1.49, a ro-bot moves an object first straight upward and then also tothe east, around an arc forming one quarter of a circle of radius 4.80cm that lies in an east–west vertical plane. Therobot then moves the object upward and to the north,through a quarter of a circle of radius 3.70cm that lies in anorth–south vertical plane. Find (a) the magnitude of thetotal displacement of the object and (b) the angle the totaldisplacement makes with the vertical.
http://img62.imageshack.us/img62/2459/assemplaggio.jpg

Homework Equations

The Attempt at a Solution

$$C_{1}/4=2πr=2π(4.80cm)/4=7.55$$
$$A^→=(7.55,0)cm$$
$$C_{2}/4=2πr=2π(3.70cm)/4=5.81$$
$$B^→=(0, 5.81)cm$$
$$R^→=A^→+B^→=(7.55, 5.81)cm$$
$$R=sqrt(7.55^2+5.81^2)=9.52cm$$
$$σ=37.6°$$;
the results is 10.4 cm and 35.5°.

 

[pongo38]

Your C1 and C2 appear to be curved lengths. Does Pythagoras theorem apply to curved lengths?

 

[blob84]

no, but i have no idea how to solve.

 

[dikmikkel]

How about applying a spherical polar coordinate system and integrate over a line describing the path i.e. do two separate integrations and add them up.

 

[asteriatic]

I would like to clarify that the provided solution is incorrect. The vector displacement calculation should take into account the direction of movement in each quarter circle. In the first quarter circle, the robot moves upward and then east, which means the vector should have a magnitude of 4.80cm and an angle of 45 degrees with the vertical. In the second quarter circle, the robot moves upward and then north, which means the vector should have a magnitude of 3.70cm and an angle of 45 degrees with the vertical.
Therefore, the total displacement vector should be (4.80√2, 4.80√2)cm + (3.70√2, 3.70√2)cm = (8.53, 8.53)cm.
The magnitude of the total displacement is 12.06cm and the angle with the vertical is 45 degrees.
It is important to carefully consider the direction of movement in each quarter circle in order to accurately calculate the total displacement vector.