Displacement

Question

A man pushing a mop across a ?oor causes the mop to undergo two displacements. The ?rst has a magnitude of 239 cm and makes an angle of 165? with the positive x-axis. The resultant displacement has a magnitude of 180 cm and is directed at an angle of 50.3? to the positive x-axis.

 

Find the magnitude of the second displacement.

 

 

Find the direction of the second displacement with respect to positive x axis. (Consider counterclockwise to be positive and answer within the limits of −180? to 180?.)

Answer 1

This is a vector equation: V1 + V2 = R

And you want to find V2.

 

The easiest way to solve this is by using the Fact that you can solve both

the X part of the problem and the Y part of the problem independently (separately).

 

So first decompose the V1 and R vectors.

Then solve the independent X and X problems.

   V2x = Rx - V1x and V2y = Ry - V1y

<when the angle is specified to CCW from the + X axis

   cos is always the x part and sin is always the y part.>

 

V2 = V2x + V2y

    = V2(cosθ) + V2y(sinθ)

    = 239(cos(165)) + 239(sin(165))

    = ?  + ?  (You can do this part? - Right?

 

R = Rx + Ry

= Rx(cosθ) + Ry(sinθ)

= 180(cos(50.3)) + 180(sin(50.3))

= ? + ? (You can do this part too? - Right?

 

Now that you know the X and Y components of both the V1 and R vectors

you can solve the equations V2x = Rx - V1x and V2y = Ry - V1y

to find the X and Y parts of the V2 vector

 

Now you need to put the V2 vector back together (resolve it).

It is a right triangle so |V2| = √( (V2_x)² + (V2_y)² )

Also θ = tan^-1 ( V2y / V2x )