Direction and force of piano.

Question

Two people are pushing a piano. One person pushes it with 175 pounds at an angle of 60 degrees To the x axis while the other pushes it with 250 pound at an angle of 30 degrees To the x axis. In what direction does the piano move and with how much force?

Answer 1

 The way I like to approach problems like these is to draw the picture. Imagine yourself as a bird flying over these two people pushing the piano. When you look down, the piano is right in the middle of your view. That's your origin point in your xy-plane. The first person is pushing the piano with 175 pounds of force at an angle of 60 degrees; you might know from Trigonometry that this angle should be more vertical than horizontal (check a unit circle if you need to review). The other person is pushing the piano at a 30 degree angle with 250 pounds of force; 30 degrees would mean the angle is more horizontal than vertical. So if the weaker person is pushing more vertical, and the stronger person is pushing more horizontal, then our intuition tells us that the piano should be moving in some direction between 30 and 60 degrees. Let's see if our intuition holds up.

To break up the forces into their components, you have to use Trigonometry again. On a simple xy-plane, multiplying the force by the cosine of the angle will give you only the force in the x-direction, and multiplying it by the sine of the angle will give you only the force in the y-direction. So,

<175*cos(60), 175*sin(60)> for the weak person, and <250*cos(30), 250*sin(30)> for the strong person. Since both people are contributing to the force, add the two forces together:

<175*cos(60), 175*sin(60)> + <250*cos(30), 250*sin(30)> (remember, these are vectors, so add the x-values (the ones with cosine) together, and then add the y values (the ones with sine) together, and keep them separated by a comma.)

This new vector you found will be the total force on the piano in x and y... but what direction is it? To find the new direction, use arctan(y/x). The y will be the y value in the vector you just found, and the x is the x value. This direction you found should be between 30 degrees and 60 degrees.

Hopefully this is enough to help you get to the answer on your own. If you need more help or there's something you need clarified, ask away.

Answer 2

get each component

y1 component  175*sin(60) = 151

y2 component  250 * sin(30)=125

 

x1 component 175*cos(60) =87

x2 component 250*cos(30) = 217

 

add components

y=151+125 = 276

x= 87+217 = 304 

vector sum is sqrt (276^2 =304^2)=411

 

angle Arctan(276/304) = 42  degres

 

 

Answer 3

 So it is a parallelogram longer side is 250 , and shorter 175, the angle between is 30° .

 

   The force is the diagonal of the parallelogram:

 

      F² = 250^2 + 175 ^2 - 2 ( 250) ( 175) cos 30° =

 

      F = √( 250 ^2 +  175^2 - 2 *175* 250) = 75 lb Direction 45° with X -axis.