3 forces each equal to P act on a body, one at (a,0,0) parallel to OY. The second at the point (0,b,0) parallel to OZ and the third at the point (0,0,c) parallel to OX, the axis being rectangular.
1)Find the resultant force and the resultant moment
2) Find the pitch of the wrench
1) So if we take the forces parellel to OX , OY, and OZ axes respectively as F1,F2 and F3 (we know they have equal magnitudes P)
Resultant force, Fr = F1+F2+F3
They are in three planes
How do we sum them?
Summing all three is a vector addition. Break up each force into three components (x,y,z). The resultant force will be the vector addition of each of these forces. Fr=F1 + F2 +F3 where each force has three directional components.
Since the forces F1, F2 & F3 passes through the points (a,0,0) , (0,b,0) & (0,0,c) and are parallel to OY, OZ, OX axes respectively,
can we take the sum as
F1=(j - ai)P
F2 =(k - bi)P
F3= (i - ck)P
F1+F2+F3 = [ (j - ai)P] + [ (k - bi)P ] + [ (i - ck)P ]= P[ (1-a)i + (1-b)j + (1-c)k ]
?
yes, that is the approach.
To sum the three forces, F1, F2, and F3, we need to consider that they are acting in three different planes. We cannot directly add forces acting in different planes together.
To find the resultant force, Fr, we need to resolve each force into its components along the OX, OY, and OZ axes. Since each force is only acting along one axis, we can directly add the components.
The components along each axis are:
- F1x = 0
- F1y = P
- F1z = 0
- F2x = 0
- F2y = 0
- F2z = P
- F3x = P
- F3y = 0
- F3z = 0
To find the resultant force, Fr, we add the components along each axis:
Fr = (F1x + F2x + F3x)i + (F1y + F2y + F3y)j + (F1z + F2z + F3z)k
= (0 + 0 + P)i + (P + 0 + 0)j + (0 + P + 0)k
= Pi + Pj +Pk
So the resultant force is Fr = Pi + Pj + Pk.
Now let's move on to finding the resultant moment.