# Thread: Vector forces on ring

1. ## Vector forces on ring

The question is:- Four given pulls and thrusts act on a ring. Find their equilibrant. The pulls and thrusts are as follows:
N = 58 - thrust
M = 105 - pull
L = 40 - thrust
K = 72 - pull
Angles are as follows:
NM - 73 deg
NL - 149 deg
NK - 194 deg

The methodology I have used is calculate the equilibrium for the pulls and thrusts separately. Then subtract the thrusts from the pulls. The answer I get is 60.36, however the answer given is 88.5. Where have I gone wrong?

2. Hello p75213

Welcome to Math Help Forum!
Originally Posted by p75213
The question is:- Four given pulls and thrusts act on a ring. Find their equilibrant. The pulls and thrusts are as follows:
N = 58 - thrust
M = 105 - pull
L = 40 - thrust
K = 72 - pull
Angles are as follows:
NM - 73 deg
NL - 149 deg
NK - 194 deg

The methodology I have used is calculate the equilibrium for the pulls and thrusts separately. Then subtract the thrusts from the pulls. The answer I get is 60.36, however the answer given is 88.5. Where have I gone wrong?
If a force $\displaystyle F$ acts outwards from the origin $\displaystyle O$ along a line that makes a direction $\displaystyle \theta$ measured anticlockwise from the positive direction of $\displaystyle Ox$, then its components along $\displaystyle Ox$ and $\displaystyle Oy$ are
$\displaystyle F\cos\theta$ and $\displaystyle F\sin\theta$, respectively
Since the angles are measured from the direction of $\displaystyle N$, I assume they are all in the same sense (e.g. all anticlockwise). So we can write down the components along and perpendicular to $\displaystyle ON$, where $\displaystyle O$ is the ring, as follows:

Along $\displaystyle ON$:
$\displaystyle N: -58$

$\displaystyle M: 105\cos73^o = 30.70$

$\displaystyle L: -40\cos149^o=34.29$

$\displaystyle K: 72\cos194^o =-69.86$
Perpendicular to $\displaystyle ON$:
$\displaystyle N: 0$

$\displaystyle M: 105\sin73^o=100.41$

$\displaystyle L: -40\sin149^o=-20.60$

$\displaystyle K: 72\sin194^o=-17.42$
We find the resultant force by summing these components:
Along ON: $\displaystyle -62.87$

Perpendicular to ON: $\displaystyle 62.4$
The magnitude of the resultant is therefore $\displaystyle \sqrt{62.87^2+62.4^2}=88.58$.