# Question Video: Using the Triangle of Forces Rule to Find the Magnitude of a Force Mathematics

The body at point π΅ is at equilibrium. Find the magnitude of the force πΉβ in newtons, rounding your answer to two decimal places.

04:22

### Video Transcript

The body at point π΅ is at equilibrium. Find the magnitude of the force πΉ sub one in newtons, rounding your answer to two decimal places.

We begin by recalling that if a body is in equilibrium, the sum of its forces in the horizontal and vertical directions must both equal zero. From the diagram, we can see there is a force acting vertically downwards of 10 newtons. We have a horizontal force πΉ sub one. We also have a third force πΉ sub two, which is not acting horizontally or vertically. Our first step is therefore to work out the horizontal and vertical components of this force.

We begin by letting the angle between the πΉ sub two force and the horizontal equal π. Using our knowledge of right-angle trigonometry, the vertical component is equal to πΉ sub two multiplied by sin π and the horizontal component is equal to πΉ sub two multiplied by cos π. Resolving horizontally, where forces acting to the right are positive, we have πΉ sub two cos π minus πΉ sub one equals zero. Adding πΉ sub one to both sides of this equation, we have πΉ sub one is equal to πΉ sub two multiplied by cos π. We will call this equation one.

Resolving vertically, where the positive direction is upwards, we have πΉ sub two sin π minus 10 is equal to zero. Adding 10 to both sides of this equation, we have πΉ sub two multiplied by sin π is equal to 10. We will call this equation two.

We now have a pair of simultaneous equations that we can solve by elimination. We can divide equation two by equation one such that πΉ sub two sin π over πΉ sub two cos π is equal to 10 over πΉ sub one. On the left-hand side, we can divide through by πΉ sub two. One of our trigonometric identities states that sin π divided by cos π is equal to tan π. This means that our equation simplifies to tan π is equal to 10 divided by πΉ sub one. Multiplying both sides of this equation by πΉ sub one and dividing through by tan π, we get πΉ sub one is equal to 10 over tan π.

At this stage, we donβt appear to know the value of tan π. However, returning to our diagram, we note that we have a right triangle where the side opposite angle π is equal to 110 centimeters and the side adjacent to angle π has length 40 centimeters. We know that the tangent of any angle is equal to the opposite over the adjacent. tan π is therefore equal to 110 over 40. And dividing both the numerator and denominator by 10, this is equal to 11 over four. We can therefore calculate πΉ sub one by dividing 10 by 11 over four or 2.75. πΉ sub one is equal to 40 over 11 or 3.6363 and so on. We are asked to round our answer to two decimal places. The magnitude of the force πΉ sub one is therefore equal to 3.64 newtons to two decimal places.