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Angular Vector Forces

Angular vector forces occur when ropes are passed through a deviation or a directional pulley. Dependent on the angle

created, this can have a multiplying effect on the forces that are felt at the deviation or directional pulley components andthe associated anchor points.

Force is an influence that has both magnitude and direction, it is usually given in the dynamic unit of Newtons (N). For ease of explanation we have used

kilograms on this page.

The Basics

Here we have a load weighing 100kg. A rope has been attached to the load and then passed through a

directional pulley which returns the rope back down to the user at ground level. The two sections of

rope are in parallel with each other so there is no internal or included angle.

In this situation the directional pulley and its anchor point are actually being loaded with 200kg, as

there is the weight of the load on one side and a effort of 100kg is being applied by the user on the

other to hold the load in place.

Deflection Angle of 20

Here the rope has been deviated away from its original line (in this illustration the vertical) by 20. While

all of the weight of the load is being exerted to the main anchor point, a percentage of the loads weight

is also being applied to the deviation / directional pulley components.

With a deviation of 20, there will be 34% of the loads weight being applied to the deviation /

directional pulley anchor point. In this situation, where we have a 100kg load this would produce a force

of 34kg on the deviation / directional pulley.

Angular Vector Forces | ropebook http://www.ropebook.com/information/vector-forces/angular-vector-forces

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point equivalent to 76% of the loads weight, so in this example where the load weighs 100kg that

would be 76kg.


With an angle of deflection measuring 60 100% of the loads weight will be applied to the deviation or

directional pulley anchor point. This can be compared to the critical angle found in vector forces

(http://www.ropebook.com/information/vector-forces) when rigging y hangs, as all three angles are equal.


A 90 angle of deflection results in a force equal to 141% of the loads weight being applied to the

deviation or directional pulley anchor point. Note that this would also be the same loading for an

included angle of 90 as they are both the same.

Deflection Angle: The Calculations

Angular Vector forces can be calculated using mathematical formula. Here is a formula that we can use

to calculate the amount of force that is being applied to the deviation or directional pulley anchor point

when measuring from the angle of deflection.


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Where:

F is the resultant force exerted to the deviation or directional pulley anchor point.

m is the mass or weight of the load.

is the angle that the rope has been deflected away from its original line.

Deflection Angle: Using the Percentage Factor

The percentage factor can be used to calculate the amount of force that is being applied to the deviation or directional pulley anchor point. This can be

achieved by using the following formula:

Angle % factor Angle % factor Angle % factor

0 0.0 65 107.5 130 181.3

5 8.7 70 114.7 135 184.8

10 17.4 75 121.7 140 187.9

15 26.1 80 128.6 145 190.7

20 34.7 85 135.1 150 193.2

25 43.3 90 141.4 155 195.3

30 51.8 95 147.4 160 197.0

35 60.1 100 153.2 165 198.3

40 68.4 105 158.7 170 199.2

45 76.5 110 163.8 175 199.8

50 84.5 115 168.8 180 200.0

55 92.3 120 173.2

60 100.0 125 177.4

Looking at it from a different angle

We can measure the angle of a deviation or directional pulley in two different ways, which ever is

easiest and more relevant to the specific set-up. Here we have detailed both methods; measuring from

the angle of deflection and measuring from the included angle.

An angle of deflection is a measurement of how many degrees the rope has been deviated or diverted

away from its original line. These angles will be easier to estimate when using small deviations.

The included angle is measured between the section of rope that passes through the deviation or

directional pulley. It is usually easier to calculate this type of angular vector force when using pulley

systems and high re-directs.

When measuring either deviations or directional pulleys, both the included angle and the angle of

deflection should always result in 180 when added together.


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equivalent to 159% of the loads weight. So in this example that is 159kg!

Included Angle of 60

Again we have measured the included angle formed where the rope passes through the directional

pulley or deviation. With an included angle of 60, a force equivalent to 173% of the loads weight will

be applied to the directional pulley or deviation.

Included Angle of 45

In this illustration the included angle measures 45. As the included angle created by the rope passing

through the pulley or deviation decreases towards 0 the force applied to the directional pulley or

deviation increases. In this example the force applied will be roughly equal 185% of the loads weight.

Included Angle: The Calculations

Angular Vector forces can be calculated using mathematical formula. Here is a formula that we can use

to calculate the amount of force that is being applied to the deviation or directional pulley anchor point

when measuring from the included angle.


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Where:

F is the resultant force exerted to the deviation or directional pulley anchor point.

m is the mass or weight of the load.

is the angle that the rope has been deflected away from its original line.

Included Angle: Using the Percentage Factor

The percentage factor can be used to calculate the amount of force that is being applied to the deviation / directional pulley anchor point. This can be

achieved by using the following formula:

Angle % factor Angle % factor Angle % factor

0 200.0 65 168.8 130 84.5

5 199.8 70 163.8 135 76.5

10 199.2 75 158.7 140 68.4

15 198.3 80 153.2 145 60.1

20 197.0 85 147.4 150 51.8

25 195.3 90 141.4 155 43.3

30 193.2 95 135.1 160 34.7

35 190.7 100 128.6 165 26.1

40 187.9 105 121.7 170 17.4

45 184.8 110 114.7 175 8.7

50 181.3 115 107.5 180 0.0

55 177.4 120 100.0

60 173.2 125 92.3

Angular Vector Force Graph

The graph below displays the force applied to the deviation or directional pulley anchor point in comparison to the relationship between the deflection

angle and the included angle.


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