How Do You Calculate the Moment in a Force-Couple System at a Pivot Point?

[zeralda21]

Homework Statement

A force F of magnitude 50 N is exerted on the automobile parking-brake lever at the position x=250 mm. Replace the force by an equivalent force-couple system at the pivot point O.

Picture: http://i.imgur.com/dJGtvAo.png

Homework Equations

M=F*d

The Attempt at a Solution

I have replaced the force by an equivalent force-system at O so I now the exact same force F(both magnitude and direction) but at O. Now I want to find the moment which is given by M=Force*distance. The distance I need is the closest distance between O and the "moment-point". For simplicity I instead try to find the distance between O and Fcos20. But I have problems finding it.

 

[tiny-tim]

hi zeralda21!

The distance I need is the closest distance between O and the "moment-point".

no, it's the closest distance between O and the line of the force

probably easiest to start by extending the left-hand part until it reaches the line of F, and finding the total length of that

 

[zeralda21]

hi zeralda21!


no, it's the closest distance between O and the line of the force

probably easiest to start by extending the left-hand part until it reaches the line of F, and finding the total length of that

Ouch, that is actually what i meant. But the closest distance is therefore the horizontal distance from O to Fcos(20), no? I'm not sure what you mean. Extend it as:

0.1m+xcot15 ?

 

[tiny-tim]

hi zeralda21!

But the closest distance is therefore the horizontal distance from O to Fcos(20), no? I'm not sure what you mean. Extend it as:

0.1m+xcot15 ?

on second thoughts, i don't think that works

try it this way …

draw a line OG parallel to the line of F

then the distance is the sum of the perpendicular distances from that dot in the middle to the line of F, and from that dot in the middle to OG

 

[Nickie]

I don't know if it is a right angled triangle or not or if I need to use some trigonometry.

I would approach this problem by first studying the concept of force systems and couples. A force system is a collection of forces acting on an object, while a couple is a pair of forces that are equal in magnitude, opposite in direction, and parallel to each other, but do not act along the same line. In this case, the force F applied on the parking-brake lever can be represented as a force system acting at the point x=250 mm.

To replace this force system with an equivalent force-couple system at the pivot point O, we need to consider the principle of transmissibility of forces. This principle states that the effect of a force on a body remains unchanged if the force is translated along its line of action. In other words, the magnitude and direction of the force remain the same, but its point of application can be changed.

In this case, we can represent the force F as a force F' acting at the pivot point O, with the same magnitude and direction. This force F' will have the same effect on the object as the original force F. Additionally, we need to add a couple at the pivot point O to account for the moment created by the original force F. The magnitude of this couple can be calculated using the formula M=F*d, where F is the magnitude of the force and d is the perpendicular distance between the force and the pivot point.

To find the distance d, we can use the concept of moments. The moment of a force about a point is defined as the product of the magnitude of the force and the perpendicular distance from the point to the line of action of the force. In this case, the moment of the force F about the pivot point O is given by M=F*d, where d is the perpendicular distance between the force and the pivot point.

To find the distance d, we can use trigonometry. The angle between the line of action of the force F and the line joining the force and the pivot point O is 20 degrees. Using trigonometry, we can find that the perpendicular distance d is given by d=x/tan(20), where x is the distance between the force and the pivot point.

In conclusion, to replace the force system with an equivalent force-couple system at the pivot point O, we need to represent the force F as