Practice Exercise S-1.3.2
Central Force Systems - Third Basic Task: Decomposition of a Force
Problem Statement
A mass with the weight force FG is hanging on two ropes.
What are the components of the force FG = 10 kN in the direction of ropes 1 and 2?
Before You Embark on This Quest: A Pre-Mission Briefing
Before you dive into this adventure, let's do a quick check to see if you're ready for the challenge.
What topic are we dealing with here? Can you explain why?
All the lines of action of the attacking forces intersect at one point. So, we're dealing with a central force system.
If you're not quite familiar with central force systems, no worries! Here are some refresher courses to get you up to speed:
Good luck!
Graphical Solution
Okay, let's figure out the magnitudes of the components of the force FG in the direction of ropes 1 and 2. First, let's give them unique labels. A good choice would be FGS1 and FGS2.
Take a close look at the picture in the problem statement first. Then, read the problem statement carefully and create a layout diagram. This will give you an overview of everything important.
Use a Cartesian coordinate system as the basis for the diagram. The common point of attack of the forces is the origin (0,0) of the coordinate system.
The position of the force FG and the lines of action of the forces FGS1 and FGS2 are given. Draw them into the diagram first:
While you're at it, you can also enter the angles of the lines of action to the two coordinate axes, as shown in Fig. 2. They are obtained from the two measured angles, similar triangles, and the fact that the coordinate axes are arranged at a right angle to each other, i.e., at a 90° angle.
Now you can add the forces FGS1 and FGS2.
You have already drawn their lines of action. The question remains, in which direction they act.
The components of the force FG are required, so FGS1 and FGS2 must form a force parallelogram whose diagonal is FG.
Continue with TechMechAcademy+
Everything. Always. Everywhere.
With TechMechAcademy+ full access to all content.
Overview of the benefits:
- Most cost-effective offer.
- Ideal if you need access to already identified content for a short period.
- Unlimited access to all existing and newly created content throughout the entire premium membership.
- Guaranteed premium membership for 24 hours. Access automatically ends at 0:00 CET (Central European Time) on the following day. No cancellation necessary.
€3.99
Overview of the benefits:
- Cost-effective offer.
- Ideal for short-term exam preparation with the content of TechMechAcademy.
- Unrestricted entry to every piece of content, both existing and newly generated, across the entirety of the premium membership.
- Guaranteed premium membership for one week. Access starts on the day of activation and ends automatically at 0:00 CET (Central European Time) on the following week's day. No cancellation necessary.
€9.99
Overview of the benefits:
- Benefit from the bestseller.
- Ideal for effective exam preparation with the content of TechMechAcademy.
- Complete access to all content, both existing and newly produced, under the umbrella of the premium membership.
- Guaranteed premium membership for one month. Access starts on the day of activation and ends automatically at 0:00 CET (Central European Time) on the following month's day. No cancellation necessary.
€14.99
Overview of the benefits:
- Cost-effective due to extended duration.
- Ideal for working alongside the semester with the content of TechMechAcademy.
- Unrestricted admission to all content, both pre-existing and recently generated, within the scope of the premium membership.
- Guaranteed premium membership for three months. Access starts on the day of activation and ends automatically at 0:00 CET (Central European Time) on the following day of the third month from the start date. No cancellation necessary.
€29.99
Overview of the benefits:
- Cost-effective due to extended duration.
- Ideal for semester preparation, concurrent semester work, and/or post-semester work with the content of TechMechAcademy.
- Unrestricted entry to a comprehensive collection of both existing and freshly generated content as part of the premium membership.
- Guaranteed premium membership for six months. Access starts on the day of activation and ends automatically at 0:00 CET (Central European Time) on the following day of the sixth month from the start date. No cancellation necessary.
€49.99
Are you already a TechMechAcademy+ premium member? Then please log in here to enjoy full access to all content.
Analytical Solution
To solve this problem numerically, a sketch of the situation is very helpful.
First, we give unique labels to the components of the force FG in the direction of ropes 1 and 2. A good choice is FGS1 and FGS2.
Take a close look at the image in the problem statement. Read the problem statement carefully and create a sketch. This will give you a clear overview of everything important.
For the sketch, use a Cartesian coordinate system. The common point of application of the forces is the origin (0,0) of the coordinate system.
The position of the force FG and the lines of action of the forces FGS1 and FGS2 are given. Draw them into the diagram first:
While you're at it, you can also enter the angles of the lines of action to the two coordinate axes, as shown in Fig. 2. They are obtained from the two measured angles, similar triangles, and the fact that the coordinate axes are arranged at a right angle to each other, i.e., at a 90° angle.
Now you can add the forces FGS1 and FGS2.
Their lines of action have already been drawn in. The question remains in which direction they act.
We are looking for the components of force FG, so FGS1 and FGS2 must form a force parallelogram, whose diagonal is FG.
Continue with TechMechAcademy+
Everything. Always. Everywhere.
With TechMechAcademy+ full access to all content.
Overview of the benefits:
- Most cost-effective offer.
- Ideal if you need access to already identified content for a short period.
- Unlimited access to all existing and newly created content throughout the entire premium membership.
- Guaranteed premium membership for 24 hours. Access automatically ends at 0:00 CET (Central European Time) on the following day. No cancellation necessary.
€3.99
Overview of the benefits:
- Cost-effective offer.
- Ideal for short-term exam preparation with the content of TechMechAcademy.
- Unrestricted entry to every piece of content, both existing and newly generated, across the entirety of the premium membership.
- Guaranteed premium membership for one week. Access starts on the day of activation and ends automatically at 0:00 CET (Central European Time) on the following week's day. No cancellation necessary.
€9.99
Overview of the benefits:
- Benefit from the bestseller.
- Ideal for effective exam preparation with the content of TechMechAcademy.
- Complete access to all content, both existing and newly produced, under the umbrella of the premium membership.
- Guaranteed premium membership for one month. Access starts on the day of activation and ends automatically at 0:00 CET (Central European Time) on the following month's day. No cancellation necessary.
€14.99
Overview of the benefits:
- Cost-effective due to extended duration.
- Ideal for working alongside the semester with the content of TechMechAcademy.
- Unrestricted admission to all content, both pre-existing and recently generated, within the scope of the premium membership.
- Guaranteed premium membership for three months. Access starts on the day of activation and ends automatically at 0:00 CET (Central European Time) on the following day of the third month from the start date. No cancellation necessary.
€29.99
Overview of the benefits:
- Cost-effective due to extended duration.
- Ideal for semester preparation, concurrent semester work, and/or post-semester work with the content of TechMechAcademy.
- Unrestricted entry to a comprehensive collection of both existing and freshly generated content as part of the premium membership.
- Guaranteed premium membership for six months. Access starts on the day of activation and ends automatically at 0:00 CET (Central European Time) on the following day of the sixth month from the start date. No cancellation necessary.
€49.99
Are you already a TechMechAcademy+ premium member? Then please log in here to enjoy full access to all content.