Equilibrium of a particle - Triangle of forces | ExamSolutions

Summary

This educational script explains how to maintain a particle in equilibrium under the influence of multiple forces. It introduces two methods: the force triangle and resolving forces into components. The force triangle uses the cosine rule to find the resultant force and angle, while resolving involves breaking down forces into horizontal and vertical components to solve for unknown forces and angles. The tutorial uses an example with forces of 8 Newtons and 6 Newtons at a 65° angle to demonstrate these concepts.

Takeaways

• 🔍 The script discusses a physics problem involving forces acting on a particle.
• 📏 It explains how to maintain equilibrium by applying equal and opposite forces.
• 📐 The script introduces a scenario with a force of 6 Newtons at 65° to an 8 Newton force.
• 🔄 It discusses the concept of resultant force and how to calculate it.
• 📈 The script presents two methods for solving the problem: using a force triangle and resolving forces.
• 📖 The force triangle method involves drawing a triangle with the forces and using the cosine rule.
• 📐 The resolving method involves breaking down forces into horizontal and vertical components and setting up equations.
• 🧮 The script provides a step-by-step calculation using both methods, including using trigonometric identities.
• 📉 It explains how to find the unknown force 'P' and angle 'Theta' using the cosine rule and trigonometric functions.
• 🔢 The script concludes with the values of P (11.9 Newtons) and Theta (27°) after calculations.
• 📘 It highlights the preference for the resolving method for problems with more than three forces.

Q & A

What is required to keep a particle in equilibrium if an 8 Newton force acts on it?

To keep the particle in equilibrium, an equal and opposite force of 8 Newtons must be applied to the particle.

What happens when a 6 Newton force is applied at an angle of 65° to an 8 Newton force?

The particle will experience a resultant force that acts somewhere between the directions of the 8 Newton and 6 Newton forces. To maintain equilibrium, a force must be applied in the opposite direction of the resultant force.

How can we calculate the force P and angle θ required for equilibrium in this scenario?

There are two methods to calculate the force P and angle θ: by using a force triangle or by resolving forces into perpendicular directions. Both methods provide the same results.

How is the cosine rule used to calculate the force P?

Using the cosine rule, P² = 8² + 6² - 2 * 8 * 6 * cos(115°). Solving this equation gives P ≈ 11.9 Newtons.

How do you determine the angle θ using the sine rule?

By applying the sine rule: sin(θ) / 6 = sin(115°) / P. Solving this gives θ ≈ 27°.

What does the force triangle method help visualize in this situation?

The force triangle method helps visualize the relationship between the three forces acting on the particle, showing how they form a closed triangle when the particle is in equilibrium.

What is the role of resolving forces horizontally in this problem?

Resolving forces horizontally allows you to break down the forces acting on the particle into horizontal components. For equilibrium, the sum of all horizontal forces must be zero.

What are the horizontal components of the forces acting on the particle?

The horizontal components are P cos(θ) to the left, -8 Newtons from the 8 Newton force, and -6 cos(65°) from the 6 Newton force. These components must sum to zero for equilibrium.

Why is resolving forces vertically also necessary in this problem?

Resolving vertically ensures that the sum of the vertical forces is zero, which is required for equilibrium. The vertical components of the forces must balance each other out.

What is the key identity used to solve for θ and P using the resolving method?

The key identity is tan(θ) = sin(θ) / cos(θ), which is derived by dividing the vertical force equation by the horizontal force equation. This identity helps solve for θ, and then P can be determined by substitution.

Outlines

00:00

🔧 Introduction to Force Equilibrium

The script begins by introducing a scenario where a particle is subjected to a force of 8 Newtons. To maintain equilibrium and keep the particle stationary, an equal and opposite force of 8 Newtons is applied. The concept is then expanded to include a third force of 6 Newtons acting at a 65-degree angle to the original 8 Newton force. The resultant force and the need for an opposing force P to maintain equilibrium are discussed. The script outlines two methods to solve such problems: using a force triangle and by resolving forces.

05:01

📐 Force Triangle Method

The force triangle method is explained by drawing a triangle with sides representing the forces. The 8 Newton force is drawn first, followed by the 6 Newton force. The angle between these two forces is marked as 65 degrees. The script then explains how to calculate the unknown force P using the cosine rule, which involves finding the square of P by summing the squares of the other two sides and subtracting twice their product times the cosine of the included angle. The calculated value of P is then rounded to three significant figures, resulting in 11.9 Newtons.

10:02

📐 Sine Rule Application

The script continues by discussing the calculation of the angle Theta using the sine rule. It explains that since the forces form a parallelogram, the angle Theta is equivalent to the angle at the base of the triangle. The sine rule is used to find the sine of Theta, which is then used to calculate the angle itself. The script provides the mathematical steps and the final value of Theta, rounded to two significant figures, which is 27 degrees.

15:05

🔍 Resolving Forces Method

The script introduces the resolving forces method, which involves breaking down the forces into horizontal and vertical components. It explains how to resolve the forces in both directions and set up equations to solve for the unknowns P and Theta. The process involves using trigonometric functions to find the components of the forces acting in the chosen directions. The script then combines the equations to solve for P and Theta using the tangent function and provides the final values for both, rounded to the appropriate significant figures.

🔄 Alternative Approaches

The final paragraph discusses alternative methods for solving the force equilibrium problem. It mentions the use of the Pythagorean identity to combine the equations and solve for P and Theta. The script also introduces a shortcut method involving the addition of squared equations to find P directly. The summary concludes by emphasizing the preference for the resolving method and acknowledges that it becomes necessary when dealing with more than three forces.

Keywords

💡Force

Force is defined as any interaction that, when unopposed, will change the motion of an object. In the context of the video, force is central to the discussion of particle dynamics. The video describes a scenario where a particle is subjected to forces of different magnitudes and directions, such as an 8 Newton force and a 6 Newton force at an angle of 65°.

💡Equilibrium

Equilibrium refers to a state in which opposing forces or influences are balanced, resulting in no net change in motion. The video discusses how to maintain a particle in equilibrium by applying equal and opposite forces, illustrating the concept with a force of 8 Newtons counteracting another force.

💡Resultant Force

The resultant force is the single force that represents the vector sum of several forces acting on an object. The video explains how to calculate the resultant force when multiple forces are acting on a particle, which is essential for determining the particle's motion or equilibrium state.

💡Cosine Rule

The cosine rule is a formula used in geometry, specifically for calculating the third side of a triangle when the lengths of the other two sides and the included angle are known. In the video, the cosine rule is applied to find the magnitude of the force 'P' acting to maintain equilibrium in a force triangle.

💡Triangle

In the context of the video, a triangle is used as a graphical tool to represent the forces acting on a particle. The force triangle method is one of the methods discussed to solve for unknown forces when multiple forces are acting on a particle.

💡Resolving

Resolving refers to the process of breaking down a force into its components along perpendicular axes. The video describes how to resolve forces into horizontal and vertical components to analyze the forces acting on a particle in different directions.

💡Sign Rule

The sign rule is a method used to determine the direction of a force relative to a chosen axis. The video uses the sign rule to find the angle 'Theta' by comparing the sine of the angle to the opposite side in a triangle formed by the forces.

💡Tangent

The tangent of an angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side. The video uses the tangent function to relate the sine and cosine of an angle, which helps in finding the angle 'Theta' when resolving forces.

💡Component

A component in physics refers to a part of a vector that lies along a specified direction. The video explains how to find the horizontal and vertical components of forces to analyze their effects in maintaining or disrupting equilibrium.

💡Parallel Lines

Parallel lines are two lines in a plane that do not meet; they are always the same distance apart and run in the same direction. The video mentions parallel lines in the context of alternate angles, which are equal when two lines are crossed by a transversal, helping to determine the angle 'Theta'.

💡Invert

In the context of the video, to invert a value means to find its reciprocal or to reverse its operation. The video discusses inverting the sine value to find the angle 'Theta' using the inverse sine function, which is necessary for determining the direction of forces.

Highlights

A particle is acted upon by a force of 8 Newtons.

To keep the particle stationary, an equal and opposite force of 8 Newtons is required.

If the force is not balanced, the particle will move in the direction of the resultant force.

A new force of 6 Newtons at 65° is introduced, creating an imbalance.

To maintain equilibrium, a force P Newtons in the opposite direction to the resultant force is needed.

Two methods are introduced to solve the problem: force triangle and resolving.

Force triangle method involves drawing a triangle with the forces as sides and using the cosine rule.

The cosine rule is applied with the known sides and angle to find the unknown force P.

The angle Theta is determined using the sine rule or by considering alternate angles in the triangle.

Resolving method involves breaking down the forces into horizontal and vertical components.

Horizontal resolution requires considering the components of forces acting to the left.

Vertical resolution involves calculating the components of forces acting downwards.

Equations are set up for the horizontal and vertical components to find P and Theta.

Trigonometric identities, such as sin(Theta)/cos(Theta) = tan(Theta), are used to solve for Theta.

The value of P is found by substituting the value of Theta back into the equations.

An alternative method of adding the equations squared is mentioned for finding P.

The final values for P and Theta are calculated with considerations for significant figures.

The tutorial concludes by comparing the force triangle and resolving methods for solving force problems.

Transcripts

hi let's suppose I've got a particle

being acted upon by a force of 8 Newtons

what's going to happen well that

particle is going to want to

move to the

right but if we wanted to keep it in

equilibrium in other words stationary

what we've got to do is apply an equal

and opposite force of 8 Newtons to

it well that's dead simple but what

happens if that 8 Newtons wasn't there

and we had say a force now of say six

Newtons acting at 65° to the 8 Newton

Force just Mark that in

there now this particle is going to want

to clearly move out here some

somewhere there'll be a resultant force

acting somewhere in between the 8

Newtons and the 6 Newton

Force now to keep this in equilibrium I

would need to apply a force then in the

opposite direction to that resultant

Force let's say it's p Newtons put it in

here P

Newtons and also what angle would that

make say with this dotted line we'll

call that angle angle

Theta so that's our question if this

particle is in equilibrium under these

three forces what would the force P

Newton be and what would the angle Theta

be well there's two ways that you can

answer questions like this when you've

got three forces acting on a

particle one of them the easiest way

really for something like this is to

draw a force triangle and the other way

is by resolving and I'll show you both

methods in this tutorial and so you can

compare which one you think is the

easier okay well if we're going to look

at the force triangle then we start off

with picking say one of these three

forces it doesn't matter which one you

pick I'm going to go first with the

eight Newtons so we'll draw 8 Newtons in

something like this okay this would be

eight units long

then we follow it with the six Newtons

so we start from the end here and this

would be six units long so we just Mark

that in as the six

Newtons now we've seen in the past that

the resultant force would act from here

to here going in this

direction but there is no resultant

Force well there is a resultant Force

it's zero because it's in

equilibrium but this force of P Newtons

would have to be in the opposite

direction to the resultant force of

these two so that Arrow would be

reversed let's just put that back in

there and reverse the direction round so

that would be P

Newton we form what is called a closed

triangle now in order to work out P from

the triangle we need to put some angles

in and we can see that this 65° is the

angle between the six and the horizontal

line here so we Mark that in is

65° so that means that this interior

angle here has to be

115° and we can work out P now quite

easily when we know two sides and the

opposite angle because we can use the

cosine rule so by the cosine rule let's

just put

that in here by the cosine rule what are

we going to have well it'll be

p^2 equals the sum of the squares of the

other two sides so that be 8^2 + 6^ 2

minus twice the product of those two

sides so that' be 2 * 8 * 6 times the

cosine of the opposite angle

115° then

and if you work this out you'll get that

p² turns out to be

140571 and so

on okay and now to get P you just need

to square root that value so square root

that

140571 and so on and it turns out that

you get

11856 and so on Newtons

Well we'd want to give that to some

degree of accuracy so I'll go for three

significant figures and that would be

11.9 Newtons to three significant

figures now as for the angle

Theta where does that appear in the

Triangle Well if I was to think of the

dotted line through there that's the

angle Theta but it's not in the

Triangle but it's equivalent to this one

down here because we've got alternate

angles two parallel lines

here so to work out Theta we could use

the cosine rule because we know all

three sides I leave it up to you to do

that if you want to use the cosine rule

but I'm not going to do that I'm going

to use the sign rule because I think

it's

quicker so if we're doing that then by

the sign rule it would be sin Theta

compared with the opposite side which

will be the six here equals the sign of

this angle

115° compared with or divided with the

opposite side the P so use the unrounded

version so that be 11. 856 and so on and

all I need to do now is just multiply

both sides by six so that gives us sin

Theta = 6 * sin of

115° and divide that by the

11.85 6 and so on and work that out in

your calculator and you should find you

get

0.45 96 and so on to get Theta we just

need to take the inverse sign of this

value and you'll find that you get

27 299 and so on

degrees I'm going to round that up to

say two significant figures and that

would be

27° to two significant

figures okay well that's one way of

getting p and our angle Theta that it

acts but it's not the only way as I said

earlier what we could do is we could

resolve and to resolve what I do is I

look at two perpendicular directions and

for a question like this it would be

sensible I would have thought to resolve

in the horizontal sense and the vertical

sense we'll just draw a dotted line down

here for the vertical

sense now when it comes to resolving say

horizontally which way do you resolve

well it doesn't matter I'm going to pick

to the left though purely because it

would make the term containing the force

P positive but that's up to

you so when we resolve to the left we're

looking at how much force acts in the

horizontal sense well if it's an

equilibrium there's going to be no

overall force there'll be zero resultant

Force then so let's look at the

components then or what forces act in

this horizontal sense to the left well

first of all taking this Force we need

to split this into two components

because not all of P acts along the

dotted line to the left so the two

components of P would be one to the left

and one downwards the one downwards has

no effect because it's at right angles

to the direction we're resolving in

we're only interested in the component

to the left which contains the angle so

it will be P cos Theta remember when it

contains an angle it's cosine when it

excludes the angle It's s so P cos Theta

that

way all of 8 Newtons acts along this

horizontal Direction but 8 Newtons acts

in the opposite sent so it's going to be

minus

8 as for the 6 Newtons though because

that's inclined to the horizontal we

need to think of splitting this into two

components and that would be one to the

right and one up

upwards the one upwards has no effect

because it's perpendicular to this

direction we want the one to the right

which will be 6 cine 65° because it

contains the angle so it's going to be -

6 cos

65° that's our resultant Force but

because it's an equilibrium is going to

equal

zero now what we do in equations like

this is we make p cos Theta the subject

so if I was to add 8 and 6 cos 65 to

both sides You' end up with P cos Theta

then equaling 8 + 6 cos

65° and if you work that out on your

calculator then that comes to a value of

10535 and so on

Newtons and I'm going to call that that

equation one we'll return to that later

on now we need another equation because

we got two unknowns here p and

Theta and we resolve then in the

perpendicular sense and it's up to you

whether you resolve upwards or downwards

I'm going to go downwards because it

will keep this term

positive so we'll resolve

downwards so again go through all the

three forces let's start with this force

of P Newton then we need to split it

into two components because P doesn't

act totally

down the component downwards would be P

sin Theta because it excludes the angle

Theta that we've got here between the

two directions here so it would be P sin

Theta now we go on to the8 newtons well

none of that acts downwards because it's

perpendicular to the

direction as for the six Newtons well

we've seen that we split that into two

components one up one to the right the

one to the right has no effect because

it's perpendicular to our Direction the

one upwards would be 6 sin 65° because

we're excluding that

65° you could if you wanted say six cos

25 but there's no point in doing that I

feel I always work off the angle that

I'm given okay so it be Min - 6 sin

65° and that's the resultant Force

downwards but because it's in

equilibrium that resultant must be

zero if we rearrange this by adding 6

sin 65 to both sides we end up with P

sin th

= 6 sin

65° work that out on your calculator and

that comes to

54378 and so on Newtons and I'm going to

call that equation

two just squeeze that in

there now we've got to work out what p

and Theta are and the way we do this

when we've got equations of this format

is to call upon a particular identity we

should know from core maths that sin

Theta over cos Theta is identical to tan

Theta and if we do equation two divided

by equation one we create that situation

because the P's would cancel one another

out and you'll just have sin Theta over

cos Theta which is tan Theta so you have

tan Theta

equals the

54378 and so on divided by

10 535 and so

on and if you work that out you end up

with

05161 and so on so to get Theta all you

need to do is do the inverse tan of

5161 and you'll end up up with

27.29 and so on which when rounded is

going to give you back that 27° to

2sf now to get P all you need to do is

substitute for Theta in either one or

two work out what C of the

27299 de is and then you should be able

to work out what p is and the applies if

you use this equation just work out what

the sign of the angle was there is an

alternative

though quite a lot of people use this

idea they do 1 2 + 2 2 equation 1 2 + 2

squ why do they do that

well what happens is that you

get p^ 2 cos 2 Theta I'll just write

this in here p^2 cos 2 Theta

plus p^ 2 sin^2 Theta p^ 2 sin^2

theta

equals 10.5 odd s+ 5.4 odd squar I

haven't really got much room to write

this in here so what I'm going to do is

just write 1^ 2 + 2^ 2 okay just to

represent those two decimals

there now what happens is you can

factorize this side and you end up with

pulling p^ s out the front and you get

cos s theta plus sin s

Theta and that's going to be equal as I

say to those two decimals squared and

added together we just put it for short

there but cos squ theta plus sin 2 Theta

well that's another identity it comes to

one so that leaves gives you with p^2 =

10.5 odd 2 + 5.4 odd 2 so to get P all

you need to do is square root that

so normally what we tend to do is go

straight to this result P will equal the

square root then of

10535 and so on

squared

plus 5

4 3 7 8 and so on

squared and I leave you to work that out

but you're going to get this number here

which when rounded is going to be your

11.9 Newtons to

3sf okay well I hope that's given you

some idea then of how you can find

out a force like P Newtons and the angle

then when you've got three forces either

by using the force triangle or by the

resolving

method as I said earlier I know that I

would generally prefer this

method but as you'll see later when

we've got more than three forces here

you're going to have to result to

resolving okay well you'll see some

examples then that will follow giving

these uh ideas so I hope you'll find

them

useful