Equilibrium of a particle - Triangle of forces | ExamSolutions
Summary
TLDRThis 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
🔧 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.
📐 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.
📐 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.
🔍 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.
Mindmap
Keywords
💡Force
💡Equilibrium
💡Resultant Force
💡Cosine Rule
💡Triangle
💡Resolving
💡Sign Rule
💡Tangent
💡Component
💡Parallel Lines
💡Invert
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
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