Equilibrium of a particle - Example : ExamSolutions
Summary
TLDRThis tutorial focuses on solving a problem involving the equilibrium of a particle subjected to multiple forces. The speaker walks through the process of resolving forces in two perpendicular directions, both horizontally and vertically. By applying trigonometry and resolving components of the forces, the tutorial guides the audience through calculating the values of P and Q. The solution involves using trigonometric identities, rearranging equations, and performing calculations to find the unknown forces. The tutorial also emphasizes checking solutions and understanding the methodology behind resolving forces in equilibrium scenarios.
Takeaways
- 🔍 The problem involves finding the magnitudes of forces P and Q acting on a particle in equilibrium under multiple forces.
- 📏 To solve, forces are resolved into horizontal and vertical components, simplifying the analysis into two perpendicular directions.
- ↔️ Forces are resolved to the right first, considering the horizontal components and their effects.
- 🔢 The 9 Newton force acts entirely to the right, contributing positively to the horizontal resolution.
- 📐 The 7 Newton force is split into horizontal and vertical components, with only the horizontal component affecting the resolution.
- 📉 The component of the 7 Newton force to the right is calculated using the cosine of the 40° angle.
- ➗ Force P is perpendicular to the horizontal resolution direction and thus does not affect the horizontal balance.
- 🔄 Force Q is split into leftward and downward components, with only the leftward component affecting the horizontal resolution.
- 🔄 The resultant horizontal force is set to zero for equilibrium, leading to the calculation of Q.
- 📊 For the vertical resolution, P acts entirely upwards, contributing positively.
- 📐 The 7 Newton force's vertical component is calculated using the sine of the 40° angle.
- ➖ The vertical component of Q is calculated using the sine of the 20° angle, affecting the vertical balance negatively.
- 🔄 The resultant vertical force is also set to zero for equilibrium, leading to the calculation of P.
Q & A
What is the main topic of the tutorial?
-The main topic of the tutorial is finding the forces P and Q acting on a particle in equilibrium under a system of forces.
What is the method used to resolve forces in this tutorial?
-The method used is resolving forces in two mutually perpendicular directions, horizontally and vertically.
Why is it necessary to resolve forces into components?
-Forces need to be resolved into components because only the components along the direction of interest affect the equilibrium of the particle.
What is the significance of the angle in resolving forces?
-The angle is significant because it determines the direction of the force component relative to the direction being resolved.
What does the cosine function represent when resolving forces?
-The cosine function represents the horizontal component of a force when resolving forces in a horizontal direction.
Why does the force P not affect the horizontal resolution in the tutorial?
-Force P does not affect the horizontal resolution because it acts perpendicular to the horizontal direction being resolved.
How is the force Q resolved into components?
-Force Q is resolved into components by considering the angle it makes with the direction being resolved, using the cosine of the angle for the horizontal component and the sine of the angle for the vertical component.
What is the resultant force acting on the particle in equilibrium?
-The resultant force acting on the particle in equilibrium is zero, as the particle is not accelerating.
How is the value of Q calculated in the tutorial?
-The value of Q is calculated by setting the sum of the horizontal components equal to zero and solving for Q.
What is the final value of Q given in the tutorial?
-The final value of Q given in the tutorial is 15.3 Newtons to one decimal place.
How is the value of P calculated in the tutorial?
-The value of P is calculated by setting the sum of the vertical components equal to zero and solving for P using the known value of Q.
What is the final value of P given in the tutorial?
-The final value of P given in the tutorial is 0.7 Newtons to one decimal place.
Outlines
🔍 Resolving Forces in Equilibrium
This paragraph introduces a problem involving the equilibrium of a particle under multiple forces. The speaker explains that to find the unknown forces P and Q, one must resolve the forces in two perpendicular directions, horizontally and vertically. The speaker then demonstrates how to resolve the forces to the right, taking into account the components of each force and using trigonometric functions to calculate the horizontal components. The force P is neglected as it acts perpendicular to the direction of resolution. The final step involves setting the resultant force to zero and solving for Q, which is calculated to be approximately 15.3 Newtons when rounded to one decimal place.
📏 Resolving Forces Perpendicularly
The second paragraph continues the tutorial by resolving forces in the vertical direction. The speaker begins with the force P acting upwards and then moves on to the force of 7 Newtons, which needs to be split into its vertical component using the sine function. The 9 Newtons force is perpendicular to the direction of resolution and thus has no effect. The force Q is also resolved into its vertical component, which is downwards. The resultant force in the vertical direction is set to zero because the particle is in equilibrium. Using the previously calculated value of Q, the speaker rearranges the equation to solve for P, which is found to be approximately 0.7 Newtons when rounded to one decimal place. The tutorial concludes with a summary of the process and a prompt for the viewer to check their work against the provided solution.
Mindmap
Keywords
💡Equilibrium
💡Resolving Forces
💡Cosine
💡Sine
💡Components
💡Force P
💡Force Q
💡Resultant Force
💡Trigonometry
💡Newton
💡Degrees of Freedom
Highlights
Introduction to a problem involving a particle in equilibrium under a system of forces.
Guidance to resolve forces in two mutually perpendicular directions: horizontally and vertically.
First step: Resolving forces horizontally, beginning with the 9 Newton force acting entirely to the right.
Explanation on splitting the 7 Newton force into two components: one horizontal and one vertical, using cosine for the horizontal component.
The P force acts perpendicularly to the horizontal, hence no contribution to the horizontal component.
Introduction to the Q force and its horizontal and vertical components, with focus on the horizontal component being negative (leftward) and calculated using cosine.
Resultant horizontal force equation is derived, which equals zero since the system is in equilibrium.
Rearrangement of the equation to solve for Q, using trigonometric functions and calculation steps to find Q = 15.3 Newtons.
Next step: resolving vertically, starting with the full P force acting upwards.
Explanation of the 7 Newton force’s vertical component using sine, as the angle doesn’t contain the angle 40°.
Clarification that the 9 Newton force has no effect on the vertical component as it acts perpendicularly.
Calculation of the downward vertical component of the Q force using sine.
Final equation for vertical forces is derived, setting it equal to zero for equilibrium.
Substitution of the previously calculated value of Q to solve for P, yielding P = 0.7 Newtons.
Conclusion: Explanation of how to tackle similar problems using force resolution techniques.
Transcripts
hi I've got a question here which you
might like to try it's a followup to my
earlier tutorial on the equilibrium of a
particle under a system of
forces so what we've got here is the
following forces act on a particle
centered here and if the particle is in
equilibrium find p and
Q so you might like to give this a go
and just come back when ready and I'll
run through uh the solution you can
check your working against
mine okay well let's see how you got on
if you had a
go well with questions like this
certainly when we've got more than three
forces acting on the particle what we do
is we resolve in two mutually
perpendicular directions
and it would seem sensible to resolve
horizontally and
vertically so what I'm going to do is
resolve to the right first of all so I'm
assuming you're familiar with resolving
if not you can always go on the website
and find the tutorials on resolving
forces but essentially if we resolve to
the right then what we've got if we go
around the forces you can see that all
of the 9 Newtons acts to the right so
that's going to be 9 + 9 because we're
going in the positive sense here to the
right if we take the 7 Newton Force then
because the seven Newtons is inclined to
this horizontal Direction here we have
to split it into two
components one to the right and one
upwards the one upwards would have no
effect because it's perpendicular to the
direction that we're resolving we're
only interested in the component to the
right and because it includes the angle
of 40° in this 90° angle then it's going
to be cosine remember it's always cosine
when you include an
angle so that would
be+ 7 cosine or cos for short of
40° as for the P Force well that acts
perpendicularly to the direction that
we're resolving in so we can neglect
this it won't have any
effect now we come on to the final force
the Q here now because it's not on the
dotted line here we have to split it
into two components and the components
of Q would be one to the left and one
downwards the one downwards has no
effect because it'll be perpendicular to
the direction but the one along the
dotted line here to the left would be Q
cos
20° because it contains the angle
between the force and the direction we
want to resolve in so that's going to
act in the negative sense here so that
be minus Q COS of
20° and this is the resultant force
acting on our particle here but it's in
equilibrium so that resultant would be
equal to
zero so all we need to do is rearrange
this now to make Q the subject so if you
add Q cos 20 to both sides and then
divide by cos 20 you'll end up with Q
equaling 9 + 7 cos
40 de and that's all divided by COS of
20° and if you work that out in your
calculator you should find you get
15.280 and so on and if we give this to
a suitable degree of accuracy let's say
one decimal place then that would be
15.3 Newtons to one DP all right so that
gives us
Q now as for p we need to resolve in the
perpendicular direction to this that can
either be up or down it's up to you I'm
going to resolve
upwards so if we resolve upwards upwards
being
positive then we can see that all of P
acts upwards so we'll start with that
Force
P when it comes to seven Newtons though
we need to split this into two
components because it doesn't lie on
this dotted line
here if you split into two components we
said earlier it' be one to the right and
one upwards the one to the right though
won't have any effect because it's
perpendicular to this direction we're
only concerned with the upward component
of the seven
Newtons so in this angle here of
90° this part doesn't contain the 40° so
it's going to be 7 sin
40° you could say it was 50° and go 7 C
50° but as I've said earlier in other
tutorials I always prefer to to use the
angle I'm given so it's 7 sin 40° and
it'd be
positive okay so we got 7 sin
40° now as for the 9
Newtons this Force acts perpendicularly
to this direction so we don't need to
worry about that one it'll have no
effect that leaves us just now with the
Q and Q can be split into two components
one to the left and one downwards the
one to the left has no effect because
it's perpendicular to the direction
we're only concerned with one
downwards and in this 90°
angle this excludes the 20° so it must
be Q sin
20° or you could say Q cos 70° up to you
but I'm going to go for the 20 and so
it'll be minus because it downwards in
the opposite sense to this Q sin
20° and this is the resultant force
acting on the particle in the vertical
sense here so it equals zero because
it's an
equilibrium now we already know Q if we
use this unrounded value here substitute
it into here we can get P so by
rearranging this equation we have that P
equal Q sin 20 minus the 7 sin 40 so in
place of Q then I'll just write 15.
284 and so on and that's being
multiplied by the S of
20° and then we've got Min - 7 sin
40 and if you work that out in your
calculator you'll end up with 0. 7279
and so on so that means that P if we
give it to one decimal place is going to
be 0.7 Newtons then to one
DP all
right so I hope you managed to get that
one if not then I hope that this
tutorial is at least giving you some
idea then how to tackle problems of this
nature okay well that brings us then to
the end of this particular example for
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