Equilibrium of a particle - Example : ExamSolutions

Summary

This 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

00:00

🔍 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.

05:01

📏 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.

Keywords

💡Equilibrium

Equilibrium refers to a state in physics where the net force acting on a body is zero, meaning it is either at rest or moving at a constant velocity. In the video, the concept is central as the tutorial aims to find the values of forces P and Q that keep a particle in equilibrium under a system of forces. The script describes how resolving forces horizontally and vertically leads to the condition of equilibrium where the sum of all forces equals zero.

💡Resolving Forces

Resolving forces involves breaking down a force into its components along perpendicular axes, typically horizontal and vertical. This concept is crucial in the video as the tutorial demonstrates how to resolve forces to the right and upwards to determine the unknown forces P and Q. The script provides a step-by-step method for resolving the 7 Newton force into its horizontal component using cosine of the angle.

💡Cosine

The cosine function is used in trigonometry to find the adjacent side of a right-angled triangle, which in physics corresponds to the horizontal component of a force. The script explains using cosine to calculate the horizontal component of the 7 Newton force inclined at 40°, illustrating how \( \cos(40°) \) is applied to find this component.

💡Sine

Sine is a trigonometric function that relates to the ratio of the opposite side over the hypotenuse in a right triangle. In the video, sine is used to calculate the vertical component of forces. The script mentions using sine to find the upward component of the 7 Newton force, where \( \sin(40°) \) is applied.

💡Components

Components in physics refer to the individual parts of a vector when it is broken down into a sum of vectors along different axes. The script discusses how to find the components of force Q to the left and downwards, and force P upwards, which are essential for determining the equilibrium conditions.

💡Force P

Force P is one of the unknown forces in the tutorial that needs to be calculated to maintain equilibrium. The script guides through the process of resolving forces to find P, emphasizing its role in the vertical equilibrium condition where its effect is considered upwards.

💡Force Q

Force Q is another unknown force that the tutorial aims to determine. The script explains how to resolve Q into horizontal and vertical components and uses the calculated value to find P, showing Q's significance in achieving equilibrium.

💡Resultant Force

Resultant force is the single force that represents the vector sum of all forces acting on a body. In the script, the tutorial calculates the resultant force in both horizontal and vertical directions, setting them to zero for equilibrium, which allows solving for the unknown forces P and Q.

💡Trigonometry

Trigonometry plays a key role in this video as it provides the mathematical tools (sine and cosine) to resolve forces into their components. The script uses trigonometric principles to calculate the horizontal and vertical components of forces, which are essential for determining equilibrium.

💡Newton

A Newton is the SI unit of force. The script mentions various forces in Newtons (e.g., 9 Newtons, 7 Newtons), and the tutorial's goal is to calculate the magnitudes of forces P and Q in Newtons to achieve equilibrium.

💡Degrees of Freedom

Although not explicitly mentioned, the concept of degrees of freedom is implicit in the tutorial. The script's approach to resolving forces in two perpendicular directions (horizontal and vertical) corresponds to the two degrees of freedom in a planar system, which is necessary for determining equilibrium.

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