WhatsApp us

my homework help logo
Profile Image
Chat with support

Basics and Statics of Particles – Introduction

Need a hand with your homework? Get expert help now and smash those grades! 💯 Submit your assignment today!

100% human expertise, no AI. 🤖

  • Experienced Tutors
  • On-Time Delivery
  • Plagiarism-Free Work
myhomeworkhelp reviews
Order Homework Help From Experts
Please enable JavaScript in your browser to complete this form.
Please enter the details of your homework requirements.
What should we call you?
Zero spam guarantee.
For order updates only.

Hire Best Homework Helpers

MyHomeworkHelp is proud to work with a team of expert professionals with verified academic credentials, selected for their proven skills in their field of study.

Confidently hire experts from the top universities across the globe.

Benefits & Assurances

subjects covered

All Subjects Covered

subjects covered

Refund Guarantee

subjects covered

Experienced Tutors

subjects covered

Unlimited Revisions

subjects covered

On Time Delivery

subjects covered

Chat With Expert

The branch of science which deals and records the behavior of a certain body when it is either is in motion or at rest is known as engineering mechanics. This field of science is divided into 2 parts.

  1. Dynamics

In case of dynamics, it usually deals with the body and studies about it when it is in motion.

  1. Statics

And when we are talking about statics, this branch of science deals with bodies and study about them when they are at rest.

Coming back to dynamics, it is again divided into 2 parts. They are:

  • Kinetics

It is a branch of classical mechanics which studies and is concerned with the relationship between motions of particle bodies. It also deals with the reasons which cause such motions, i.e., torque and forces.

  • Kinematics

Kinematics defines the field of science which studies a body which is in motion but does not know the reason which is the cause of this motion.

If stated in simplified words, in engineering mechanics, dynamics means body in motion and the various forces which cause it, and statics means equilibrium of a body at rest.

There are few important terms that are important in mechanics. Those are as follows:

  1. Scalar quantity

Any quantity whose complete specification is based on magnitude is known as magnitude. Some of the widely used scalar quantities are:

  • Length
  • Mass
  • Temperature
  • Time
  1. Vector quantity

Any quantity which is specified fully by direction and magnitude and has a one dimensional measurement like weight or temperature is known as avector quantity.Depiction of vector quantity in figurative aspect is shown as a straight line from one point to another with an arrow between that line. Length of that straight line illustrates magnitude whereas the arrow in it shows direction of that vector. Some of the widely used vector quantities are:

  • Force
  • Velocity
  • Momentum
  • Acceleration
  1. Particle

Possessing an indefinite and extremely small volume of a body which has a negligible dimension is known as a particle. In this case, there is an assumption that mass of the body is concentrated at a certain point. Therefore, on the basis of this information, particle is believed to be that point,and the mass of it are concentrated at that point of particle.

  1. Law of Triangle of Forces

According to this law, when 2 forces act a point, which are direction and magnitude of a triangle (2 adjacent sides), that triangle’s closing side is taken in reverse order. That order of equilibrium depicts the resultant of both the forces representing direction and magnitude.

The figure representing the law of triangle of forces is in a slanted diamond pattern with the angle between the forces R and P as. The other angle  depicts the angle between the forces Q and P.

  1. Law of parallelogram of forces

When 2 forces actat a certain point in a plane, to discern the result of those forces,we take the help of law of parallelogram of forces. As per the definition in engineering mechanics, “2 forces which are depicted by direction and magnitude are acting at a certain point by 2 adjacent sides of a parallelogram. Its outcome is discerned by direction and magnitude of the parallelogram’s diagonal section which passes through that point.”

Suppose we take a figure of a parallelogram which has 2 forces with names P and Q. both the forces are joined at a point which is depicted by O. the horizontal line depicting the force P represents magnitude with the direction from point O to A. similar to this, an angled vertical line which depicts the force Qrepresents magnitude and is towards the direction OQ.

Angle between the forces P and Q is represented as ‘.’When these 2 forces pass through O, which is the diagonal of the parallelogram, OP and OQ are the adjacent sides of it. The outcome of these 2 forces is acquired in magnitude.

After the creation of the parallelogram, the other point where A and B joins (opposite of O) is named C. resultant of both these forces which is illustrated in the direction from O to C is named as R.

As we have already constructed the diagram of aparallelogram and have named the forces, we will find out the magnitude of resultant.

Magnitude of Resultant (R)

Taking the help of the drawn parallelogram with different names in its various point, we will find the magnitude of the resultant which is depicted by R.

We will draw a horizontal line from A and a vertical line from G downwards so that it joins at a point of their cross section E. this will form a right angle triangle with points AD and CD.

Now, let angle between P and Q be ‘’.this angle will be some what like ∠AOB.
∠DAC = ∠AOB

This depicts corresponding angles.

In case of the portion of the parallelogram with OACB as its 4 points, you will find AC to be equal and parallel to OB. Therefore, aforce parallel to AC will be Q.

And when it is about triangle ACD,

CD =AC sin  = Q sin

AD = AC cos  = Q cos

In case of triangle OCD,

OC2 = OD2 + DC2

But when equating for OC,

OC = R

OD = OA +AD = P + Q cos

DC =Q sin

The next step for R will be,

R 2 = (P + Q cos )2 + (Q sin )2

= p2 + Q2 cos2 a+ 2PQ cos < X+ Q2 sin2α

= p2 + Q2 (cos2α+ sin2α)+ 2PQ cos α

= P2 + Q2 + 2PQ cos α

So finally, we will find,

R = √ p2 + Q2 + 2PQ cos α (equation 1.1)

This equation describes the magnitude of R (resultant force).

After finding the equation for the magnitude of the resultant, we will find the direction of it.

 Direction of Resultant
As per the figure of aparallelogram, let us consider the angle made by OA to be θ.

After this, we will find the value of θ from OCD (triangle)

Tan θ = CD / OD = Q sin α / P + Q cos α

θ = tan -1 (Q sin α / P + Q cos α) (equation 1.2)

This value gives the direction of R.

As we have seen that the resultant is found out with the help of (tan) rule, we can also find the direction taking the help of sine rule.

According to this rule, incase of triangle OAC,

OC = R

AC = Q

OA = P

∠OACis (180 – α)

Now the value of ∠ACO is,

= 180- [θ + 180 – α]

= [(α – θ)]

On calculating with the help of sin θ,

sin θ / AC = sin (180 –α) OC = sin (α – θ) / OA

sin θ / Q = sin (180 – α) / R = sin (α – θ) / P

Now there are 2 cases that are very important.

Case #1

You are already aware of the 2 forces of the parallelogram which are acting at an angle and are equal.

According to this information, direction and magnitude of a resultant are showcased in equational form. So the equation in this case is,

R = √(P2 + Q2 + 2PQ cos α)

= √(P2 +P2 +  2P  x  P  x  cos α)

= √(2P2 +  2P2  cos α)

= √(2P2 (1 +  cos α))

= √(2P2  x  2 cos2 α/2)

= √(4P2  x  cos2 α/2)

= 2P cos α/2

On equating further,

θ = tan -1((Q sin⁡α)/(P+Q cos⁡α ))

= tan -1((P sin⁡α)/(P+P cos⁡α ))

= tan -1(P sin⁡α)/(P (1+cos⁡〖 α〗))

= tan -1sin⁡α/(1+cos⁡〖 α〗 )

= tan -1(〖2 sin〗⁡〖α/2〗  cos⁡〖α/2〗)/(2 cos2⁡〖α/2〗 )

= tan -1sin⁡〖α/2〗/cos⁡〖α/2〗

= tan -1(tan⁡〖α/2〗 )

=  α/2

Case #2

In the second case, at right angles, both forces P and Q act together. When this happens, the angle between the 2 which is depicted by ‘α’ is 90°.

With the help of previously found equation (1.1), we can easily get the resultant of the parallelogram’s magnitude. Its calculation will stand at,

R = √(P2 + Q2 + 2PQ cos α)

= √(P2 + Q2 + 2PQ cos 90°)

= √(P2 + Q2)

This is because the value of cos 90° is zero.

Now with the help of equation (1.2), we will find the calculation of resultant’s direction. In this case, it stands to be,

θ = tan -1 (Q sin α / P + Q sin α)

= tan -1 (Q sin 90° / P + Q cos 90°)

= tan -1 Q /P

It is with the help of both the stated cases above;we can come to a conclusion that it is not mandatory for any of the forces to be on the x axis.According to the equations and the figure created with the assistance of law of triangle of forces, it is clear that the forces P and Q can be in any direction.From thesecases, it is also evident that resultant of the 2 forces which are ‘α’ is equation (1.1), i.e.,

R = √ p2 + Q2 + 2PQ cos α

Now for obtaining the direction of that resultant, we need to take the help of equation (1.2), i.e.,

θ = tan -1 (Q sin α / P + Q cos α)

But the resultant angle made with the help of the direction of force P will be angle ‘e’ in this respect.

6.    Lami’s theorem

Lami’s theorem is stated as an equation that interconnects 3 forces which keep a body in static equilibrium. Three forces in this respect are:

A. Concurrent forces

B. Coplanar forces

C. Non-collinear forces

As per the working of these forces, when these acts at a certain point which is in equilibrium, each of these forces become proportional to the sine of angle between remaining 2 forces.

To understand the concept clearly, let us consider 3 forces facing towards different directions. Also, let us give a figurative name of these 3 forces as P, Q and R.Their conjoining point is O. all the forces at that fixed point are in equilibrium.

After a figure is created with the help of this information, we can see the 3 differentangles formed with each other.

1. Angle between 2 forces R and P is represented byγ

2. Angle between 2 forces Q and R is represented byβ

3. Angle between 2 forces P and Q is represented byα

As it is already stated above in Lami’s theorem, P is directly proportional to the sine angle between sin β (2 forces Q and R). as per this information,

P / sin β is found constant

Likewise,

R / sin α is constant

Q / sin γ is also constant

Therefore,

P / sin β = R / sin α = Q / sin γ

This equation gave the value to Lami’s theorem. But in initial case, the proof of this equation being genuine was being questioned upon. So, an equational proof was also given of this theorem.

Proof of Lame’s Theorem

To produce Lami’s theorem, all the 3 forces were constructed in such a way that it looked like a triangle. In this case, all the forces are in equilibrium and are joined at a certain point.

P and R are joined with O, P and Q are joined with M, and R and Q are joined with N.

1. The angle at O is (180 – γ)

2. Angle at N is (180 – β)

3. Angle at M is (180 – α)

It can also be depicted as,

P / sin (180 – β) = Q / sin (180 – γ) = R / sin (180 – α)

Or its representation can be,

P / sin β = Q / sin γ = R / sin α

Hence the theorem is proved.

An important fact about this Lami’s theorem is that its forces act either towards their fixed point or away from it.

 

Links of Previous Main Topic:-

Links of Next Mechanical Engineering Topics:-

Your Price: $15.00 $13.99 🔥

Pricing may change based on requirements

Get Homework Help In

Maths and Statistics

Science

Humanities & Arts

l

Essay Writing

How Our Homework Service Works

FAQ number 1

Share your requirements

FAQ number 2

Make secure payment

FAQ number 3

Get homework answers

FAQ - Our Homework Help Services

We have put together some commonly asked questions.

K
L

Is there a website that can do my homework?

Yes, if you are looking for such a site, MyHomeworkHelp is one of the leading homework help websites that can do your homework and assist with various academic needs. Here's all you need to know about our team and how they provide flawless homework help.

If you find yourself thinking, "I need to pay someone to do my homework," our team is ready to assist. It's common for students to seek help with their homework, and our experts are prepared to provide personalized support tailored to your needs.

You can visit www.myhomeworkhelp.com to get all types of homework-related help from our experts.

K
L

What is the best homework help website?

The best homework help website enhances students' experience of getting homework assistance from experts. MyHomeworkHelp, recognized as one of the best online homework help platforms, serves as both a guide and a solution provider for students. We've been rocking the homework help scene since 2012, assisting over 20k students, and we must be doing something right because more than 70% of them keep coming back for more. Whether you need help understanding concepts, completing assignments, or getting your homework done, MyHomeworkHelp is here to support your academic journey as the best college homework website.

K
L

How can I get help with my homework?

1. Tell us your homework details

To get started with online homework help, simply choose the type of homework assistance you need. When filling out the order form, make sure to provide detailed instructions, including the deadline, number of pages, grading rubric, and any other relevant information.

2. Confirm your order

Review the details of your homework order and proceed to make a secure payment using VISA, Mastercard, or PayPal. Once the payment is processed, our experts will begin working on your homework immediately.

3. Receive your order

The completed homework will be delivered directly to your email. This ensures that students receive their homework assistance on time and can meet their academic requirements without stress.

K
L

Do you guarantee plagiarism-free homework answers?

We take academic integrity very seriously and understand the importance of providing homework answers free of plagiarism. Our tutors are committed to delivering original and unique content for every assignment. Here are the steps we take to ensure your homework is plagiarism-free:

  1. Original Work: Each assignment is completed from scratch based on your specific requirements and guidelines.
  2. Expert Tutors: Our tutors are highly qualified and experienced in their respective fields, ensuring the work they produce is both original and high-quality.
  3. Plagiarism Detection: We use advanced plagiarism detection software to scan every assignment before delivery. This ensures that the content is unique and free of any copied material.
  4. No AI Usage: We guarantee that all work is 100% human-crafted expertise, with no AI-generated content involved. This ensures the authenticity and quality of the work provided.
  5. Quality Assurance: Our quality assurance team reviews each assignment for originality and adherence to academic standards before it is sent to you.

By following these stringent measures, we guarantee that the homework answers you receive are completely free of plagiarism and AI involvement. You can trust us to provide you with original and high-quality academic assistance.

For more information on our guarantees, please visit our guarantees page.

K
L

Is your homework service confidential?

Absolutely! Confidentiality is one of the basic guarantees of our service at MyHomeworkHelp.com.

When our customers make their homework orders, they know their information remains 100% confidential.

Even your homework writer will not be provided with your personal details. So, when you hire a professional homework helper on our website, rest assured that your information will remain private and safe.

For a comprehensive understanding of how we protect your privacy, please refer to our Privacy Policy.

K
L

How much do I need to pay to get my homework done?

We do not have monthly fees or minimum payments for our services. Each homework assignment is quoted a unique price, based on the tutor’s estimate of the time required to complete your work. There are no obligations, and you are free to discuss the price quote with the tutor.

When you decide to pay someone to do my homework, our pricing starts at $13.99 per page. The final pricing depends on various factors including the page count, deadline, nature of the task, and the homework writer’s level of expertise.

We offer flexible payment options, allowing you to make a partial payment to start working on your assignment and pay the remaining amount upon completion. This ensures no pressure of up-front payment and no hidden order costs.

Our dynamic pricing model takes into account the type of assignment, its urgency, and the complexity of the task. For instance, shorter deadlines and more complex assignments may increase the price. You can get an instant quote by selecting your assignment type, deadline, and page count on our website.

Get the help with your homework at competitive rates, tailored to your specific needs and deadlines. For more detailed information on pricing, please visit our pricing page.

K
L

Who are the experts who will help me with my homework?

We work with subject experts from all over the world. We hire them with the help of complex tests in the disciplines of their expertise. We also verify the IDs and academic transcripts of our homework writers to be sure that our experts have a bachelor's or master’s or a PhD degree. This is how we can be sure of the proficiency of our panel of experts who will help you with your homework. Here's all you need to know about our team and how they provide flawless homework help.

  1. Our team comprises top-tier homework helpers who excel in addressing and resolving questions on various topics. Each tutor effectively utilizes their expertise to concentrate on the core subject matter and thoroughly analyze it.
  2. Our professional tutors specifically craft each assignment tailored to students' requirements, including preferred formats, citation styles, and essential elements.
  3. Whether it’s coursework, dissertations, essays, or case studies, our platform guarantees timely college homework help across a wide range of academic papers. Don’t hesitate any longer; connect with our tutors when you seek help with your homework for perfect assignment solutions and improved grades.

K
L

Can I chat with my homework helper for updates?

Yes, you can certainly chat with your homework helper on our secure chatboard platform. MyHomeworkHelp.com provides a direct communication channel between you and your assigned expert (while maintaining privacy). This feature allows you to ask questions, clarify instructions, and provide additional information about your homework order to ensure that the help you receive is tailored to your specific needs.

You'll be notified by email whenever there are new messages for you, but we advise regularly checking your personal page to ensure no critical updates from your writer are overlooked.

Should you have an urgent matter to address, do not hesitate to contact the support team for assistance. This system ensures that students can stay updated on their homework paper progress and maintain clear communication with their experts.

K
L

What makes you the best homework helper?

MyHomeworkHelp is ranked as one of the best online homework helpers, providing exceptional support to students. We've been excelling in the homework assistance arena since 2012, assisting over 200k students, and consistently achieving high satisfaction rates, with more than 70% of our students returning for additional help. But do not take our word for it! See our independent reviews on google, facebook, sitejabber, and on Trustpilot. For all collated reviews, see here

We ensure our prices remain competitive and offer attractive discounts to make our services accessible. Our commitment to delivering plagiarism-free solutions guarantees the integrity and originality of your assignments. These attributes, combined with our dedication to personalized support and timely delivery, establish MyHomeworkHelp as the premier homework helper for all your academic needs.

K
L

How will I receive my homework order?

You'll receive the initial version of your paper as a secure, non-editable PDF within your specified deadline. This allows you to review the content and decide if any revisions or edits are required. Our homework service will refine the homework paper based on your feedback.

Should the document meet your expectations and not require any changes, you can simply approve the order and access the editable Word .doc file.