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Listen: we know homework isn’t fun, but it is a good way to reinforce the ideas and concepts you’ve learned in class. But what if you’re really struggling with your homework assignments?

If you’ve looked online for a little extra help with your take-home assignments, you’ve probably stumbled across websites claiming to provide the homework help and answers students need to succeed . But can homework help sites really make a difference? And if so, which are the best homework help websites you can use? 

Below, we answer these questions and more about homework help websites–free and paid. We’ll go over: 

• The basics of homework help websites
• The cost of homework help websites 
• The five best homework websites out there 
• The pros and cons of using these websites for homework help 
• The line between “learning” and “cheating” when using online homework help 
• Tips for getting the most out of a homework help website

So let’s get started! 

The Basics About Homework Help Websites–Free and Paid

Homework help websites are designed to help you complete your homework assignments, plain and simple. 

What Makes a Homework Help Site Worth Using

Most of the best sites allow users to ask questions and then provide an answer (or multiple possible answers) and explanation in seconds. In some instances, you can even send a photo of a particular assignment or problem instead of typing the whole thing out! 

Homework help sites also offer more than just help answering homework questions. Common services provided are Q&A with experts, educational videos, lectures, practice tests and quizzes, learning modules, math solving tools, and proofreading help. Homework help sites can also provide textbook solutions (i.e. answers to problems in tons of different textbooks your school might be using), one-on-one tutoring, and peer-to-peer platforms that allow you to discuss subjects you’re learning about with your fellow students. 

And best of all, nearly all of them offer their services 24/7, including tutoring! 

What You Should Should Look Out For

When it comes to homework help, there are lots–and we mean lots –of scam sites out there willing to prey on desperate students. Before you sign up for any service, make sure you read reviews to ensure you’re working with a legitimate company. 

A word to the wise: the more a company advertises help that veers into the territory of cheating, the more likely it is to be a scam. The best homework help websites are going to help you learn the concepts you’ll need to successfully complete your homework on your own. (We’ll go over the difference between “homework help” and “cheating” a little later!) 

You don't need a golden piggy bank to use homework help websites. Some provide low or no cost help for students like you!

How Expensive Are the Best Homework Help Websites?

First of all, just because a homework help site costs money doesn’t mean it’s a good service. Likewise, just because a homework help website is free doesn’t mean the help isn’t high quality. To find the best websites, you have to take a close look at the quality and types of information they provide! 

When it comes to paid homework help services, the prices vary pretty widely depending on the amount of services you want to subscribe to. Subscriptions can cost anywhere from $2 to $150 dollars per month, with the most expensive services offering several hours of one-on-one tutoring with a subject expert per month.

The 5 Best Homework Help Websites 

So, what is the best homework help website you can use? The answer is that it depends on what you need help with. 

The best homework help websites are the ones that are reliable and help you learn the material. They don’t just provide answers to homework questions–they actually help you learn the material. 

That’s why we’ve broken down our favorite websites into categories based on who they’re best for . For instance, the best website for people struggling with math might not work for someone who needs a little extra help with science, and vice versa. 

Keep reading to find the best homework help website for you! 

Best Free Homework Help Site: Khan Academy

• Price: Free!
• Best for: Practicing tough material 

Not only is Khan Academy free, but it’s full of information and can be personalized to suit your needs. When you set up your account , you choose which courses you need to study, and Khan Academy sets up a personal dashboard of instructional videos, practice exercises, and quizzes –with both correct and incorrect answer explanations–so you can learn at your own pace. 

As an added bonus, it covers more course topics than many other homework help sites, including several AP classes.

Runner Up: Brainly.com offers a free service that allows you to type in questions and get answers and explanations from experts. The downside is that you’re limited to two answers per question and have to watch ads. 

Best Paid Homework Help Site: Chegg

• Price: $14.95 to $19.95 per month
• Best for: 24/7 homework assistance  

This service has three main parts . The first is Chegg Study, which includes textbook solutions, Q&A with subject experts, flashcards, video explanations, a math solver, and writing help. The resources are thorough, and reviewers state that Chegg answers homework questions quickly and accurately no matter when you submit them.  

Chegg also offers textbook rentals for students who need access to textbooks outside of their classroom. Finally, Chegg offers Internship and Career Advice for students who are preparing to graduate and may need a little extra help with the transition out of high school. 

Another great feature Chegg provides is a selection of free articles geared towards helping with general life skills, like coping with stress and saving money. Chegg’s learning modules are comprehensive, and they feature solutions to the problems in tons of different textbooks in a wide variety of subjects. 

Runner Up: Bartleby offers basically the same services as Chegg for $14.99 per month. The reason it didn’t rank as the best is based on customer reviews that say user questions aren’t answered quite as quickly on this site as on Chegg. Otherwise, this is also a solid choice!

Best Site for Math Homework Help: Photomath

• Price: Free (or $59.99 per year for premium services) 
• Best for: Explaining solutions to math problems

This site allows you to t ake a picture of a math problem, and instantly pulls up a step-by-step solution, as well as a detailed explanation of the concept. Photomath also includes animated videos that break down mathematical concepts to help you better understand and remember them. 

The basic service is free, but for an additional fee you can get extra study tools and learn additional strategies for solving common math problems.

Runner Up: KhanAcademy offers in-depth tutorials that cover complex math topics for free, but you won’t get the same tailored help (and answers!) that Photomath offers. 

Best Site for English Homework Help: Princeton Review Academic Tutoring

• Price: $40 to $153 per month, depending on how many hours of tutoring you want 
• Best for: Comprehensive and personalized reading and writing help 

While sites like Grammarly and Sparknotes help you by either proofreading what you write via an algorithm or providing book summaries, Princeton Review’s tutors provide in-depth help with vocabulary, literature, essay writing and development, proofreading, and reading comprehension. And unlike other services, you’ll have the chance to work with a real person to get help. 

The best part is that you can get on-demand English (and ESL) tutoring from experts 24/7. That means you can get help whenever you need it, even if you’re pulling an all-nighter! 

This is by far the most expensive homework site on this list, so you’ll need to really think about what you need out of a homework help website before you commit. One added benefit is that the subscription covers over 80 other subjects, including AP classes, which can make it a good value if you need lots of help!  

Best Site for STEM Homework Help: Studypool

• Best for: Science homework help
• Price: Varies; you’ll pay for each question you submit

When it comes to science homework help, there aren’t a ton of great resources out there. The best of the bunch is Studypool, and while it has great reviews, there are some downsides as well. 

Let’s start with the good stuff. Studypool offers an interesting twist on the homework help formula. After you create a free account, you can submit your homework help questions, and tutors will submit bids to answer your questions. You’ll be able to select the tutor–and price point–that works for you, then you’ll pay to have your homework question answered. You can also pay a small fee to access notes, lectures, and other documents that top tutors have uploaded. 

The downside to Studypool is that the pricing is not transparent . There’s no way to plan for how much your homework help will cost, especially if you have lots of questions! Additionally, it’s not clear how tutors are selected, so you’ll need to be cautious when you choose who you’d like to answer your homework questions.  

What Are the Pros and Cons of Using Homework Help Sites?

Homework help websites can be a great resource if you’re struggling in a subject, or even if you just want to make sure that you’re really learning and understanding topics and ideas that you’re interested in. But, there are some possible drawbacks if you don’t use these sites responsibly. 

We’ll go over the good–and the not-so-good–aspects of getting online homework help below. 

3 Pros of Using Homework Help Websites 

First, let’s take a look at the benefits. 

#1: Better Grades Beyond Homework

This is a big one! Getting outside help with your studies can improve your understanding of concepts that you’re learning, which translates into better grades when you take tests or write essays. 

Remember: homework is designed to help reinforce the concepts you learned in class. If you just get easy answers without learning the material behind the problems, you may not have the tools you need to be successful on your class exams…or even standardized tests you’ll need to take for college. 

#2: Convenience

One of the main reasons that online homework help is appealing is because it’s flexible and convenient. You don’t have to go to a specific tutoring center while they’re open or stay after school to speak with your teacher. Instead, you can access helpful resources wherever you can access the internet, whenever you need them.

This is especially true if you tend to study at off hours because of your extracurriculars, work schedule, or family obligations. Sites that offer 24/7 tutoring can give you the extra help you need if you can’t access the free resources that are available at your school. 

#3: Variety

Not everyone learns the same way. Maybe you’re more of a visual learner, but your teacher mostly does lectures. Or maybe you learn best by listening and taking notes, but you’re expected to learn something just from reading the textbook . 

One of the best things about online homework help is that it comes in a variety of forms. The best homework help sites offer resources for all types of learners, including videos, practice activities, and even one-on-one discussions with real-life experts. 

This variety can also be a good thing if you just don’t really resonate with the way a concept is being explained (looking at you, math textbooks!).

Not so fast. There are cons to homework help websites, too. Get to know them below!

3 Cons of Using Homework Help Websites 

Now, let’s take a look at the drawbacks of online homework help. 

#1: Unreliable Info

This can be a real problem. In addition to all the really good homework help sites, there are a whole lot of disreputable or unreliable sites out there. The fact of the matter is that some homework help sites don’t necessarily hire people who are experts in the subjects they’re talking about. In those cases, you may not be getting the accurate, up-to-date, and thorough information you need.

Additionally, even the great sites may not be able to answer all of your homework questions. This is especially true if the site uses an algorithm or chatbot to help students…or if you’re enrolled in an advanced or college-level course. In these cases, working with your teacher or school-provided tutors are probably your best option. 

#2: No Clarification

This depends on the service you use, of course. But the majority of them provide free or low-cost help through pre-recorded videos. Watching videos or reading info online can definitely help you with your homework… but you can’t ask questions or get immediate feedback if you need it .

#3: Potential For Scamming 

Like we mentioned earlier, there are a lot of homework help websites out there, and lots of them are scams. The review comments we read covered everything from outdated or wrong information, to misleading claims about the help provided, to not allowing people to cancel their service after signing up. 

No matter which site you choose to use, make sure you research and read reviews before you sign up–especially if it’s a paid service! 

When Does “Help” Become “Cheating”?

Admittedly, whether using homework help websites constitutes cheating is a bit of a grey area. For instance, is it “help” when a friend reads your essay for history class and corrects your grammar, or is it “cheating”? The truth is, not everyone agrees on when “help” crosses the line into “cheating .” When in doubt, it can be a good idea to check with your teacher to see what they think about a particular type of help you want to get. 

That said, a general rule of thumb to keep in mind is to make sure that the assignment you turn in for credit is authentically yours . It needs to demonstrate your own thoughts and your own current abilities. Remember: the point of every homework assignment is to 1) help you learn something, and 2) show what you’ve learned. 

So if a service answers questions or writes essays for you, there’s a good chance using it constitutes cheating. 

Here’s an example that might help clarify the difference for you. Brainstorming essay ideas with others or looking online for inspiration is “help” as long as you write the essay yourself. Having someone read it and give you feedback about what you need to change is also help, provided you’re the one that makes the changes later. 

But copying all or part of an essay you find online or having someone write (or rewrite) the whole thing for you would be “cheating.” The same is true for other subjects. Ultimately, if you’re not generating your own work or your own answers, it’s probably cheating.

5 Tips for Finding the Best Homework Help Websites for You

Now that you know some of our favorite homework help websites, free and paid, you can start doing some additional research on your own to decide which services might work best for you! Here are some top tips for choosing a homework help website. 

Tip 1: Decide How You Learn Best 

Before you decide which site or sites you’re going to use for homework help, y ou should figure out what kind of learning style works for you the most. Are you a visual learner? Then choose a site that uses lots of videos to help explain concepts. If you know you learn best by actually doing tasks, choose a site that provides lots of practice exercises.

Tip 2: Determine Which Subjects You Need Help With

Just because a homework help site is good overall doesn’t mean that it’s equally good for every subject. If you only need help in math, choose a site that specializes in that area. But if history is where you’re struggling, a site that specializes in math won’t be much help. So make sure to choose a site that you know provides high-quality help in the areas you need it most. 

Tip 3: Decide How Much One-On-One Help You Need 

This is really about cost-effectiveness. If you learn well on your own by reading and watching videos, a free site like Khan Academy is a good choice. But if you need actual tutoring, or to be able to ask questions and get personalized answers from experts, a paid site that provides that kind of service may be a better option.

Tip 4: Set a Budget

If you decide you want to go with a paid homework help website, set a budget first . The prices for sites vary wildly, and the cost to use them can add up quick. 

Tip 5: Read the Reviews

Finally, it’s always a good idea to read actual reviews written by the people using these homework sites. You’ll learn the good, the bad, and the ugly of what the users’ experiences have been. This is especially true if you intend to subscribe to a paid service. You’ll want to make sure that users think it’s worth the price overall!

What’s Next?

If you want to get good grades on your homework, it’s a good idea to learn how to tackle it strategically. Our expert tips will help you get the most out of each assignment…and boost your grades in the process.

Doing well on homework assignments is just one part of getting good grades. We’ll teach you everything you need to know about getting great grades in high school in this article.

Of course, test grades can make or break your GPA, too. Here are 17 expert tips that’ll help you get the most out of your study prep before you take an exam.

Ashley Sufflé Robinson has a Ph.D. in 19th Century English Literature. As a content writer for PrepScholar, Ashley is passionate about giving college-bound students the in-depth information they need to get into the school of their dreams.

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9 Best Homework Help Websites

July 22, 2023

When I was in high school, resources for extra homework help weren’t exactly abundant. If you were struggling with a Shakespeare sonnet, you could always run to the bookstore and pick up a CliffNotes guide. SparkNotes was also gaining in popularity. But these early homework help resources had limited catalogs and were focused primarily on literature. Today, I imagine students suffer from the opposite problem—having too many choices when it comes to homework help websites. When the options are seemingly endless, knowing what to look out for takes on an added importance. Below, I’ll go through a list of 9 stand-out homework help websites and briefly discuss what makes them worth a visit.

Homework Help Websites – The Basics

The best homework help websites do more than just spit out an answer to that tricky math problem. They actually help students learn the material. Common features of homework help websites are educational videos and lectures, practice tests and quizzes, study tools like flashcards, and Q&As with experts. Many sites offer features that allow students to ask specific questions and get real-time feedback. There are also a number of services that offer one-on-one tutoring. Some homework help sites are free, while others require a paid subscription.

1) Khan Academy

Khan Academy is an amazing resource for students of all ages. It’s free, and it really is an academy—it offers full courses in a wide array of subjects, from pre-K math to high school physics. The courses consist of readings, video lectures, practice exercises, and quizzes. The breadth of material is impressive. In math alone, I see course listings for Algebra 1 and 2, Geometry, Trigonometry, Precalculus, Statistics, Multivariable calculus—you get the idea. Khan Academy also offers a wide variety of AP courses, state-specific curricula, test-prep programs, and life skill courses, like personal finance.

It’s important to note that Khan Academy isn’t a one-on-one tutoring platform. But because of their extensive library of material, the search function is especially powerful. Try it out. I did a search for argumentative essay help, and found a comprehensive guide to writing argumentative essays that was a part of a larger writing course.

Chegg is a paid homework help service. Unlike Khan Academy, Chegg isn’t built around specific courses. Rather, it offers a variety of homework-support resources. Among those resources are plagiarism and grammar checkers, a proofreading service, and a “math solver”, which allows students to enter a problem and get back both a solution and a detailed step-by-step explanation of how the problem was solved. Perhaps the most powerful tool Chegg offers is its “Expert Q&A” feature. This service allows students to take a picture of their homework problem, upload it to the site, and get a detailed response in return. Chegg’s emphasis on process and explanation make it a valuable educational resource for students—not just a way to get a quick answer.

Best Homework Help Websites (Continued)

Quizlet is a well-known and worthwhile study resource. It offers a variety of courses, and it also has an expert-response feature. But Quizlet’s best feature, in my option, is the flashcards tool. Students can create their own digital decks of cards and practice them on Quizlet—just like an old fashion set of index cards. I had a ton of success using Quizlet’s flash card feature to help me memorize words for my foreign language requirement in college. It’s a simple but powerful tool. Although often maligned as a learning method, rote rehearsal and spaced repetition are effective ways to encode information . Quizlet’s flashcard feature is a great way to put those techniques into practice.

4) Socratic

is an AI-powered homework support app that allows students to type or take pictures of questions and receive solutions right away. Since it works with AI, it relies on the web’s vast stores of accumulated knowledge—you’re not interacting with a human tutor. Nonetheless, I found it to be an extremely helpful tool. I tried it out first using a specific math problem. In just a few seconds I was provided with the solution and an explainer with relevant formulas, plus a graphic to help visualize the underlying logic. There were also suggested links to additional resources. For example, when I asked Socratic to explain how the German genitive case works, it suggested a YouTube video and a number of articles from blogs and other language-learning sites.

Since Socratic doesn’t feature courses or one-on-one tutoring support, I wouldn’t lean on it if I were really struggling in a particular class. But as a tool to check your work, make sure you’re on the right track, and become aware of additional resources, it’s worth a download.

5) Photomath

Photomath is, as you might have guessed, a site for math homework help. Like other homework help websites, Photomath allows students to take a picture of a problem and receive an instant, step-by-step solution. Included along with the solution is an explanation of relevant concepts and formulas, plus videos covering mathematical concepts. Photomath does offer a few basic courses, too. So if in addition to homework-specific help you want to brush up on the basics, they’ve got you covered in arithmetic, algebra, and calculus crash courses.

6) Studypool

Studypool is a paid homework support service that provides solutions to specific questions. Studypool offers support in all the major subjects, with a particular emphasis on science. Students can ask questions on everything from anatomy to physics. Like other services, students upload their exact questions or problems directly to the site. But Studypool’s payment model is a bit different: instead of paying for tutoring time or a monthly subscription, students pay for solutions to each question they submit. When a student submits a question, tutors submit bids to answer them. The student then can select which tutor/price option works best. After students select the price and tutor they want, they’re connected with the tutor and given the solution and explanation via messenger.

The draw of Studypool is that it gives students access to real (i.e., human) tutors who are experts in their field. The downside is that pricing isn’t transparent, and students pay per question.

7) College Info Geek

College Info Geek is the study-support website that I wish I knew about when I was in high school and college (they didn’t pay me to write that, I swear). The site focuses not on specific courses or questions, but on how to become a more effective learner. Here it’s all about “learning how to learn”—study tips, memorization and note-taking techniques, and much more. The articles are well-researched, clearly-communicated, practical, and comprehensive. For example, the article on how to improve your memory includes a breakdown of the different types of memory processes, memorization techniques, and even a discussion of how nutrition affects memory. College Info Geek is a great resource for everyone, not just high school and college students.

8) SparkNotes

Yes, Sparknotes made the list! The site offers lessons in a whole bunch of subjects—biology, chemistry, computer science, history, philosophy, math—but its specialty is literature. SparkNotes provides summaries and analyses of novels, short stories, poetry, and non-fiction, from The Canterbury Tales to Toni Morrison, Saul Bellow, and Junot Diaz. SparkNotes breaks down books into sub-sections and provides synopses and analyses for each section. There are also separate pages for character breakdowns, discussions of themes and motifs, and explanations of important quotes. I’d caution against using SparkNotes if you’re trying to “hack” a novel or poem and get simple answers about what it “means.” But as a way to supplement your own understanding and interpretation, it’s a great resource. Shmoop is also worth checking out for extra support in literature, poetry, mythology, and the history of literary movements.

9) Grammarly

I’m not sure if Grammarly is an obvious or unexpected choice to round out the list. Either way, it deserves a mention here. Grammarly is a writing tool. It checks and suggests corrections for incorrectly spelled words and misused punctuation. But Grammarly also scans and corrects for things like clarity and vocab usage. It flags sentences that are vague, or overly wordy, and alerts you if you’re using that flashy vocab word incorrectly. It even gives suggestions if it thinks your writing is a bit bland. I don’t see Grammarly as a crutch, but rather as a tool. It can help you master those pesky recurring grammar and usage issues. Always mix up effect and affect? Grammarly will continue to course correct until you’ve got it down yourself.

Homework Help Websites – Final Thoughts

None of the above homework help websites should be seen as a panacea. Each has benefits and drawbacks, strengths and weak points. The list is far from exhaustive. And the sites don’t have to be used in isolation. Try a few out, mix and match. College Info Geek is an excellent supplement to any study regimen. Socratic can be used as a tool to check answers for math homework, and at the same time you can use Grammarly to describe your problem to a tutor on Chegg. At their best, these sites are more than quick fixes to stubborn homework problems—they’re aids to genuine learning.

Additional Resources

You should also check out College Transitions’ “ High School Success ” blogs for help with a number of common high school assignments, including:

• Lord of the Flies Summary & Analysis 
• The Great Gatsby and The American Dream
• Analysis of Hamlet’s “To be or not to be” Speech
• Robert Frost’s Road Not Taken Analysis 
• High School Success

Dane Gebauer

Dane Gebauer is a writer and teacher living in Miami, FL. He received his MFA in fiction from Columbia University, and his writing has appeared in Complex Magazine and Sinking City Review .

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HackerRank Lily's Homework problem solution

In this HackerRank Lily's Homework problem solution we have given the array arr, determine and return the minimum number of swaps that should be performed in order to make the array beautiful.

Problem solution in Python.

Problem solution in java., problem solution in c++..

Posted by: YASH PAL

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Maybe I am missing something but I think this is not quite correct.. long [] reverseValue = IntStream.rangeClosed(1, values.length).mapToLong( i -> values[values.length - i]).toArray(); System.out.println(Math.min(swaps(values), swaps(reverseValue))); providing "values" and "reverseValue" arrays as parameters to "swaps" method is the same thing.. both of them are sorted to the same array in the "swaps" method.. I think the idea was to do a reverse sort (descending) INSIDE the "swaps" method..

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Statistics 110: Probability

Strategic Practice and Homework Problems

Actively solving practice problems is essential for learning probability. Strategic practice problems are organized by concept, to test and reinforce understanding of that concept.  Homework problems  usually do not say which concepts are involved, and often require combining several concepts. Each of the Strategic Practice documents here contains a set of strategic practice problems, solutions to those problems, a homework assignment, and solutions to the homework assignment. Also included here are the exercises from the  book that are marked with an s, and solutions to those exercises. 

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ME 274: Basic Mechanics II

Homework h6.k - sp24.

DISCUSSION THREAD

Any questions?? Please ask/answer questions regarding this homework problem through the "Leave a Comment" link above.

24 thoughts on “Homework H6.K - Sp24”

In order to solve this problem, it would be useful to review base excitation which can be found on page 416 in the book. It would also be helpful to look at problem 6.C.4 which we covered in class.

For this question, its important to consider the moment from the no slip point in order to cancel out the effects of friction in the moment equation.

This problem is almost identical to Example 6.C.4 from the lecturebook. It is important to remember the usage of kinematics to relate the x coordinate and its derivatives to theta and its derivatives. When dealing with the sum of moments around a no-slip point, make sure to use the parallel axis theorem to find the moment of inertia around that point.

It can be nice in the problem to take the moment about the bottom of the disk to avoid having friction as a force that effects it. We have done this many other homework's.

Based on the FBD, it is smart to sum up moments about the no slip point, which allows you to ignore the friction force at that point. For my kinematics equation, related the no slip point to O in order to relate the theta and x variables and their derivatives to each other.

Converting the x-components of the differential equation to theta using x = R*theta helps derive the EOM in terms of theta which helps when proceeding to theta_p in the particular solution.

A good way to make sure the direction of the base extension spring force is correct is to once again draw a system where displacement in x_b is larger than x. Once this is done, it is easy to tell the force will be to the right.

Is the base B able to move in the y direction? The problem doesn't say it has any vertical motion but the diagram makes it look like it's not being supported in the y direction.

The base B is assumed to not move in the y direction as there are no forces in the y while also not having an equation of motion in the y direction.

The base B is more of an idea than a real object in the system. The rectangle represents a displacement of the spring due to some other factor which is not known within the context of the problem. You could think of this as another body that is infinitely larger than the bodies in the system, so much that this body's motion is unaffected by the spring force.

I appreciate what you are saying; however, the base B is actually a body. That body "B" is given a prescribed displacement in time of x_B(t). The actual size or mass of the body is not relevant in terms of the motion of disk. The size does matter in terms of the force that must be applied to B to get this prescribed displacement vs. time. Typically, the more mass that B has, the larger the force that must be applied to it in order get that motion.

In the Kinetics, it can be helpful to sum the moments about the no slip point. This is useful since it allows for friction to be ignored.

In order to solve this problem, I applied the techniques used to solve 6.C.4, the most important of which was taking the sum of the moments around the no-slip point, as this removed many of the troublesome factors.

As stated above this problem is very similar to 6.C.4 from the lecture book. One thing to keep in mind is that x_b(t) is already given and the particular solution is given in the form of Asin(omega t) so we can easily point out that there is no cosine term in the answer.

When solving this problem, the moment is considered about the no-slip point. Due to this, it is important to use the parallel axis theorem to find the value of I.

When drawing the Free Body Diagram for this problem, it is not important whether you draw the friction force to the left or the right.

Yes I agree with this, because later on you will take the moments around the no-slip point which will eliminate the frictional forces. That being said this is not universally applicable and in future problems care should be taken to make sure it is set up correctly based on the direction of motion of the desired object.

The plot for the excitation frequency has a notable remark as there is a change in the plot at the solved for frequency of the problem in part (d). Revealing, the significance of the natural frequency of the system.

For this problem it helps if you take the moment about the contact point to get rid of the friction term and also remember to use the parallel axis theorem for calculating I

For this problem, solving using the moment about O will also work as long as you also sum forces in the x direction to eliminate the friction force.

In class, we have been told that the Euler equation only works when taken at fixed points. In this problem, it is possible, although harder, to take the moment instead around the center of mass, O. This is not a fixed point of rotation of course, but why does it still work? Is it just a convenient situation?

In class, we have discussed THREE possibilities, in general, for the reference point in Euler's equation: 1) the center of mass; 2) a non-accelerating point; and, 3) a point whose acceleration vector points in the direction of the center of mass.

For this problem, you have options 1) and 3) possible: the center of mass O, and point C (note that the acceleration of point C points toward the center of mass O).

Let me know if this is not clear.

Similar to the example done in class, taking the moment about the where the disk touches the ground will allow you to avoid friction.

This problem is very similar to problem 6.C.4 that was done in class. After drawing FBDs, take the moment about the no-slip point to negate the friction force acting there. Make sure to then use the parallel axis theorem when find the moment of inertia. Use kinematics to relate x values and theta values to find the EOM.

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