Block and wedge problem -- Where am I going wrong?

[NTesla]

Homework Statement

Kindly see the attached pic for the Question. I'm solving part b of the question.

Relevant Equations

F = ma

This question is from the book Kleppner and Kolenkow Q 3.8:

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My attempt is posted here:

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The value of a(minimum) that I'm getting is different from the official solution. The official solution is posted here:

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Here's 2 problems that I'm encountering:

(a) The value of a(minimum) that I'm getting is different from the official answer. If my calculations are wrong, I'm not able to find out where am I going wrong. Kindly help.

(b) From the equation of N(in my calculation), if theta tends to zero, then N tends to mg. This makes sense to me.
However, in the official answer(posted above), if we try to see what happens to value of N as theta tends to zero, then N becomes mg/mu(the symbol of coefficient of friction). This doesn't make sense to me. So, which equation of N is correct ?

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I understand that in the official answer, the equations have been written from inertial frame of reference. I understand those equations and they are right according to me. This is causing a conundrum, that if I'm finding the official equations right(which I do), then the value of a(minimum) must be right as given in the official answer. But I'm also not able to find where my equations are wrong(if they are wrong), leading to different value of a(minimum).

 

[TSny]

Your solution looks correct to me. The error in the official solution is here:

 

[kuruman]

The solution is wrong. Look at the equation, part (b) vertical direction, $$N\sin\theta+f_s\cos\theta-mg=0.\tag{1}$$Now look at the solution's two FBDs on the right.

In the first FBD, angle $\theta$ is defined as the angle between the horizontal and the incline. Equation (1) was derived from the second FBD. In it angle $\theta$ is labeled as the angle between the vertical and the incline which makes the two FBDs inconsistent. Equation (1) should have the sine and cosine swapped to be consistent with the definition of $\theta$ as the angle of the incline.

By the way, you have posted here enough times to appreciate the advantage of using LaTeX to write equations. To learn how, please click "LaTeX Guide", lower left above "Attach files". It's easy to learn and will serve you well.

I see that @TSny beat me to the punch.

 

[TSny]

I see that @TSny beat me to the punch.

Only by a second or two. Amazing.

 

[kuruman]

Only by a second or two. Amazing.

It could have been simultaneous if both of us were at rest relative to the PF server when we posted.

 

[NTesla]

@kuruman, @TSny : Thanks for the prompt help.

Earlier when I used to upload pics of my hand written equations in this forum, it used to be sharp and clear enough to be read. But this time, after reading your advice of using latex, I was checking whether the uploaded pics were clear and sharp or not. I observed that the pics that I have on my phone (which I uploaded here) are quite sharp, but after having uploaded, the pics are not sharp at all. I don't know why that has happened. The picture quality has been reduced to the point of not being clear enough to be read.

I'm typing this on phone, don't have access to desktop/laptop, & as there were a full page of equations and calculations, that was the reason why I preferred to upload the pic of written page, rather than typing each line of calculation. It was efficient compared to typing the whole page in latex, that too on the phone. I hadn't encountered low upload quality earlier, but only this time. In some of my earlier posts, I did write in LaTeX when there were only a few equations to write. I also hadn't anticipated that the compression algorithm in this forum would render the image quality too difficult to read.

If uploading pics in original quality is possible, then kindly let me know the way to do so. If not, then I'll try to write using latex from next time onwards, wherever necessary.

 

[kuruman]

Here is a test scribbled work for comparison posted directly from my phone. It’s pretty well-defined. Using a black pen and enlarging the posted photo to fill the page helps.

 

[haruspex]

as there were a full page of equations and calculations, that was the reason why I preferred to upload the pic of written page, rather than typing each line of calculation.

One reason we prefer typed in equations is for ease of quoting. We may need to make reference to a particular one of your equations. It will help, therefore, if you number them all.
We may also want to post an edited version of an equation, which again is easier if we can copy the original. Since it often happens that there are multiple posts in response to the one original, it can be less work in total if the original is copyable line by line.