Here are some problems.if you have solved any one or more please post it's solutions.
The question number $20, 28,31$ are resolved.
If anyone want to see my attempt for a particular problem, they can ask me I will happily show the attempt.
Thanks in advance.

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Consider the force $F$ to have a horizontal and vertical component $F_x$ and $F_y$ respectively. Drawing a free body diagram will give you the following equations:

Now for the sphere to just start rolling upwards, it must just start linearly accelerating. In the limiting case, this happens when $a=0$ and $f = \mu N$. Note that $a$ and $\alpha$ are unrelated. I was initially treating this as a pure rolling problem, but the problem does not specify that. This is a case of 'not pure rolling'. So plugging this understanding into the second equation above gives:

$F_y = \mu F_x + mg$

Now, the resultant force is:

$F = \sqrt{F_x^2+F_y^2}$

The force $F$ is minimum when the expression in the square root is minimum. Therefore, we have a constrained minimisation problem. We are required to minimise the following function $G$:

$G(F_x,F_y) = F_x^2+F_y^2$

Such that:

$F_y = \mu F_x + mg$

Solving this optimisation problem is something you should try yourself. The answer comes out to be:

@Karan Chatrath Thanks for the solution.
I am trying to understand it.
By the way ,the last 3rd step of your solution seems me incorrect, according to dimension. Isn't it.?

@Karan Chatrath
–
@Karan Chatrath Yes i got it. The next step is differentiation and then just squaring.
But the most important thing is the moment of inertia is not used in this problem which seems me bit surprising.

@Talulah Riley
–
I used moment of inertia when I treated this problem as a case of pure rolling. That is why I got a factor of 49 multiplied with $\mu^2$. But since it is not a pure rolling case, $a \ne R\alpha$ and therefore, the moment of inertia dependence is lost.

@Karan Chatrath
–
@Karan Chatrath i didn't understand the solution of this problem
Can you post a very explanatory solution.
After reading word rolling, my brain stops working . :)

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## Comments

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TopNewestIs the answer to problem 27 $1.5 mg$?

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@Karan Chatrath $\Huge YES$

I am even not able to understand what the question wants to say?

Can you help me to understand what it is saying?

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I will post a note later.

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@Steven Chase @Karan Chatrath Please sir can you help me.. Thanks in advance.

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I'm getting approximately $0.35$ for Problem 23. Is that the correct answer?

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@Steven Chase $\Huge YES$

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My answer for problem 13 is:

$F_{min} = \frac{mg}{\sqrt{49 \mu^2 + 1}}$

Is it correct?

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@Karan Chatrath No. In the answer the coefficient of $\mu^{2}$ is $1$ instead of $49$.

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I am managing to get that answer but somehow that result is not making sense to me.

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Okay, I have now understood the problem. What I did is also not wrong. I analysed a slightly different case.

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@Karan Chatrath Yeah Nice. Can you show the solution?

OR would you like to see my attempt?

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Problem 13 solution:

Consider the force $F$ to have a horizontal and vertical component $F_x$ and $F_y$ respectively. Drawing a free body diagram will give you the following equations:

$N = F_x$ $F_y - mg - f = ma$ $fR = \frac{2mR^2}{5} \alpha$

Now for the sphere to just start rolling upwards, it must just start linearly accelerating. In the limiting case, this happens when $a=0$ and $f = \mu N$. Note that $a$ and $\alpha$ are unrelated. I was initially treating this as a pure rolling problem, but the problem does not specify that. This is a case of 'not pure rolling'. So plugging this understanding into the second equation above gives:

$F_y = \mu F_x + mg$

Now, the resultant force is:

$F = \sqrt{F_x^2+F_y^2}$

The force $F$ is minimum when the expression in the square root is minimum. Therefore, we have a constrained minimisation problem. We are required to minimise the following function $G$:

$G(F_x,F_y) = F_x^2+F_y^2$

Such that:

$F_y = \mu F_x + mg$

Solving this optimisation problem is something you should try yourself. The answer comes out to be:

$F_{min} = \frac{mg}{\sqrt{\mu^2 + 1}}$

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@Karan Chatrath Thanks for the solution.

I am trying to understand it. By the way ,the last 3rd step of your solution seems me incorrect, according to dimension. Isn't it.?

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I have used $\alpha$ for angular acceleration and $a$ for linear acceleration. Both symbols look similar.

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@Karan Chatrath LAST 3rd step sir?

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@Karan Chatrath g is a gravitational acceleration and F is a force.?

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$G$ of $F_x$ and $F_y$. I have modified the solution.

Ah, good catch! I meant to define a functionLog in to reply

@Karan Chatrath i didn't understand how did you optimised that at the last. Please elaborate the solution. Thanks in advance.

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Maxima and minima problem. Find the minimum value of the function:

$G = F_x^2 + F_y^2$

Such that: $F_y = \mu F_x + mg$

Write $F_y$ in terms of $F_x$ and you have a function of $F_x$ only. Do you know the next step?

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@Karan Chatrath Yes i got it. The next step is differentiation and then just squaring.

But the most important thing is the moment of inertia is not used in this problem which seems me bit surprising.

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$\mu^2$. But since it is not a pure rolling case, $a \ne R\alpha$ and therefore, the moment of inertia dependence is lost.

I used moment of inertia when I treated this problem as a case of pure rolling. That is why I got a factor of 49 multiplied withLog in to reply

@Karan Chatrath Exactly. It means friction is acting in the ball. Thanks By the way try the 14th question.

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@Karan Chatrath i didn't understand the solution of this problem

Can you post a very explanatory solution.

After reading word rolling, my brain stops working . :)

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