I need help with Physics Motion, Forces, and Work problems. Thank you in advance

Question

I need help with Physics Motion, Forces, and Work problems. Thank you in advance (:

Problem 1 A cylinder is cut out of a large block with mass M sitting on a table. A small block of negligible size and mass m is placed within the hole (see Fig. 1). The coefficient of static friction between the small block and the large block is , and you may assume that the friction between the large block and the table is negligible. What is the minimum and maximum value of F so that does not change? Figure 1 Problem 2 A ball is attached by a string to a post and is being whirled around with some speed. When the post is held still, ° = 70° (Fig. 2, left). When the post is accelerated upward with acceleration a, this angle decreases and is now Vaccel -60° (Fig. 2, right). What is a? You can assume that the string has negligible mass, that the speed of the ball does not change, that the ball does not bobble up and down, and that the post is still vertical to the ground. Figure 2

Explanation / Answer

1).

let a be the acceleration of the entire system; a = F/(m + M)

Taking M as our reference we can apply a pseudo force on m in opposite direction.

Let N be the normal force

Let Fs be the frictional force

Now to find the minimum F let us assume that Fs is in upward direction

N = masin + mgcos ...1

ma cos + Fs = mgsin => ma cos + N = mgsin....2

putting 1 in 2

=> macos + (masin + mgcos ) = mgsin

=> a = (mgsin - mgcos)/(mcos +msin)

Minimum force = (m+M)a = (m+M) *((mgsin - mgcos)/(mcos +msin))

For maximum force F let us assume that Fs is in downward direction

N = masin + mgcos ...i

ma cos = mgsin + Fs => ma cos = mgsin + N...ii

putting i in ii

=> macos = mgsin + (masin + mgcos)

=> a = (mgsin + mgcos)/(mcos - msin)

Force = (m+M)a = (m+M)*((mgsin + mgcos)/(mcos - msin)) {max F}

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