A copy of The History of the Decline and Fall of the Roman Empire by Edward Gibbon lies exactly in the middle of an otherwise empty 6.1m long bookshelf in Robarts Library. The coefficient of static friction for the book on the shelf is μS = 11/60 and the shelf has been polished (so kinetic friction may be neglected). A tall, disgruntled librarian slowly lifts up the one end of the bookshelf at a constant rate, so that it pivots upward, anchored at the opposite end (see Figure 2). After 5.5 seconds, she sees the book begin to slide down the
tilted shelf. She continues to raise her end of the shelf at the same rate, gleefully watching until the book falls onto the floor.
How long does it take the book to fall off the shelf (measured from the time it
The Attempt at a Solution
So I am pretty sure this problem involves differentiation, since the angle is constantly changing, and which would mean the acceleration of the book is also changing. This relationship is described mathematically as a=gsinθ, where θ is the angle of inclination.
And so the angle is then changing with time, which in turn means the acceleration is changing with time. But how can i relate these quantities in such a way to solve for t?
I tried a number of different approaches (none of which seemed to get me anywhere)
For instance, I took the derivative of t=sqrt(6.1/9.81sinθ) with respect to θ, and tried to use the pythagorean theorem by treating the problem as a related rate, but that didn't get me anywhere.
Oh and sorry for the uninformative title, I realized only after i submitted the post, that I didn't finish the title.