After the delicious snack is already served in the cabin, the pilot of an airlin

Question

After the delicious snack is already served in the cabin, the pilot of an airline flight gets directed to climb to a higher altitude. During one phase of the ascent, the plane experiences a constant acceleration a at an angle of ? with respect to the horizontal. The tray table remains horizontal at all times. What does the minimum coefficient of static friction µS,min between a cup of highly staining grape juice and the tray table have to be, so the juice does not spill all over you? Hints: The juice does not spill if it follows the acceleration of the airplane. Consider extreme angles (horizontal and vertical).
1. µS,min=(asin?)/(acos?+g)
2. µS,min=a/g
3. µS,min=(acos?+g)/(asin?)
4. µS,min=(acos?)/(asin?+g)
5. µS,min=tan?
6. µS,min=(atan?)/g

Explanation / Answer

Following the hint, the juice does not spill if its acceleration is equal to plane's. Breaking the acceleration into x- and y- components, we get ax = a*cos? and ay = a*sin?. Applying Newton's 2nd Law, we get
SFy = n - mg = ma*sin?    ---> n = m(a*sin? + g) SFx = µsn = ma*cos?    ---> µs = ma*cos?/n = a*cos?/(a*sin? + g) So the correct answer is choice 4.

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