When a slice of buttered toast is accidentally pushed over the edge of a counter, it rotates as it falls. If the distance to the floor is `80 cm` and for rotation less than `1 rev`, what are the
(i) smallest and
(ii) largest angular speeds that cause the toast to hit and then topple to be butter-side down ?

Options

A. smallest angular speed is `(5 pi)/(4) rad//s` and largest angular speed is `(15 pi)/(4) rad/s`.
B. smallest angular speed is `(pi)/(4) rad//s` and largest angular speed is `(3 pi)/(4) rad//s`
C. smallest angular speed is `(pi)/(2) rad//s` and largest angular speed is `(3 pi)/(2) rad//s`.
D. smallest angular speed is `(5 pi)/(2) rad//s` and largest angular speed is `(15 pi)/(2) rad//s`.

Answer

A

Explanation

(a) The time it takes for the buttered toast to hit floor is.
`Delta t = sqrt((2 h)/(g)) = sqrt((2(8 m))/(10 m//s^2)) =0.40 s`
The smallest angle turned for the toast to land butter - side down is `Delat theta_(min) = (1)/(4) rev = pi//2`. This corresponds to an angular speed of
`omega_(min) = (Delta theta_(min))/(Delta t) = (pi//2 rad)/(0.4 s) = (5 pi)/(4) rad//s`
The largest angle (less than `1` revolution) turned for the toast to land butter-side down is `Delta theta_(max) = 0.75 rev = 3 pi//2 rad`. This corresponds to an angular speed of
`omega_(max) = (Delta theta_(max))/(Delta t) =(pi//2 rad)/(0.4 s) =(15 pi)/(4) rad//s`.