Question

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 78 cm and for rotation less than 1 rev, what are the (a) smallest and (b) largest angular speeds that cause the toast to hit and then topple to be butter-side down? Assume free-fall acceleration to be equal to 9.81 m/s2.

Answer 1

a) 3.94 rad/s

b) 11.84 rad/s

Step-by-step explanation:

Given:

Distance, s = 78 cm = 0.78 m

Now the time taken, t

we know

`s = ut +(1)/(2)gt^2`

where,

s = distance

u = initial speed

g = acceleration due to gravity

since it is a free fall. thus, u = 0

thus, we get

`0.78 = 0* t +(1)/(2)9.8* t^2`

or

`t=\sqrt{(2* 0.78)/(9.8)}=0.398 s`

a) now, the smallest angle will be 1/4 of the revolution so as to fall on the butter side i.e 90° or (
`(\pi)/(2)`)

also,

`angular\ speed\ (\omega) = (Change\ in\ angle)/(time)`

thus, we have

`angular\ speed\ (\omega) = ((\pi)/(2))/(0.398) = 3.94rad/s`

b)now, the largest angle will be 3/4 of the revolution so as to fall on the butter side i.e 270° or (
`(3\pi)/(2)`) contributing to the largest angular speed

`angular\ speed\ (\omega) = (Change\ in\ angle)/(time)`

thus, we have

`angular\ speed\ (\omega) = ((3\pi)/(2))/(0.398) =11.84rad/s`