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 79 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.

Explanation:

Given:

Distance, S = 79 cm

= 0.79 m

Acceleration due to gravity, g = 9.81 m/s^2

Theta = <1 rev

= < 2pi rad

A.

Using equation of motion,

S = vo × t + 1/2 × g × t^2

Initially at rest, vo = 0 m/s

t = sqrt((0.79 × 2)/9.81)

= 0.40 s

The smallest and largest angle that can cause the toast to hit and then topple to be butter-side down is guven by the range;

0.75 rev > theta > 0.25 rev

A.

Angular velocity = angular displacement/time

= (0.25 × 2pi)/0.4

= 3.92 rad/s

B.

Angular velocity = angular displacement/time

= (0.75 × 2pi)/0.4

= 11.74 rad/s

the minimum constant speed at which she can catch the train is 2.7 m/s

given:

acceleration = 0.6

time = 4.5 s

to find:

velocity = ?

formula used:

from 1st kinematic equation we get,

v = u + at

u = initial speed of train = 0 m/s

v = final speed of the train

a = acceleration of the train

t = time

solution:

from 1st kinematic equation we get,

v = u + at

u = initial speed of train = 0 m/s

v = final speed of the train

a = acceleration of the train

t = time

v = 0 + (0.6 × 4.5)

v = 2.7 m/s

the minimum constant speed at which she can catch the train is 2.7 m/s