Question

When a slice of buttered toast is accidentally pushed over the edge of a counter, it rotates as it falls. Suppose the distance to the floor is 89 cm and the toast rotates less than 1 rev. (a) What is the smallest angular speed that causes the toast to hit and then topple to be butter-side down?

Answer 1

Answer:

7.38 rad/s

Explanation:

Assume no air resistance, we can first calculate the time it takes for the toast to be dropped 0.89m to the floor. Since we have

$h = \frac{gt^2}{2}$

where $g = 9.81m/s^2$ and h = 0.89 m

$t^2 = \frac{2h}{g} = \frac{2*0.89}{9.81} \approx 0.181$

$t = \sqrt{0.811} \approx 0.426 s$

This is also the time for the toast to flip one time at a constant angular speed. The angle it covered would be at $\theta = \pi$ radian.

So the smallest angular speed it needs to hit and topple butter-side down is

$\omega = \frac{\theta}{t} = \frac{\pi}{0.426} \approx 7.38 rad/s$