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 59 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.

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Answer:

a) 4.49rad/s

b) 13.46rad/s

Explanation:

The distance from the counter to the floor =59m.

The toaster rotates at a constant speed less than 1 rev.

Using cinematic equation to calculate the time taken by the toast to hit the ground:

d = Vot + 1/2gt^2

59 = 0 + 1/2 × 9.81 × t^2

t = Sqrt(2 ×0. 59)/ 9.8)

t = Sqrt(1.18/9.8)

t = Sqrt(0.12041)

t =0. 347secs

As the toast is accidentally pushed over the counter with the side up, the toast rotates as it falls. If it hits the ground and topples to the butter side down, the smallest angle is 1/4 of A revolution.

W(min) = ΔΦ/Δt

W(min) = 0.25 rev/ 0.35

W(min) = 0.25 * 2π/0.35

W(min) = 1.57/0.35

W(min) = 4.49 rad/s

Same with the first part, the largest angle is 3/4 of a revolution

W(max) = ΔΦ/Δt

W(max) = 0.75 rev/0.35

W(max) = 0.75 * 2π/0.35

W(max) = 4.71/0.35

W(max) = 13.46 rad/s

Answer: a) 4.49 rad/s

b) 13.46 rad/s

Explanation:

distance from the counter to the floor, d = 59cm = 0.59m

Rotation is less than 1rpm

we use kinematics equation of motion to calculate the time taken by the toast to hit the floor

S = ut + 1/2at²

0.59 = 0 + 1/2*9.8*t²

1.18 = 9.8t²

t² = 1.18/9.8

t² = 0.12

t = 0.35s

As the toast is accidentally pushed over the counter with the side up, the toast rotates as it falls. If it hits the ground and topples to the butter side down, the smallest angle is 1/4 of A revolution.

W(min) = ΔΦ/Δt

W(min) = 0.25 rev/ 0.35

W(min) = 0.25 * 2π/0.35

W(min) = 1.57/0.35

W(min) = 4.49 rad/s

Same with the first part, the largest angle is 3/4 of a revolution

W(max) = ΔΦ/Δt

W(max) = 0.75 rev/0.35

W(max) = 0.75 * 2π/0.35

W(max) = 4.71/0.35

W(max) = 13.46 rad/s