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 52 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

4.83rad/s and 14.50 rad/s

Explanation:

The distance from the center to the floor d = 52cm = 0.52m

Rotation is less than 1 rev

The toast rotates at a constant angular speed

Using kinematic d = Vit + 1/2gt^2

d = 0+1/2gt^2

t = (2d/g) square root

t = square root of 2× 0.52/ 9.8

t = 0.325s

The toast is accidentally pushed over the edge of the centre with butter side up, then the toast rotates as it falls . If the toast hits the ground and topples, the smallest angle will be less than 1/4 rev with correspondence to the smallest angular speed

Wmin = change in tetha/ change in time

= 0.25 rev/ change in t

= 0.25×2pii / change in t

= 0.5pii/0.325

= 0.5×3.142/0.325

= 4.83 rad/s

The toast is accidentally pushed over the edge of the centre with butter side up, then the toast rotates as it falls . If the toast hits the ground and topples, the maximum angle will be less than 3/4 rev with correspondence to the maximum angular speed

Wmax = change in tetha/ change in time

= 0.75 rev/ Change in time

= 0.75 ×2pii/change in tetha

= 1.5pii/0.325

= 1.5 ×3.142/0.325

= 14.50 rad/s

Note the following

Vi is the initial speed of the toast which is zero because it was initially at rest

t is the time taken for the toast to hit the floor

g is the gravitational acceleration

d is the distance between the counter and the floor. There is an attachment for you as well.

Thanks

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the bal will not change its velocity.