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in Physics by (71.9k points)

An electric lamp is placed in a frosted-glass sphere of radius 20 cm and is suspended from a height of 5 m over a field. A ball of radius 10 cm is held under the lamp at a height of 1 m. Find the size of the umbra and penumbra formed by the ball. At what height should the ball be placed for its umbra to vanish? At this height, what will be the size of the penumbra? What should be the diameter of the ball for the size of its umbra to be the same, regardless of the distance from the ball to the field? 

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The radius of the penumbra is 17.5 cm, the radius of the total shadow is 7.5 cm. The umbra vanishes if the ball is at a height of 2.5 m. The dimensions of the umbra are constant if the radius of the ball is equal to the radius of the sphere.

The radius of the penumbra can be found from the similar triangles CC2D, BB1D, and AA2D (see the schematic diagram in Fig.). The radius of the umbra is obtained by considering triangles OA1A, OBBl, and OCC1 Line C2C1 is equal to the diameter of the sphere; line BB1 is equal to the radius of the ball; segments AC and AB are equal to the respective heights of the sphere and ball over the field.

  

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