After a particularly cold winter, a chunk of ice sits a top the hemispherical to

Question

After a particularly cold winter, a chunk of ice sits a top the hemispherical top a metal silo on a farm in South Dakota. On a sunny spring day, the chunk of ice begins to slide down the frictionless surface. If the radius of the cap is R = 18.0 m and the bottom of the cap is at height H = 36.0 m above the ground, find:

a) the height of the chunk of ice, above the ground, at the point that it loses contact with the cap. (answer = 48.0 m)

b)the speed of the chunk of ice at the point that it looses contact with the cap. (answer = 10.8 m/s)

c) the distance d from the slide of the silo where the chunk of ice impacts the ground. (answer = 11.4 m)

d) the speed of the chunk of ice when it impacts the ground. (answer = 3.72 m/s)

See picture.

Explanation / Answer

Given radius of semicircular path = 18 m Height = 36 m _____________________________________________________________________________ (b) consider the cap at rest , here v is the velocity of chunk of ice with mass m relative to the cap v = sqrt(2/3gR) = sqrt(2/3*9.8*18) = 10.8 m/s _____________________________________________________________________________ (a) By conservation of mechanical energy mgR = 1/2mv^2+mgh R-h = v^2/g 18-h = (10.8)^2/2*9.8 18-h = 6 h = 12 m total height from the ground = 12+36 = 48 m _____________________________________________________________________________ (c) The trajectory of the chunk is a parabola Range R = u sqrt(2H/g) = 10.8*sqrt(2*48/9.8) = 33 m _____________________________________________________________________________- (d) time of decent = td = sqrt(2H/g) = 3.1 sec velocity of chunk of ice to hit the ground = v = gt = 3.1*9.8 = 30.6 m/s

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