A Man stands at the very top of the Sky Dump Stadium roof to pitch the opening ball for the Blew Jaws game. He slips a bit left of the centre and slides down along the frictionless roof surface as shown. For what vertical distance, measured from the ground, will he slide before leaving the roof? Assume the roof has a circualr cross section of radius 125 m.
Sample Problem 8-36 Solution
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it doesnt let me put the website on so i spaced between each letter just delete the spaces or in google type "A boy is seated on top of a hemispherical ice mound of radius
R = 13.8m. Starting from rest, he begins to slide down the
mound. At what height does he lose contact with the mound?
Assume frictionless conditions."
To find the vertical distance the man will slide before leaving the roof, we can break down the problem into two parts:
1. Find the horizontal distance the man slides along the roof.
2. Use the circular cross section of the roof to calculate the vertical distance.
Let's start with the first part:
1. Find the horizontal distance:
Since the man slips left of the center, we can assume he starts sliding from the edge of the circular roof. The circumference of the circular cross section of the roof can be calculated using the formula:
Circumference = 2 * π * radius
Given that the radius is 125 m, the circumference is:
Circumference = 2 * π * 125 = 250π (approximately 785.4 m)
Therefore, the man will slide a total horizontal distance of 250π (approximately 785.4 m) along the frictionless roof surface.
Now, let's move on to the second part:
2. Calculate the vertical distance:
Since the roof is round, we can imagine the circular cross section as a ramp. The vertical distance the man slides will be equal to the height difference between the starting point and the point where he leaves the roof.
To calculate this height, we need to find the length of the arc on which the man slides. This can be done by finding the angle that subtends the arc on the circle.
Considering that the circumference of a circle is 2πr, where r is the radius, and the arc length is related to the angle θ and the circle's circumference by the formula:
Arc Length = θ * circumference / 360
In this case, the arc length is 250π. So, we can rearrange the formula and solve for θ:
θ = (Arc Length * 360) / Circumference
θ = (250π * 360) / (250π) = 360
Therefore, we find that θ = 360 degrees.
Since the man slips, he will slide along a straight line until he leaves the roof. This means the angle between the horizontal plane and the line joining the starting point with the point where he leaves the roof will be 90 degrees.
Using basic trigonometry, we can find the vertical distance (h) from the ground:
sin θ = Opposite / Hypotenuse
sin 90 = h / 125
1 = h / 125
h = 125
Therefore, the man will slide a vertical distance of 125 m before leaving the roof.
In conclusion, the man will slide a vertical distance of 125 m from the ground before leaving the roof of the Sky Dump Stadium.