You plan to throw stones by using a sling of length 0.6 m which you whirl over your head. Suppose you wish to throw a stone a distance of 32 m. What must be the centripetal acceleration of the stone just before its release if it is to reach this distance? Assume that the release height is 2.2 m.
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To determine the required centripetal acceleration of the stone just before its release, we first need to analyze the motion of the stone while it is being whirled in the sling.
Given:
Length of the sling (L) = 0.6 m
Distance to be thrown (d) = 32 m
Release height (h) = 2.2 m
To solve this problem, we can break it down into two parts: the horizontal motion of the stone and the vertical motion of the stone.
Horizontal Motion:
The horizontal motion of the stone can be considered as a uniform circular motion due to the sling being whirled over your head. In uniform circular motion, the centripetal acceleration (ac) is given by:
ac = (v^2) / r
Where:
v = linear velocity of the stone
r = radius of the circular path
Since we are given the length of the sling (L), we can consider it as the radius of the circular path. Therefore, r = L.
Vertical Motion:
The vertical motion of the stone can be treated independently since it is not affected by the horizontal motion. We need to determine the initial vertical velocity (vi) of the stone just before it is released. To do that, we can use the principles of projectile motion.
The vertical distance traveled by the stone can be given by the equation:
d = vi * t + (1/2) * g * t^2
Where:
d = vertical distance (h = 2.2 m)
vi = initial vertical velocity
t = time of flight
g = acceleration due to gravity (-9.8 m/s^2)
Since the stone is released at an angle of 45 degrees, the time of flight (t) can be calculated using the formula:
t = (2 * v * sinθ) / g
Where:
θ = angle of release (45 degrees)
Substituting this value of t back into the equation of vertical distance, we can solve for vi.
Once we have both the horizontal centripetal acceleration (ac) and the initial vertical velocity (vi), we can find the required centripetal acceleration just before releasing the stone.
It is important to note that the calculation of vi and t involve trigonometric functions and quadratic equations, which may require solving simultaneously.