A boy whirls a stone in a horizontal circle of radius1.5 m and at height 2.0 m above level ground. The string breaks,and the stone flies off horizontally and strikes the ground aftertraveling a horizontal distance of 10 m. What is the magnitudeof the centripetal acceleration of the stone during the circularmotion?
Question
A boy whirls a stone in a horizontal circle of radius1.5 m and at height 2.0 m above level ground. The string breaks,and the stone flies off horizontally and strikes the ground aftertraveling a horizontal distance of 10 m. What is the magnitudeof the centripetal acceleration of the stone during the circularmotion?
Solution
To solve this problem, we need to find the speed of the stone first, and then use that to find the centripetal acceleration.
Step 1: Find the speed of the stone
We know that the stone travels a horizontal distance of 10 m after the string breaks. We also know that the stone is 2.0 m above the ground when the string breaks. We can use these two pieces of information to find the time it takes for the stone to hit the ground.
The time it takes for an object to fall a certain distance under the acceleration due to gravity is given by the equation:
t = sqrt(2h/g)
where h is the height and g is the acceleration due to gravity. Plugging in the given values:
t = sqrt(2*2.0 m/9.8 m/s^2) = 0.64 s
The stone travels a horizontal distance of 10 m in this time, so its speed is:
v = d/t = 10 m / 0.64 s = 15.6 m/s
Step 2: Find the centripetal acceleration
The centripetal acceleration of an object moving in a circle of radius r at a speed v is given by the equation:
a = v^2 / r
Plugging in the given values:
a = (15.6 m/s)^2 / 1.5 m = 162.4 m/s^2
So, the magnitude of the centripetal acceleration of the stone during the circular motion is 162.4 m/s^2.
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