A girl whirls a stone in a horizontal circle 1.9 m above the ground by means of a string 111 cm long. The string breaks, and the stone flies of horizontally and strike the ground 11.2 m away. What is the centripetal acceleration of the stone while in circular motion?
To find the centripetal acceleration of the stone while in circular motion, we need to use the given information about the height of the circle, the length of the string, and the distance the stone flew after the string broke.
The centripetal acceleration (ac) of an object moving in a circular path can be calculated using the following formula:
ac = (v^2) / r
Where:
- ac is the centripetal acceleration
- v is the velocity of the object
- r is the radius of the circular path
In this case, since we are given the distance the stone flew after the string broke (11.2 m), we can calculate the velocity (v) of the stone just before the string broke. We can assume that the horizontal velocity just before the string broke is the same as the velocity of the stone when it hit the ground.
Now, let's calculate the velocity of the stone:
v = d / t
Where:
- v is the velocity
- d is the distance
- t is the time
Since we are given the distance (11.2 m) and the height (1.9 m), we can use the following formula to find the time it took for the stone to fall:
t = sqrt(2h / g)
Where:
- h is the height
- g is the acceleration due to gravity (approximately 9.8 m/s^2)
Now, let's calculate the time it took for the stone to fall:
t = sqrt(2 * 1.9 / 9.8) ≈ 0.616 s
Now that we have the time and the distance, we can calculate the velocity of the stone:
v = 11.2 / 0.616 ≈ 18.182 m/s
Next, we need to find the radius of the circular path. The radius is equal to the length of the string.
r = 111 cm = 1.11 m
Finally, we can calculate the centripetal acceleration using the formula we mentioned earlier:
ac = (v^2) / r
= (18.182^2) / 1.11 ≈ 297.67 m/s^2
Therefore, the centripetal acceleration of the stone while in circular motion is approximately 297.67 m/s^2.
To find the centripetal acceleration of the stone while in circular motion, we can use the formula:
a = (v^2) / r
where:
a = centripetal acceleration
v = velocity
r = radius of the circular path
First, let's find the velocity of the stone before the string breaks. To do this, we can use the fact that the stone traveled 11.2 m horizontally in a circular path. The distance traveled along the circular path is equal to the circumference of the circle:
C = 2πr
where:
C = circumference
r = radius of the circular path
Given the radius of the circular path (111 cm or 1.11 m), we can calculate the circumference:
C = 2π(1.11 m)
= 2π(1.11 m) ≈ 6.97 m
Since the stone traveled a distance of 11.2 m horizontally, it made more than one complete revolution before the string broke.
Let's estimate the number of revolutions by dividing the total distance traveled (11.2 m) by the circumference of the circle (6.97 m):
number of revolutions = 11.2 m / 6.97 m ≈ 1.60 revolutions
Since the stone made approximately 1.60 revolutions, we can conclude that its initial velocity is equal to the circumference of the circle times the number of revolutions per second. To find the velocity, we can divide the distance traveled (11.2 m) by the time it took to make the rotations:
v = 11.2 m / (1.60 revolutions)
≈ 7.00 m/rev
So, the velocity of the stone before the string broke is approximately 7.00 m/rev.
Finally, we can calculate the centripetal acceleration using the formula:
a = (v^2) / r
= (7.00 m/rev)^2 / 1.11 m
≈ 44.76 m^2/s^2
Therefore, the centripetal acceleration of the stone while in circular motion is approximately 44.76 m^2/s^2.