A ball on the end of a string is whirled around

in a horizontal circle of radius 0.33 m. The
plane of the circle is 1.26 m above the ground.
The string breaks and the ball lands 2.4 m
away from the point on the ground directly
beneath the ball’s location when the string
breaks.
The acceleration of gravity is 9.8 m/s
^2
.
Find the centripetal acceleration of the ball
during its circular motion.
Answer in units of m/s
^2

To find the centripetal acceleration of the ball during its circular motion, we can use the formula for centripetal acceleration:

a = v^2 / r

where:
a = centripetal acceleration
v = velocity of the ball in the circular motion
r = radius of the circular motion

However, the given information does not provide the velocity directly. We need to find the velocity first using the available information.

We are given that the ball lands on the ground 2.4 m away from the point beneath its location when the string breaks. This horizontal distance represents the radius of the circular motion.

Using this information, we can calculate the velocity in terms of distance and time.

First, let's calculate the time it takes for the ball to reach the ground after the string breaks. We can use the equation of motion:

y = (1/2) * g * t^2

where:
y = height of the circle above the ground = 1.26 m
g = acceleration due to gravity = 9.8 m/s^2
t = time taken to reach the ground

Rearranging the equation, we get:

t = √(2 * y / g)

Substituting the values, we have:

t = √(2 * 1.26 / 9.8)
t = √(2 * 0.1286)
t ≈ √(0.2572)
t ≈ 0.506 s (rounded to three decimal places)

Now, we can calculate the velocity using the distance traveled and the time taken:

v = d / t

where:
v = velocity of the ball
d = horizontal distance traveled = 2.4 m
t = time taken to reach the ground ≈ 0.506 s

Substituting the values, we have:

v = 2.4 / 0.506
v ≈ 4.739 m/s (rounded to three decimal places)

Now that we have the velocity, we can calculate the centripetal acceleration using the formula:

a = v^2 / r

where:
a = centripetal acceleration
v = velocity of the ball ≈ 4.739 m/s
r = radius of the circular motion = 2.4 m

Substituting the values, we have:

a = (4.739)^2 / 2.4
a ≈ 9.365 m/s^2 (rounded to three decimal places)

Therefore, the centripetal acceleration of the ball during its circular motion is approximately 9.365 m/s^2.