A stone tied to the end of a string 80cm long is whirled in a horizontal circle with a constant speed. If the stone makes 14 revolutions in 22s then the acceleration of the stone is ?
We can start by using the formula for centripetal acceleration:
a = (v^2) / r
where a is the centripetal acceleration, v is the velocity of the stone, and r is the radius of the circle.
To find the velocity, we need to use the formula for angular velocity:
w = 2πn / t
where w is the angular velocity, n is the number of revolutions, and t is the time taken.
Plugging in the given values, we get:
w = (2π * 14) / 22 = 2.8π rad/s
To convert this to linear velocity (v), we can use the formula:
v = rw
where r is the length of the string. Plugging in the given value, we get:
v = 0.8 * 2.8π = 2.22 m/s
Now we can use the formula for centripetal acceleration:
a = (v^2) / r
Plugging in the values we calculated, we get:
a = (2.22^2) / 0.8 = 6.16 m/s^2
Therefore, the acceleration of the stone is 6.16 m/s^2.
To find the acceleration of the stone, we can use the formula for centripetal acceleration, which is given by:
a = (v^2) / r
Where:
a = acceleration
v = linear speed of the stone
r = radius of the circular path
First, let's find the linear speed of the stone. We know that the stone makes 14 revolutions in 22 seconds.
The linear distance traveled by the stone in one revolution is equal to the circumference of the circle:
2πr
Since the length of the string is equal to the circumference of the circle, we can write:
2πr = 80 cm
Dividing both sides by 2π, we get:
r = 80 cm / 2π
Now, let's calculate the linear speed of the stone:
Linear speed (v) = distance / time
= (14 * 2πr) / 22 s
Substituting the value of r, we get:
v = (14 * 2π * 80 cm / 2π) / 22 s
v = 560 cm / 22 s
v ≈ 25.45 cm/s
Now we can use the formula for centripetal acceleration:
a = (v^2) / r
= (25.45 cm/s)^2 / (80 cm)
Simplifying the expression, we get:
a ≈ 8.09 cm/s^2
Therefore, the acceleration of the stone is approximately 8.09 cm/s^2.