conical pendulum is formed by attaching a 0.200 ball to a 1.00 -long. It has a radius of 40.0cm . What is the angular velocity of the ball if the tension in the string is 2.14N?

To find the angular velocity of the ball in a conical pendulum, we can use the formula for centripetal force:

F = (m * v^2) / r

Where:
F is the tension in the string,
m is the mass of the ball,
v is the velocity of the ball, and
r is the radius of the circular path.

First, let's find the mass of the ball. We are given that the ball has a mass of 0.200 kg.

Next, we need to find the velocity of the ball. The velocity can be calculated using the formula for angular velocity:

v = r * ω

Where:
v is the velocity of the ball,
r is the radius of the circular path, and
ω is the angular velocity of the ball.

We are given that the radius of the circular path is 40.0 cm, which is 0.40 m.

Now, let's rearrange the equation for velocity, v = r * ω, to solve for ω:

ω = v / r

Substituting the given radius, we have:

ω = v / 0.40

Now, let's substitute the formula for velocity, v = r * ω, into the equation for centripetal force:

F = (m * (r * ω)^2) / r

F = m * ω^2 * r

We are given that the tension in the string is 2.14 N, so we can write:

2.14 = m * ω^2 * r

Now, we can substitute the known values into the equation and solve for the angular velocity, ω:

2.14 = (0.200) * ω^2 * (0.40)

Simplifying:

2.14 = 0.080 * ω^2

Now, divide both sides by 0.080:

ω^2 = 2.14 / 0.080

ω^2 = 26.75

Finally, take the square root of both sides to solve for ω:

ω = √26.75

Using a calculator, we find that ω ≈ 5.17 rad/s.

Therefore, the angular velocity of the ball is approximately 5.17 rad/s.