a .150kg object tied to a string is whirled in a horizontal circle ata constant angular speed of 3.50 revolutions per second. the tension is the string is 75newtons. what is the length of the string?

To find the length of the string in this scenario, we can use the centripetal force equation:

F = (m * v^2) / r

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

First, let's convert the given angular speed from revolutions per second to radians per second. Since 1 revolution is equal to 2π radians, we have:

Angular speed = 3.50 revolutions per second * 2π radians per revolution = 7π radians per second.

Next, we can find the velocity of the object using the formula:

v = r * ω

Where ω is the angular velocity (7π radians per second) and r is the radius of the circular path.

Now, rearranging the formula F = (m * v^2) / r, we can determine the radius of the circular path:

r = (m * v^2) / F

Substituting the given values, we have:

r = (0.150 kg * (r * ω)^2) / 75 N

Simplifying the equation, we get:

1 = (0.150 kg * (r * (7π))^2) / 75 N

To solve for the radius (r), we need to isolate it. Cross-multiplying and simplifying, we have:

(r * (7π))^2 = (75 N * 1) / 0.150 kg

r^2 * 49π^2 = 500 N / 0.150 kg

r^2 = (500 N / 0.150 kg) / 49π^2

Now, we can calculate the square root of both sides to find the radius:

r = √((500 N / 0.150 kg) / 49π^2)

Evaluating this expression using a calculator, we can determine the value of r. This will give us the length of the string, as the string's length is equal to the radius of the circular path.