A small 0.533-kg object moves on a frictionless horizontal table in a circular path of radius 1.10 m. The angular speed is 6.41 rad/s. The object is attached to a string of negligible mass that passes through a small hole in the table at the center of the circle. Someone under the table begins to pull the string downward to make the circle smaller. If the string will tolerate a tension of no more than 105 N, what is the radius of the smallest possible circle on which the object can move

Question

A small 0.533-kg object moves on a frictionless horizontal table in a circular path of radius 1.10 m. The angular speed is 6.41 rad/s. The object is attached to a string of negligible mass that passes through a small hole in the table at the center of the circle. Someone under the table begins to pull the string downward to make the circle smaller. If the string will tolerate a tension of no more than 105 N, what is the radius of the smallest possible circle on which the object can move

Answer 1

To determine the radius of the smallest possible circle on which the object can move, we need to find the tension in the string when the circle is at its smallest.

The tension in the string can be found using the centripetal force equation:

Tension = (mass x radial acceleration)

Radial acceleration can be calculated using the formula:

Radial acceleration = (angular speed)^2 x radius

Substituting the given values:

Tension = (0.533 kg) x ((6.41 rad/s)^2) x radius

Now, since the tension must not exceed 105 N, we can solve for the minimum radius:

105 N = (0.533 kg) x ((6.41 rad/s)^2) x radius

Now we can rearrange the equation to solve for the radius:

radius = 105 N / ((0.533 kg) x ((6.41 rad/s)^2))

Simplifying the expression:

radius = 105 N / (0.533 kg x 41.0081 rad^2/s^2)

radius = 105 N / 21.86635 kg·m²/s²

Using a calculator, we can find the value of the radius:

radius ≈ 1.108 m

Therefore, the smallest possible radius on which the object can move is approximately 1.108 meters.