A small 0.469-kg object moves on a frictionless horizontal table in a circular path of radius 1.42 m. The angular speed is 6.31 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?
To find the radius of the smallest possible circle, we need to determine the tension in the string when the radius is the smallest.
Let's break down the problem step by step:
1. Determine the initial tension in the string.
Since the object is moving in a circular path with constant speed, there must be a centripetal force acting towards the center of the circle. This centripetal force is provided by the tension in the string. We can use the formula:
Tension = mass * centripetal acceleration
The centripetal acceleration can be calculated using the formula:
Centripetal acceleration = (angular speed)^2 * radius
Substituting the given values:
Centripetal acceleration = (6.31 rad/s)^2 * (1.42 m)
2. Calculate the initial tension.
Using the above formula for tension and substituting the given mass:
Tension = (0.469 kg) * (6.31 rad/s)^2 * (1.42 m)
3. Determine the maximum tension the string can tolerate.
Given that the maximum tension is 105 N, we can set up an equation:
Tension <= 105 N
4. Solve for the smallest possible radius.
Rearrange the equation in step 2 to solve for radius:
Radius = Tension / [(mass * centripetal acceleration)]
Substitute the values from step 3 to find the smallest possible radius.