A small 0.499 kg object moves on a frictionless horizontal table in a circular path of radius 0.96 m. The angular speed is 6.37 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 on which the object can move, we need to first understand the forces acting on the object.

When the string is pulled downward, it will exert a tension force on the object in the upward direction. This tension force will act as the centripetal force necessary to keep the object moving in a circular path.

The centripetal force required for an object moving in a circle is given by the formula:
Fc = (m * v^2) / r

Where:
Fc is the centripetal force,
m is the mass of the object,
v is the velocity of the object, and
r is the radius of the circle.

In this case, since the object is moving with an angular velocity (ω) instead of linear velocity (v), we can relate the two using the formula:
v = ω * r

Substituting this relation into the centripetal force formula, we get:
Fc = (m * (ω * r)^2) / r
= m * ω^2 * r

Now we can solve for the radius of the smallest possible circle.

Rearranging the formula, we have:
r = Fc / (m * ω^2)

Given that the mass of the object (m) is 0.499 kg, the maximum tension the string can tolerate (Fc) is 105 N, and the angular velocity (ω) is 6.37 rad/s, we can substitute these values into the formula to find the radius (r).

r = 105 N / (0.499 kg * (6.37 rad/s)^2)

Calculating this expression will give us the radius of the smallest possible circle on which the object can move.

To find the smallest possible circle on which the object can move, we need to determine the maximum tension in the string.

1. We start by determining the centripetal force acting on the object. The centripetal force is given by the equation:

F = m * w^2 * r

Where:
F is the centripetal force,
m is the mass of the object,
w is the angular speed,
r is the radius of the circle.

2. Plugging in the given values, we have:

F = 0.499 kg * (6.37 rad/s)^2 * 0.96 m

3. Calculating the centripetal force, we find:

F = 0.499 kg * 40.5769 m^2/s^2 * 0.96 m
= 19.699 N

4. The tension in the string should not exceed 105 N. Therefore, we have:

T <= 105 N

5. Since the tension in the string is equal to the centripetal force, we can set T = F:

T = F

6. Substituting the value of F, we get:

T = 19.699 N

7. Now we can use the maximum tension in the string to calculate the smallest possible radius. The maximum tension occurs when the object moves in a circle with the smallest radius.

T = m * w^2 * r_min

Solving for r_min, we have:

r_min = T / (m * w^2)

Plugging in the given values, we get:

r_min = 19.699 N / (0.499 kg * (6.37 rad/s)^2)

8. Calculating r_min, we find:

r_min = 19.699 N / (0.499 kg * 40.5769 m^2/s^2)
= 0.9685 m

Therefore, the radius of the smallest possible circle on which the object can move is approximately 0.9685 m.