You swing a 2.40kg stone in a circle, using the full length of a thin 60.0cm rope.

Really? Then what?

To find the answer, we need to calculate the tension in the rope as the stone swings in a circle. The tension in the rope provides the necessary centripetal force to keep the stone moving in its circular path.

We can start by finding the velocity of the stone as it swings in the circle. The rope forms the circumference of the circle, so we can use the formula for circumference (C) to calculate the distance traveled by the stone in one revolution:

C = 2πr,

where r is the radius of the circle. In this case, the radius is half the length of the rope, so r = 0.5 * 60.0cm = 30.0cm = 0.3m.

Substituting the values into the formula, we have:

C = 2π * 0.3m = 1.89m.

Now we can find the velocity (v) of the stone using the formula for linear speed:

v = distance/time,

where distance is the circumference (C) and time is the period (T) of one revolution.

Since the stone is performing uniform circular motion, we can express the period (T) in terms of the number of revolutions per unit time (f), using the relation T = 1/f.

Let's assume the stone completes one revolution in 1 second, so f = 1 revolution/1 second = 1 Hz. Therefore, T = 1/1 = 1s.

Plugging the values into the formula, we have:

v = 1.89m/1s = 1.89m/s.

The velocity (v) represents the speed at which the stone is moving around the circle.

Next, we can calculate the centripetal force (F) required to keep the stone in its circular path using the formula:

F = (m * v^2) / r,

where m is the mass of the stone.

Given that the mass (m) of the stone is 2.40kg and the velocity (v) is 1.89m/s, and the radius (r) is 0.3m, we can substitute the values into the formula:

F = (2.40kg * (1.89m/s)^2) / 0.3m.

Evaluating the expression, we have:

F = (2.40kg * 3.5721m^2/s^2) / 0.3m = 28.572N.

Therefore, the tension in the rope when swinging the stone in a circle using the full length of a thin 60.0cm rope is approximately 28.572N.