You swing a 2.00 kg stone in a circle, using the full length of a thin 80.0 cm rope. At what speed should you swing it so its acceleration will be 9.8 m/s^2?

To determine the speed at which you should swing the stone so that its acceleration is 9.8 m/s^2, we can use the centripetal acceleration formula:

a = v^2 / r

where:
a = centripetal acceleration
v = linear velocity
r = radius of the circular motion

In this case, we are given the acceleration (a) as 9.8 m/s^2 and the radius (r) as 80.0 cm. However, before we can proceed with the calculation, we need to convert the radius to meters because the units must be consistent.

Converting the radius:
Given: r = 80.0 cm
1 meter = 100 centimeters, so 80.0 cm = 80.0/100 = 0.8 m

Now we can substitute the values into the formula and solve for v:

a = v^2 / r
9.8 m/s^2 = v^2 / 0.8 m

Rearranging the equation to solve for v:

v^2 = a * r
v = sqrt(a * r)

Substituting the values:

v = √(9.8 m/s^2 * 0.8 m)
v ≈ √7.84 m^2/s^2
v ≈ 2.8 m/s

Therefore, to achieve an acceleration of 9.8 m/s^2, you should swing the stone at a speed of approximately 2.8 m/s.