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?
Well, if you swing a 2.00 kg stone around on a thin 80.0 cm rope to get an acceleration of 9.8 m/s^2, you might end up with a rather interesting situation. It's like giving a stone the ultimate amusement park ride!
Now, when you swing something in a circle, you're dealing with centripetal acceleration. This acceleration is given by the equation a = v^2 / r, where v is the speed and r is the radius of the circle (in this case, the length of the rope). We want to find the speed, so we can rearrange the equation to v = √(a * r), where √ denotes the square root.
Plugging in the given numbers, we have v = √(9.8 m/s^2 * 0.8 m). I'll do the math for you... *calculating*... and we get approximately 3.13 m/s. So, you'll need to swing the stone with a speed of about 3.13 meters per second to achieve that acceleration.
Now, just make sure to hold on tight and enjoy the wild ride, my friend! Just don't let go of that rope or you'll have yourself a flying stone situation, and I doubt you want to deal with that cleanup!
To find the speed at which the stone should be swung, we can use the formula for centripetal acceleration:
a_c = (v^2) / r
Where:
a_c is the centripetal acceleration
v is the velocity of the stone
r is the radius of the circle
In this case, the centripetal acceleration is given as 9.8 m/s^2, and the radius is equal to half the length of the rope (40.0 cm or 0.40 m).
Substituting these values into the formula, we have:
9.8 m/s^2 = (v^2) / 0.40 m
Rearranging the formula to solve for v, we get:
v^2 = 9.8 m/s^2 * 0.40 m
v^2 = 3.92 m^2/s^2
To find v, we take the square root of both sides:
v = √(3.92 m^2/s^2)
v ≈ 1.98 m/s
Therefore, the stone should be swung at a speed of approximately 1.98 m/s to achieve a centripetal acceleration of 9.8 m/s^2.