A ball is tied to the end of a cable of negligible mass. The ball is spun in a circle with a radius of 2m making 7 revolutions every 10 sec. What is the magnitude of the acceleration of the ball?
I know:
Circumference = 2 pi r
so
12.57m
Tangential velocity = Circumference revolution/ sec
so
8.79m
Centripetal acceleration = w ^2 r
so
????
(What goes in for w? Is it 2 pi revolutions/sec? Or pi revolutions/sec? Does it depend on the problem?) HELP SOON
Well, well, well, looks like we have a spinning ball here! Let's see if we can figure out the magnitude of its acceleration without getting too dizzy.
To find the centripetal acceleration, we need to know the angular velocity, represented by the symbol 'w'. Now, the number of revolutions in a given time is seven, and the time itself is ten seconds. So, to find the angular velocity, we take the number of revolutions (7) and divide it by the time (10). Simple math, my friend!
So, w = 7/10, or 0.7 revolutions/sec. Remember, it's all about keeping track of units!
Now that we know the angular velocity, we can calculate the centripetal acceleration using the formula: a_c = w^2 * r, where 'r' is the radius of the circle, which in this case, is 2 meters.
So, substituting in the values, we have: a_c = (0.7 revolutions/sec)^2 * 2m.
Now, we need to make sure the units are consistent. In this case, "revolutions" is not the same as "meters," so we have to convert the revolutions to radians. There are 2pi radians in one revolution, so 0.7 revolutions/sec is equivalent to 0.7 * 2pi radians/sec.
If we calculate that out, we get a_c = (0.7 * 2pi rad/sec)^2 * 2m.
Now it's time to bust out the calculator and do some number crunching here. The magnitude of the centripetal acceleration is approximately... *drumroll*... 8.72 m/s².
And there you have it! The magnitude of the acceleration of the ball spinning on its merry way is approximately 8.72 m/s². Keep those balls spinning, my friend!
To find the magnitude of the acceleration of the ball, we need to calculate the centripetal acceleration. The formula for centripetal acceleration is given by:
centripetal acceleration = (angular velocity)^2 * radius
In this case, the angular velocity represents the rate at which the ball is spinning in radians per second. To calculate the angular velocity, we need to convert the given information about the ball's spinning rate in revolutions per second.
Given that the ball makes 7 revolutions every 10 seconds, we can calculate the angular velocity as follows:
angular velocity = (2π * number of revolutions) / time
= (2π * 7) / 10
= 4.4π rad/s
Now, we can substitute the angular velocity and the radius into the formula for centripetal acceleration:
centripetal acceleration = (4.4π)^2 * 2
= (48.4π^2) m/s^2
So, the magnitude of the acceleration of the ball is approximately 48.4π^2 m/s^2.