A tire 2.0 ft in diameter is placed on a balancing machine where it is spun so that its tread is moving at a constant speed of 51 mph. A small stone is stuck in the tread of the tire. What is the acceleration of the stone as the tire is being balanced?

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To determine the acceleration of the stone, we need to find the centripetal acceleration acting on it as the tire rotates.

The centripetal acceleration (ac) can be calculated using the formula:

ac = (v^2) / r

Where:
v is the tangential velocity of the object (in this case, the tread of the tire)
r is the radius of the circular path (in this case, half the diameter of the tire)

First, convert the given speed of the tire from miles per hour to feet per second, since the radius of the tire is given in feet.

1 mile = 5280 feet
1 hour = 3600 seconds

So, the speed of the tire is:

51 mph = (51 * 5280) / 3600 ft/s = 74.8 ft/s (rounded to one decimal place)

Next, calculate the radius of the tire by dividing its diameter by 2:

Radius = diameter / 2 = 2.0 ft / 2 = 1.0 ft

Now, substitute the values into the formula:

ac = (v^2) / r = (74.8^2) / 1.0 = 5590.4 ft/s^2

Therefore, the acceleration of the stone, as the tire is being balanced, is approximately 5590.4 ft/s^2.