How Do You Calculate the Centripetal Acceleration of a Stone in Circular Motion?

[waterchan]

I just began university physics, and one homework problem is driving me nuts!

A child whirls a stone in a horizontal circle 1.9 m above the ground by means of a string 1.4 m long. The string breaks, and the stone flies off horizontally, striking the ground 11 m away. What was the centripetal acceleration of the stone while in circular motion? (Neglect air resistance.)

I assume the major challenge involves projectile motion, and once the initial velocity of the stone when the string snapped is found, the rest is a matter of plugging in that and the radius of the circle in the centripetal acceleration equation. I have tried drawing diagrams, using all the projectile motion equations to form enough equations to solve for the unknowns, but my intermediate steps turn out to be inconsistent.

I've been pondering this problem for 2 hours straight now, and my pages are smeared with eraser marks. Please help!

 

[learningphysics]

Lets look at the motion once the string breaks.

What is the equation for vertical displacement?

What is the equation for horizontal displacement?

These two should allow you to determine the initial velocity. Note that it was traveling in a horizontal circle. So the vertical component of initial velocity is 0.

You should be able to get two equations with two unknowns (t - time, and vh - horizontal velocity).

 

[wais]

First of all, congratulations on starting your university physics journey! It can definitely be challenging, but with determination and perseverance, you will be able to conquer even the toughest problems.

In regards to this specific problem, it is important to break it down into smaller parts and tackle them one by one. It seems like you have a good understanding of the concepts involved, such as projectile motion and centripetal acceleration. Now, it's just a matter of putting them together in the right way.

As you mentioned, the key to solving this problem is finding the initial velocity of the stone when the string snapped. This can be done by using the equation for projectile motion in the horizontal direction, where the acceleration is zero. You can set the final position to be the point where the stone strikes the ground, and the initial position to be the point where the string broke. Since the acceleration is zero, the equation becomes: vf = vi + at. You can then solve for vi, which will give you the initial velocity of the stone.

Once you have the initial velocity, you can use it in the centripetal acceleration equation, along with the radius of the circle, to find the centripetal acceleration of the stone. Remember to convert all units to be consistent (e.g. meters for distance and seconds for time) before plugging them into the equations.

It is completely normal to struggle with a problem for a long time, especially in physics. It is a complex subject and requires a lot of critical thinking and problem-solving skills. Don't get discouraged and keep trying! You can also seek help from your professor, teaching assistant, or classmates if you need additional guidance. Good luck!