# A ball is swung in a circular path

• 234jazzy2
In summary: You should not have two answers, since there is only one value of x that makes x^2 + a x + b = 0 for fixed a and b. You might have x = cos(theta) = some value > 1. In that case there is no real solution, but the solutions in the complex plane correspond to two real solutions for theta.In summary, a 0.5 kg ball is swung in a circular path from a 80 cm long rope, attached to a vertical pole. The ball has a speed of 2.056 m/s at theta = 40 deg, with a kinetic energy of 1.06 J. To find the new value of theta when the ball's kinetic energy drops
234jazzy2

## Homework Statement

A 0.5 kg ball is swung in a circular path from a 80 cm long rope, attached to a vertical pole.
A. What is the speed of the ball theta (between the pole and rope) is 40 deg?
B. What is the KE of the ball at theta = 40 deg?
C. The ball slows down and the KE drops to 50% of the value in (B). What is the new value of theta?

## Homework Equations

F = ma
centripetal acceleration = V^2/r

## The Attempt at a Solution

A.
Fy = 0
Tcos(theta) = mg
Fx = mv^2/r
Tsin(theta) = mv^2/r - > V = sqrt(g*r*tan(theta)) = 2.056 m/s
B. KE = 0.5 *m*v^2 = 1.057151179
C. I get all the have to new velocity but i don't know how to get the angle... I need some pointers.

Also, i am not sure if this is the right approach.

Thanks

Hello jazzy,

Looks like the right approach. A few remarks: KE = 1.06 J (don't forget the units and don't give many more digits than the given variables have -- but if the first digit is a one, then give one more).

For C, you have the same equilibrium equation (##\ v^2 = g\, r \tan\theta\ ##), only now v is given and ##\theta## has to be determined. Your problem is then the goniometric equation when you put in ##r = L \sin\theta## (L is the length of the rope).

If you have no way to solve this, perhaps you are supposed to find the answer with trial and error ?

Yea, i get suck at the trig. And, it's definitely not trial and error. Trying different reference frame to see if I can get rid of a trig.

234jazzy2 said:
Yea, i get suck at the trig. And, it's definitely not trial and error. Trying different reference frame to see if I can get rid of a trig.
What trig equation do you get? Something like sin(θ)tan(θ)=value? There is an analytic way to solve that.

Last edited:
BvU
(Sin^2(theta))/( cos(theta)) = some number. I tried using some trig identities but nothing seemed to work. As I write this, I think I could've solved it, because that also equals (1 - cos^ 2(theta))/ cos(theta) = something and set x = cos(theta) and sove the quadratic. But that will give two answers... I'll solve it later. But if you have any othersuggestions, please let me know.

234jazzy2 said:
set x = cos(theta) and solve the quadratic.
That is the method I had in mind.

## 1. What is the centripetal force acting on the ball?

The centripetal force acting on the ball is the force that keeps the ball moving in a circular path. It is directed towards the center of the circle and is equal to the mass of the ball multiplied by its velocity squared divided by the radius of the circle.

## 2. How does the speed of the ball affect the centripetal force?

The centripetal force is directly proportional to the square of the speed of the ball. This means that as the speed increases, the centripetal force also increases.

## 3. Can the ball maintain a circular path if there is no centripetal force acting on it?

No, the ball cannot maintain a circular path without a centripetal force. According to Newton's First Law of Motion, an object will continue moving in a straight line at a constant speed unless acted upon by a force. Without a centripetal force, the ball would continue moving in a straight line instead of a circular path.

## 4. How does the radius of the circle affect the centripetal force?

The centripetal force is inversely proportional to the radius of the circle. This means that as the radius increases, the centripetal force decreases. This is why it is easier to swing a ball on a longer string compared to a shorter string.

## 5. Is there a difference between centripetal force and centrifugal force?

Yes, there is a difference between centripetal force and centrifugal force. Centripetal force is the force that keeps an object moving in a circular path, while centrifugal force is the imaginary force that appears to push an object away from the center of the circle. Centrifugal force is not a real force, but rather a result of inertia and the tendency of objects to continue moving in a straight line.

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