AP Physics CH 5: Centripetal Motion

[mitchmcsscm94]

Homework Statement

Derive an expression for V_min, the minimum speed the ball can have at point Z without leaving the circular path.

Homework Equations

V≤(x/t)
V≤(2∏r)/(t)
V=√(GM₁)/(r)

The Attempt at a Solution

I set 2∏r = to x since its in a circle. i then had V≤(2∏r)/(t)
then i had (2∏r)/(t)=√(GM₁)/(r)
in the end i got t=2∏√(r³)/(Gm₁)
it didnt make sense to me so i came here for help /:

 

[lightgrav]

The key equation here are v.v=-a.r , and ΣF=ma .
Is "z" the top of a circular loop? if so, then a fast ball would be pushed down by the track (helping mg to accelerate the ball's mass downward). too slow, and the radius of the ball's path, given by r = -v.v/g , would be too small to stay on the track. you want the radius of the ball's path to be the same as the Radius of the track.

They want the speed, not the time.

 

[mitchmcsscm94]

point "z" is about 90 degrees there are also points "M, P, and Q". "Q" is about 270 degrees. "P" is 180 and m is about 60.


idk if that helps

 

[lightgrav]

90 degrees? from where? from horizontal? then that is the TOP. are you trying to hide important aspects of the question from yourself?

 

[rwisz]

Hello,

I am a scientist and I am happy to provide a response to your question.

The expression you have derived is incorrect. To find the minimum speed at point Z, we need to consider the forces acting on the ball. The ball is moving in a circular path at point Z, which means there must be a centripetal force acting on it, directed towards the center of the circle. This force is provided by the tension in the string attached to the ball.

The minimum speed at point Z is the speed at which the centripetal force is just enough to keep the ball moving in a circular path, without it leaving the circular path. This means that the centripetal force must be equal to the weight of the ball, which is given by the equation Fc = mg, where m is the mass of the ball and g is the acceleration due to gravity.

We can also use the equation for centripetal force, Fc = mv²/r, where m is the mass of the ball, v is its speed, and r is the radius of the circular path.

Setting these two equations equal to each other, we get mg = mv²/r. Solving for v, we get v = √(gr). This is the minimum speed the ball must have at point Z in order to stay in the circular path. This equation does not involve time, as the speed at point Z is constant.

I hope this helps in your understanding of centripetal motion. If you have any further questions, please feel free to ask. Keep up the good work in your AP Physics class!