The discussion revolves around determining the velocity of a body at the highest point in vertical circular motion after it breaks off from an inclined groove. Participants explore the application of energy conservation principles and the conditions for break-off in the context of classical mechanics.
Discussion Character
Exploratory
Technical explanation
Mathematical reasoning
Debate/contested
Main Points Raised
Some participants suggest using the law of energy conversion to find the velocity at the highest point.
A participant emphasizes the need to identify the condition satisfied at break-off, specifically the relationship between gravitational force and centripetal force.
Another participant proposes that the maximum height after break-off can be determined by conservation of energy, equating initial potential energy to the sum of potential energy at the highest point and kinetic energy at break-off.
One approach involves defining the break-off point in terms of the angle and using trigonometric relationships to express the forces acting on the body.
There is a suggestion that the break-off height can be calculated as \( x = \frac{5}{6} h \) based on the derived equations.
Areas of Agreement / Disagreement
Participants express various viewpoints on the method to determine the velocity and conditions at break-off, indicating that multiple competing views remain. The discussion does not reach a consensus on a single approach or solution.
Contextual Notes
Some limitations include the assumptions made regarding negligible friction and the need for clarity on the definitions of variables used in the equations. The discussion also highlights unresolved mathematical steps in deriving the break-off height and velocity.
#1
DrunkenOldFool
20
0
A small body $A$ starts sliding from the height $h$ down an inclined groove passing through into a half circle of radius $h/2$. Assuming friction to be negligiable, find the velocity of the body at the highest point of its trajectory(after breaking off the groove).
A small body $A$ starts sliding from the height $h$ down an inclined groove passing through into a half circle of radius $h/2$. Assuming friction to be negligiable, find the velocity of the body at the highest point of its trajectory(after breaking off the groove).
https://www.physicsforums.com/attachments/268
1. What is the condition that is satisfied at break-off?
2. What is the velocity and height at break-off?
3. After break-off the horizontal component of velocity is constant, and at greatest height the vertical component of the velocity is zero.
4. The haximum height is determined by conservation of energy. You know the initial energy, it is the potential energy at A. The final energy is the potential energy at the greatest height plus the KE corresponding to the horizontal component of velocity at break-off.
CB
#4
TheBigBadBen
79
0
Here's one approach to framing the problem:
Let $x$ be the height of the ball at its break-off point
Let
We note that an object will leave its circular orbit when the radial component of the force of gravity becomes greater than the centripetal force required to keep the ball in the circle. That is, the ball will stay on track as long as
$$
\frac{mv^2}{r} \geq mg \sin \theta
$$
Where $v$ is the speed of the ball, $m$ is its mass (which we can cancel out), $r$ is the radius of the circle, $g$ is the acceleration due to gravity near earth, and $\theta$ is the angle that a radius pointing to the ball would make with the horizontal. From the above equation, we can deduce that the break-off point will be the point at which
$$
\frac{v^2}{r} = g \sin \theta
$$
Now, we can make the above solvable for $x$ by using the following substitutions:
$$
\textbf{conservation of energy: }
\frac12 mv^2 = g(h-x)\Rightarrow v^2 = 2g(h-x)\\
\textbf{definition of our angle: }
\sin \theta = \frac{(x-h/2)}{h/2}=4\left(1-\frac x h\right) \\
\textbf{the radius given: }
r = \frac h2
$$
You should find $x = \frac56\, h$. Where could you go from there, using CB's approach? Hint:
At the point of break-off, the ball's trajectory is perpendicular to the radius. What is the vertical component of velocity at the time of break-off if the ball makes an angle of $\frac{\pi}2-\theta$ with the horizontal?
