Frictionless Bead Sliding Down A Parabola

[devon cook]

Homework Statement

It seems simple, doesn't it? A bead starts from (0,1/2) and simply slides down the parabola y=(1/2)(x-1)2 under gravity. The problem is to get its position rel. to origin in terms of time, ie.
r = [x(t),y(t)]. Anyone into this?

Homework Equations

y' = x-1 = tan A where A is angle tangential veloc. (v) (and tang. accel.(a)) makes with x axis.
a = -g sin A .

The Attempt at a Solution

For starters, can I say that sin A = -(x-1)/sqr(1+(x-1)^2) ?

 

[WiFO215]

What quantities do you think are conserved? Can you use them?

 

[thomate1]

I find this problem intriguing and worth exploring. The concept of a frictionless bead sliding down a parabola under the influence of gravity presents an interesting scenario for studying motion and dynamics.

To approach this problem, we can use the basic principles of motion and Newton's laws of motion. The equation y=(1/2)(x-1)^2 represents the parabolic path of the bead, and we can use this to determine the position of the bead at any given time.

Firstly, we can consider the tangential velocity and acceleration of the bead. As mentioned in the problem, y' = x-1 = tan A, where A is the angle that the velocity vector makes with the x-axis. This means that the tangential velocity of the bead is given by v = dx/dt = (x-1)dx/dt. Similarly, the tangential acceleration can be expressed as a = d^2x/dt^2 = (x-1)d^2x/dt^2.

Using the equation of motion, we can determine the position of the bead as a function of time. r = [x(t), y(t)] = [x(t), (1/2)(x(t)-1)^2]. This equation will give us the position of the bead at any given time.

To further analyze the motion of the bead, we can also consider the forces acting on it. The only force acting on the bead is the gravitational force, given by F = mg. This force is always acting vertically downwards, and we can use it to determine the acceleration of the bead along the parabolic path.

In conclusion, the problem of a frictionless bead sliding down a parabola under gravity presents an interesting scenario for studying motion and dynamics. By using the principles of motion and Newton's laws, we can determine the position of the bead at any given time and analyze its motion in detail. Further exploration and calculations can be done to understand the behavior of the bead and its motion along the parabolic path.