Why can't we do this by using radius of curvature?

[Nimarjeet Bajwa]

Homework Statement

A small particle attach to the one end of string. The other end of the string is attached to the fixed point. The mass is raised such that the string is horizontal released. The locus of the tip of acceleration vector is a circle with radius (a/b)g. Find the value of a+b?

Relevant Equations

a+b is an integer. Also what exactly does Homework equation means

I have tried this question thrice. and for 3 days. I will try to explain My attempts as best as i can

Attempt-1--> This is fairly basic. I found X(t) and Y(t) in polar form and put them in the equation of circle. After that diffrentiated both sides with respect to "x" however the answer came out to be wrong. And i still don't completely understand why i diffrentiated

Attempt-2--> in this i just found the co-ordinates of the particle when it was at the bottom most point and at the horizontals. Then using the property of circles. I obtained a determinant which when simplified gave the wrong answer.

Attempt-3--> I found out that if i plot the graph of the pendulum of Displacement in X and Y directions it comes out to be f(x)= -R|sinx|. Where "R" is the length of the string. After obtaining this function i found the radius of curvature of the function . And since the radius of curvature and Length of the string should be equal I equated both after solving it Nothing of importance was obtained.

 

[tnich]

First of all, the problem is poorly posed. If the radius of the circle $r = \frac a b g$, then it is also true that $r = \frac {ca} {cb} g$ for any $c \in \mathbb R$ except $0$. So you can make $ca + cb$ any real number (or integer) except $0$ you want by choosing an appropriate value of $c$. I suggest that you point that out in your answer.

That aside, there is a fairly straightforward way to solve the problem. I can't make any sense of your diagram, but I would suggest starting with a free body diagram showing all of the forces acting on the particle and the resultant acceleration vector as a function of the string angle. That vector involves the speed of the particle, so you will need to use conservation of energy to get the particle speed as a function of the string angle. With that you can write out the acceleration vector components as functions of the string angle and notice that they form a parametric equation of a circle.

 

[haruspex]

Further to @tnich 's reply, in your thread title you ask why it cannot be solved using radius of curvature. You will use that to get the centripetal component of acceleration, but there is also a tangential component.