How do I find the speed of a particle at a given position of an arc?

Jun 2017
229
6
Lima, Peru
The problem is as follows:

A particle of mass $m$ slides with no friction over an arc $AB$ of a circular surface of radius $R$ as shown in the figure from below. The particle has on $A$ the speed $v$ and the acceleration due gravity is $g$. Find the speed on $B$ in terms of the variables given.



The alternatives are as follows:

$\begin{array}{ll}
1.&\sqrt{v^2+2gR\left(1-\cos\alpha\right)}\\
2.&\sqrt{v^2-2gR\left(1+\cos\alpha\right)}\\
3.&\sqrt{v^2-2gR\cos\alpha}\\
4.&\sqrt{v^2+2gR\left(1-\sin\alpha\right)}\\
5.&\sqrt{v^2+2gR\cos\alpha}\\
\end{array}$

I'm lost in this question on where should I put the vectors. Can somebody help me?. I think that the solution will require that there is a conservation of mechanical energy, such as at $B$ the potential energy on $A$ will be also transformed into kinetic energy. Am I right with this?. What would be the right equation?.
 

skeeter

Math Team
Jul 2011
3,135
1,698
Texas
Use energy principles ...

$E_A = E_B$

$mgR + \dfrac{1}{2}mv_A^2 = mgR\sin{\alpha} + \dfrac{1}{2}mv_B^2$

solve for $v_B$
 
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