Circular motion in a magnetic field

[kaspis245]

Homework Statement

Small mass $m$ ball has a negative charge $q$ and is hanging on an inelastic string which has a length of $l$. What is the smallest velocity that one need to impart on this ball for it to make one revolution? There is also a uniform magnetic field $B$ as shown in the drawing.

Homework Equations

$F=Bqv$
$F_c=\frac{mv^2}{r}$

The Attempt at a Solution

We need to find $v_o$

Conservation of energy:
(1) $\frac{mv_o^2}{2}=2mgl+\frac{mv^2}{2}$

The force $F=Bqv$ always points into the center of the circle. When the ball reaches the top of the circle, it will be affected by two forces $F=Bqv$ and $mg$. Both point downwards, hence the sum of those forces must be the centripetal force.
(2) $Bqv+mg=\frac{mv^2}{l}$

Now I can express v from this equation and place it into (1). Is that correct?

 

[SammyS]

Homework Statement

Small mass $m$ ball has a negative charge $q$ and is hanging on an inelastic string which has a length of $l$. What is the smallest velocity that one need to impart on this ball for it to make one revolution? There is also a uniform magnetic field $B$ as shown in the drawing.

Homework Equations

$F=Bqv$
$F_c=\frac{mv^2}{r}$

The Attempt at a Solution

We need to find $v_o$

Conservation of energy:
(1) $\frac{mv_o^2}{2}=2mgl+\frac{mv^2}{2}$

The force $F=Bqv$ always points into the center of the circle. When the ball reaches the top of the circle, it will be affected by two forces $F=Bqv$ and $mg$. Both point downwards, hence the sum of those forces must be the centripetal force.
(2) $Bqv+mg=\frac{mv^2}{l}$

Now I can express v from this equation and place it into (1). Is that correct?

Yes. That looks to be correct.

 

[kaspis245]

Please check if my final answer is correct.

$mv^2-Bqlv-mgl=0$
$v=\frac{Bql+\sqrt{B^2q^2l^2+4m^2gl}}{2m}$

$v_o=\sqrt{4lg+\frac{Bql+\sqrt{B^2q^2l^2+4m^2gl}}{2m}}$

 

[TSny]

Please check if my final answer is correct.

$mv^2-Bqlv-mgl=0$
$v=\frac{Bql+\sqrt{B^2q^2l^2+4m^2gl}}{2m}$

I think this expression for $v$ is correct.

$v_o=\sqrt{4lg+\frac{Bql+\sqrt{B^2q^2l^2+4m^2gl}}{2m}}$

Did you forget to square $v$ when you substituted for $v$ inside the radical?