- #1

Lotto

- 231

- 13

- Homework Statement
- A small charged ball is attached to a string of a lenght ##l=10 \, \mathrm{cm}##. The ball have a charge of ##Q=10\,\mathrm {\mu C}##. The rope is attached to a point with the same charge ##Q##. What minimum speed do we need to give the ball for it to move in a full circle?

Solve for ##m=150\,\mathrm g## and ##m=50\,\mathrm g##.

- Relevant Equations
- ##m\frac {{v_2}^2 }{l}=T_2+mg-\frac{1}{4\pi \epsilon}\frac{Q^2}{l^2}##

##\frac 12 m{v_2}^2+2mlg=\frac 12 m{v_1}^2##

First, if we sign the speed in the highest point of the ball's trajectory to be ##v_2##, we can write

##m\frac {{v_2}^2 }{l}=T_2+mg-\frac{1}{4\pi \epsilon}\frac{Q^2}{l^2}##.

Now, depending on the ball's particular mass, the electrostatic force can be bigger of smaller than its gravity force. So:

1) ##m=150\,\mathrm g##: ##mg-\frac{1}{4\pi \epsilon}\frac{Q^2}{l^2}>0##

2) ##m=50\,\mathrm g##: ##mg-\frac{1}{4\pi \epsilon}\frac{Q^2}{l^2}<0##.

In the first case, the gravity force is bigger, so (from the non-inertial point of view) there has to be a centrifugal force that ballances them. But the tension force can be zero, so then we can calculate the speed ##v_2## and by using ##\frac 12 m{v_2}^2+2mlg=\frac 12 m{v_1}^2## we can determine ##v_1##.

In the second case, the gravity force is smaller than the electrostatic force, so there has to be a tension force so that the ball can move. But I think that the centrifugal force can be zero, we don't need it. So ##v_2=0## and by using ##\frac 12 m{v_2}^2+2mlg=\frac 12 m{v_1}^2## we can determine ##v_1## again.

Are my thoughts correct?

##m\frac {{v_2}^2 }{l}=T_2+mg-\frac{1}{4\pi \epsilon}\frac{Q^2}{l^2}##.

Now, depending on the ball's particular mass, the electrostatic force can be bigger of smaller than its gravity force. So:

1) ##m=150\,\mathrm g##: ##mg-\frac{1}{4\pi \epsilon}\frac{Q^2}{l^2}>0##

2) ##m=50\,\mathrm g##: ##mg-\frac{1}{4\pi \epsilon}\frac{Q^2}{l^2}<0##.

In the first case, the gravity force is bigger, so (from the non-inertial point of view) there has to be a centrifugal force that ballances them. But the tension force can be zero, so then we can calculate the speed ##v_2## and by using ##\frac 12 m{v_2}^2+2mlg=\frac 12 m{v_1}^2## we can determine ##v_1##.

In the second case, the gravity force is smaller than the electrostatic force, so there has to be a tension force so that the ball can move. But I think that the centrifugal force can be zero, we don't need it. So ##v_2=0## and by using ##\frac 12 m{v_2}^2+2mlg=\frac 12 m{v_1}^2## we can determine ##v_1## again.

Are my thoughts correct?