A spherical ball of mass m with charge q can revolve in a vertical pla...
Analysis of the problem
The given problem involves a spherical ball of mass m with charge q revolving in a vertical plane at the end of string of length l. At the centre of revolution, there is a second ball with a charge identical in sign and magnitude to that of the revolving ball. We are required to find the minimum horizontal velocity that must be imparted to the ball in the lowest position to enable it to make a full revolution.
Solution
Step 1: Finding the gravitational forceThe gravitational force acting on the revolving ball is given by:
F
g = mg
where m is the mass of the ball and g is the acceleration due to gravity.
Step 2: Finding the electrostatic forceThe electrostatic force acting on the revolving ball due to the second ball at the centre is given by:
F
e = (1/4πε
0) * ((q
2)/l
2)
where ε
0 is the permittivity of free space, q is the charge on each ball and l is the length of the string.
Step 3: Finding the net forceThe net force acting on the ball is the vector sum of the gravitational force and the electrostatic force. Since the electrostatic force is always directed towards the centre, the net force is given by:
F
net = F
e - F
gStep 4: Finding the velocityThe minimum horizontal velocity required to enable the ball to make a full revolution is given by the condition that the net force acting on the ball is always perpendicular to the string. Hence, we equate the net force to the centripetal force:
F
net = mv
2/l
where v is the velocity of the ball.
Step 5: Solving for velocitySubstituting the values of F
e and F
g from steps 1 and 2 and equating to the centripetal force, we get:
((1/4πε
0) * ((q
2)/l
2)) - mg = mv
2/l
Solving for v, we get:
v = sqrt(((1/4πε
0) * ((q
2)/l)) - g)
Step 6: Final answerHence, the minimum horizontal velocity that must be imparted to the ball in the lowest position to enable it to make a full revolution is given by the above equation.