A spherical ball of mass m with charge q can revolve 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. what minimum horizontal velocity must be imparted to the ball in the lowest position to enable it to make a full revolution?

Question

Answer

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 force

The 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 force

The electrostatic force acting on the revolving ball due to the second ball at the centre is given by:

F_e = (1/4πε₀) * ((q²)/l²)

where ε₀ 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 force

The 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_g

Step 4: Finding the velocity

The 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 = mv²/l

where v is the velocity of the ball.

Step 5: Solving for velocity

Substituting the values of F_e and F_g from steps 1 and 2 and equating to the centripetal force, we get:

((1/4πε₀) * ((q²)/l²)) - mg = mv²/l

Solving for v, we get:

v = sqrt(((1/4πε₀) * ((q²)/l)) - g)

Step 6: Final answer

Hence, 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.

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