To solve this problem, we need to use the principle of conservation of energy. This principle states that the total energy of an isolated system remains constant if no external forces act on it. In this case, the system is the bead and the particle, and we can assume that no external forces are acting on it.

Step 1: Initial Energy
Initially, the particle is held in contact with the wire with the string taut. So, the initial potential energy of the system is m*g*L (mass of the particle times gravity times the length of the string). The bead is at rest, so its initial kinetic energy is 0.

Step 2: Final Energy
When the string is vertical, the particle has fallen a distance L, so its final potential energy is 0. The bead has moved, so it has some kinetic energy, which we'll call K.

Step 3: Conservation of Energy
According to the conservation of energy, the initial energy of the system equals the final energy. So, we have:

m*g*L = 0 + K

Solving for K, we get:

K = m*g*L

However, the kinetic energy of the bead is not K, but a fraction of K, because the bead has mass 3m. The kinetic energy of an object is given by the formula (1/2)*mass*velocity^2. Since the bead and the particle are connected by a string, they have the same velocity. So, the kinetic energy of the bead is (1/2)*3m*velocity^2.

Step 4: Equating Energies
We can equate the kinetic energy of the bead to the potential energy lost by the particle to find the velocity:

(1/2)*3m*velocity^2 = m*g*L

Solving for velocity, we get:

velocity = sqrt((2*g*L)/3)

Step 5: Kinetic Energy of the Bead
Substituting this velocity back into the kinetic energy equation, we get:

Kinetic Energy of the bead = (1/2)*3m*velocity^2 = (1/2)*3m*((2*g*L)/3) = m*g*L

So, the kinetic energy of the bead when the string is vertical is m*g*L.

Question

To solve this problem, we need to use the principle of conservation of energy. This principle states that the total energy of an isolated system remains constant if no external forces act on it. In this case, the system is the bead and the particle, and we can assume that no external forces are acting on it.

Step 1: Initial Energy
Initially, the particle is held in contact with the wire with the string taut. So, the initial potential energy of the system is m*g*L (mass of the particle times gravity times the length of the string). The bead is at rest, so its initial kinetic energy is 0.

Step 2: Final Energy
When the string is vertical, the particle has fallen a distance L, so its final potential energy is 0. The bead has moved, so it has some kinetic energy, which we'll call K.

Step 3: Conservation of Energy
According to the conservation of energy, the initial energy of the system equals the final energy. So, we have:

m*g*L = 0 + K

Solving for K, we get:

K = m*g*L

However, the kinetic energy of the bead is not K, but a fraction of K, because the bead has mass 3m. The kinetic energy of an object is given by the formula (1/2)*mass*velocity^2. Since the bead and the particle are connected by a string, they have the same velocity. So, the kinetic energy of the bead is (1/2)*3m*velocity^2.

Step 4: Equating Energies
We can equate the kinetic energy of the bead to the potential energy lost by the particle to find the velocity:

(1/2)*3m*velocity^2 = m*g*L

Solving for velocity, we get:

velocity = sqrt((2*g*L)/3)

Step 5: Kinetic Energy of the Bead
Substituting this velocity back into the kinetic energy equation, we get:

Kinetic Energy of the bead = (1/2)*3m*velocity^2 = (1/2)*3m*((2*g*L)/3) = m*g*L

So, the kinetic energy of the bead when the string is vertical is m*g*L.

Knowee AI · Accepted Answer