In the Figure, the bead has a speed of    at the point A, 180 cm high. Force of friction is neglected. Calculate its speed at point C which is 120 cm above ground level.

The problem you're describing is a classic physics problem involving the conservation of mechanical energy. Here's how you can solve it:

1. First, we need to understand that the total mechanical energy of the bead is conserved because the only forces doing work are conservative forces (gravity in this case, as friction is neglected). The total mechanical energy of the bead is the sum of its kinetic energy (KE) and potential energy (PE).

2. At point A, the bead has a certain speed v1 (which is not given in the problem) and a potential energy due to its height h1 (180 cm or 1.8 m). So, the total energy at A (E1) is:

   E1 = KE1 + PE1 E1 = 0.5 * m * v1^2 + m * g * h1

3. At point C, the bead has a speed v2 that we want to find out and a potential energy due to its height h2 (120 cm or 1.2 m). So, the total energy at C (E2) is:

   E2 = KE2 + PE2 E2 = 0.5 * m * v2^2 + m * g * h2

4. Since the total mechanical energy is conserved, E1 = E2. We can set the two equations equal to each other and solve for v2:

   0.5 * m * v1^2 + m * g * h1 = 0.5 * m * v2^2 + m * g * h2

5. We can simplify this equation by canceling out the mass m from both sides:

   0.5 * v1^2 + g * h1 = 0.5 * v2^2 + g * h2

6. Now, we can solve for v2:

   v2^2 = 2 * g * (h1 - h2) + v1^2 v2 = sqrt[2 * g * (h1 - h2) + v1^2]

7. Plug in the given values (g = 9.8 m/s^2, h1 = 1.8 m, h2 = 1.2 m, and v1 is the initial speed at point A) to find v2, the speed at point C.

Please note that without the initial speed at point A (v1), we cannot calculate the exact speed at point C (v2).

The problem seems to be missing some information, such as the initial speed of the bead at point A. However, assuming we have that information, we can solve this problem using the principle of conservation of mechanical energy, which states that the total mechanical energy in a closed system—i.e., both kinetic and potential energy—remains constant.

Here are the steps:

1. Calculate the potential energy (PE) at point A using the formula PE = mgh, where m is the mass of the bead, g is the acceleration due to gravity (9.8 m/s²), and h is the height (1.8 m).

2. Since there's no friction, the total mechanical energy at point A is equal to the total mechanical energy at point C. Therefore, the kinetic energy (KE) plus potential energy (PE) at point A equals the KE plus PE at point C.

3. Calculate the PE at point C using the same formula as in step 1, but with the height at point C (1.2 m).

4. Subtract the PE at point C from the total mechanical energy at point A to find the KE at point C.

5. Finally, calculate the speed at point C using the formula KE = 0.5 * m * v², where v is the speed. Solve for v to get the speed at point C.

Remember, this solution assumes that the bead's initial speed at point A is known and that the bead's mass is either known or can be cancelled out.