A bead slides on the frictionless wire as shown in the figure attached. If the speed of the beead is 2.0 m/s when its at A, how fast will the bead be going at point B and at point C?

To determine the speed of the bead at points B and C, we can use the principle of conservation of mechanical energy. This principle states that in the absence of non-conservative forces (such as friction), the total mechanical energy of a system remains constant.

In this case, the only force acting on the bead is gravity, which is a conservative force and does not change the mechanical energy of the system.

We can analyze the system at points A, B, and C individually:

1. Point A:
At point A, the bead has a speed of 2.0 m/s. Let's denote the height of point A as hA. The mechanical energy at point A consists of both kinetic energy (due to the bead's speed) and potential energy (due to its height above a reference point).

2. Point B:
At point B, the bead has descended to a lower height, hB. Due to the conservation of mechanical energy, the total mechanical energy at point B is equal to the total mechanical energy at point A. However, at point B, the bead only has kinetic energy, as it has descended to the same height as the reference point. Therefore, the potential energy is zero.

3. Point C:
At point C, the bead has descended even further to a lower height, hC. Again, due to the conservation of mechanical energy, the total mechanical energy at point C is equal to the total mechanical energy at point A. Similar to point B, the bead only has kinetic energy at point C.

To calculate the speed at points B and C, we'll need to use the conservation of mechanical energy equation:

(EkA + EpA) = (EkB + EpB) = (EkC + EpC)

Since the potential energy at points B and C is zero (as the heights are the same as the reference point), we can simplify the equation:

EkA = EkB = EkC

Therefore, the bead will have the same kinetic energy at points A, B, and C.

To calculate the speed at points B and C, we need to know the height of each point relative to the reference point.