A bead of mass m = 42.5 kg is released from point A which is located 5 m above the ground and slides on the frictionless track as shown in the figure. Determine the beads speed when it reaches point C which is located 2.0 m above the ground.

dropped 3 meters

if no friction
(1/2) m v^2 = m g h

v = sqrt (2 g h)

almost 8 m/s (in my head)

To determine the bead's speed when it reaches point C, we can apply the principle of conservation of mechanical energy. According to this principle, the total mechanical energy of the system (in this case, the bead) remains constant throughout its motion.

At point A, the bead only has potential energy due to its position above the ground. The potential energy is given by the formula: Potential energy = mass * acceleration due to gravity * height.

So, at point A, the potential energy is (42.5 kg) * (9.8 m/s^2) * (5 m) = 2053.5 J.

At point C, the bead has both potential and kinetic energy. The potential energy at point C is (42.5 kg) * (9.8 m/s^2) * (2 m) = 823 J.

To find the kinetic energy at point C, we subtract the potential energy at point C from the total mechanical energy at point A: Kinetic energy = Total mechanical energy at point A - Potential energy at point C.

Total mechanical energy at point A (consisting only of potential energy) is 2053.5 J.

Kinetic energy at point C = 2053.5 J - 823 J = 1230.5 J.

The kinetic energy of an object is given by the formula: Kinetic energy = (1/2) * mass * velocity^2.

We can rearrange this equation to solve for velocity: velocity = sqrt((2 * kinetic energy) / mass).

Plugging in the values we have: velocity = sqrt((2 * 1230.5 J) / 42.5 kg).

Evaluating this expression, the speed of the bead when it reaches point C is approximately 10.78 m/s.