A bead of mass m = 40.5 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.
initial PE + initial KE=finalKE+finalPE
mg*5+ 0=1/2 m v^2 +mg*2
solve for v
To determine the bead's speed when it reaches point C, we can apply the principle of conservation of mechanical energy. At point A, the bead has potential energy due to its height above the ground, and as it slides down the track, this potential energy will be converted into kinetic energy.
1. First, let's find the potential energy of the bead at point A. The formula for potential energy is given by:
Potential Energy (PE) = mass (m) * gravitational acceleration (g) * height (h)
PE = m * g * h
In this case, the mass of the bead (m) is 40.5 kg, the gravitational acceleration (g) is approximately 9.8 m/s^2, and the height (h) is 5 m. Therefore,
PE = 40.5 kg * 9.8 m/s^2 * 5 m
PE = 1989 J (joules)
2. Next, let's find the potential energy of the bead at point C, which is located 2.0 m above the ground. Using the same formula as above:
PE = m * g * h
PE = 40.5 kg * 9.8 m/s^2 * 2.0 m
PE = 794.4 J
3. The difference in potential energy between points A and C will be converted into kinetic energy. Therefore, to find the bead's speed at point C, we need to equate the change in potential energy to the kinetic energy:
(PE at A) - (PE at C) = 1/2 * mass * velocity^2
1989 J - 794.4 J = 1/2 * 40.5 kg * velocity^2
1194.6 J = 20.25 kg * velocity^2
Divide both sides by 20.25 kg to isolate velocity^2:
velocity^2 = 1194.6 J / 20.25 kg
velocity^2 ≈ 59.0556
4. Finally, take the square root of both sides to find the bead's speed:
velocity ≈ √59.0556
velocity ≈ 7.68 m/s
Therefore, the bead's speed when it reaches point C is approximately 7.68 m/s.