A bead slides without friction around a loop- the-loop. The bead is released from a height of 12.9 m from the bottom of the loop-the- loop which has a radius 4 m.
The acceleration of gravity is 9.8 m/s2 . A
4m
12.9 m
What is its speed at point A ? Answer in units of m/s
Where is point A?
To find the speed of the bead at point A, we can use the principle of conservation of mechanical energy. At the top of the loop, all the potential energy is converted into kinetic energy. We can set the potential energy at the top equal to the kinetic energy at point A.
Potential Energy at the top (PE_top) = Kinetic Energy at point A (KE_A)
The potential energy at the top is given by:
PE_top = m * g * h
where m is the mass of the bead, g is the acceleration due to gravity, and h is the height from the top of the loop to point A.
Given that the height from the top of the loop to point A is 12.9 m, we have:
PE_top = m * g * 12.9
The kinetic energy at point A is given by:
KE_A = (1/2) * m * v^2
where v is the speed of the bead at point A.
Setting the potential energy at the top equal to the kinetic energy at point A:
m * g * 12.9 = (1/2) * m * v^2
Simplifying and solving for v:
v^2 = 2 * g * 12.9
v = sqrt(2 * g * 12.9) = sqrt(2 * 9.8 * 12.9) m/s
v ≈ 17.07 m/s
Therefore, the speed of the bead at point A is approximately 17.07 m/s.
To find the speed of the bead at point A, we can use the principle of conservation of mechanical energy. At the highest point (point A), all of the bead's initial potential energy is converted into kinetic energy.
1. Determine the potential energy at the initial height:
The potential energy (PE) at a certain height is given by the formula PE = mgh, where m is the mass of the bead, g is the acceleration due to gravity, and h is the height. In this case, the height is 12.9 m, and the acceleration due to gravity is 9.8 m/s^2.
PE = mgh = mg(12.9) = 12.9mg
2. Determine the kinetic energy at point A:
At point A, all of the potential energy is converted into kinetic energy (KE). The formula for kinetic energy is KE = (1/2)mv^2, where m is the mass of the bead and v is its velocity.
PE (initial) = KE (final)
12.9mg = (1/2)mv^2
3. Solve for the speed (v):
We can cancel out the mass (m) on both sides of the equation, and solve for v:
12.9g = (1/2)v^2
v^2 = (2 * 12.9g)
v = √(2 * 12.9g)
Now we can plug in the values. Given that g is 9.8 m/s^2, we can calculate:
v = √(2 * 12.9 * 9.8) ≈ 17.79 m/s
Therefore, the speed of the bead at point A is approximately 17.79 m/s.