A bead slides without friction around a loopthe-loop.
The bead is released from a height
of 18.6 m from the bottom of the loop-theloop
which has a radius 7 m.
The acceleration of gravity is 9.8 m/s
2
.
18.6 m
7 m
A
What is its speed at point A ?
Answer in units of m/s.
Well, well, isn't this a merry-go-round of a problem? The bead is sliding around a loop-the-loop, gravity is doing its thing, and we need to find its speed at point A. How exciting!
Now, we know that when the bead reaches point A, it's at the highest point on the loop. At this point, all of its gravitational potential energy has been converted to kinetic energy. So, we can use the conservation of energy to solve this clownish riddle.
The gravitational potential energy at the starting point is given by mgh, where m is the mass of the bead, g is the acceleration due to gravity, and h is the height. In this case, m and g don't matter since we're only trying to find the speed. So, let's focus on h.
The starting point is 18.6 m above the bottom of the loop. But we need to find the height from the top of the loop to point A. Since the loop has a radius of 7 m, we can use some circus math to figure out the height from the top to A.
The height from the top to A is the same as the radius minus the height from the bottom to A. So, the height from the top to A would be 7 m - (18.6 m - 7 m) = 7 m - 11.6 m = -4.6 m. Oops, seems like we made a clownish mistake there! The height from the top to A should actually be 7 m + (18.6 m - 7 m) = 7 m + 11.6 m = 18.6 m.
Now that we've got the height from the top to A sorted out, we can use the conservation of energy to find the speed. The total mechanical energy at the top of the loop is equal to the kinetic energy at point A. So, we have:
mg(2R) = 0.5mv^2
Now we need to solve for v. Let's cancel out the m's and plug in the values:
9.8 m/s^2 (2 * 7 m) = 0.5 * v^2
After some silly math, we find that v = √(19.6 * 7) ≈ 19.6 m/s.
So, the bead's speed at point A is like a speedy clown flying out of a cannon - approximately 19.6 m/s.
To find the speed of the bead at point A, we can use the principle of conservation of mechanical energy.
When the bead is released from a height of 18.6 m, it has potential energy which will be converted into kinetic energy as it travels down the loop-the-loop. At point A, the bead will have both kinetic energy and potential energy.
According to the conservation of mechanical energy, the total mechanical energy at any point remains constant, i.e., the sum of kinetic energy and potential energy is constant.
At the highest point of the loop (point A), all the potential energy is converted into kinetic energy because the bead has no potential energy at this point.
The potential energy at the highest point (point A) can be calculated using the formula:
Potential Energy = mass * gravity * height
Since the mass of the bead is not given, and we are only interested in its speed at point A, we can cancel out the mass term by dividing both sides of the equation by mass.
So, the potential energy at point A is:
Potential Energy (A) = gravity * height
Substituting the given values:
Potential Energy (A) = 9.8 m/s² * 18.6 m
Now, the total mechanical energy at point A is equal to the kinetic energy:
Total Mechanical Energy (A) = Kinetic Energy (A)
Kinetic Energy (A) = 1/2 * mass * velocity²
Canceling out the mass term, we can write:
Total Mechanical Energy (A) = 1/2 * velocity²
Since the total mechanical energy at point A is equal to the potential energy at point A, we can equate the two equations:
1/2 * velocity² = Potential Energy (A)
Solving for velocity:
velocity² = 2 * Potential Energy (A)
velocity = √(2 * Potential Energy (A))
Substituting the calculated potential energy at point A:
velocity = √(2 * 9.8 m/s² * 18.6 m)
Simplifying:
velocity = √(2 * 9.8 * 18.6) m/s
Calculating:
velocity ≈ √(364.32) m/s
velocity ≈ 19.08 m/s
Therefore, the speed of the bead at point A is approximately 19.08 m/s.
To find the speed of the bead at point A, we can use the law of conservation of energy. At point A, all of the potential energy (due to the initial height of 18.6 m) is converted into kinetic energy (due to the bead's speed).
The potential energy at the initial height is given by the formula:
PE = m * g * h
Where:
PE = Potential energy
m = Mass of the bead (which we can assume to be canceled out in this case)
g = Acceleration due to gravity (9.8 m/s^2)
h = Height of the release point (18.6 m)
The kinetic energy at point A is given by the formula:
KE = (1/2) * m * v^2
Where:
KE = Kinetic energy
m = Mass of the bead (again, we assume it cancels out)
v = Velocity of the bead at point A
Since we canceled out the mass of the bead in both equations, we don't need to consider it for finding the speed.
The total mechanical energy (E) of the bead is conserved throughout the motion:
E = PE + KE
At the release point, point B (where the height is 0), the bead has only kinetic energy:
E = KE
Since the mechanical energy is conserved, we can equate these two equations:
m * g * h = (1/2) * m * v^2
Simplifying the equation, we have:
g * h = (1/2) * v^2
Now we can solve for v:
v^2 = 2 * g * h
Taking the square root of both sides:
v = sqrt(2 * g * h)
Plugging in the given values:
v = sqrt(2 * 9.8 * 18.6)
Calculating this expression, we find:
v ≈ sqrt(362.16)
v ≈ 19.01 m/s
Therefore, the speed of the bead at point A is approximately 19.01 m/s.
I do not know where point A is
however if it is at height h
then
mg(18.6) - mg(h) = (1/2) m v^2
so
v = sqrt [ 2 * 9.8 * (18.6-h) ]