What Is the Minimum Height to Release a Marble for a Successful Loop Completion?

[cherche86]

Question from this site! --> http://www3.tsl.uu.se/~tengblad/Energisystem/kursinfo-student/Mekanik/Lab5_prehw.pdf

you will be using a track for a marble shaped into a loop. These questions help
you calculate the minimum height to release a marble (assuming no energy loss) so that
the marble makes it around a loop without falling off the track.

When the ball is at the top of the loop, after it has rolled
down the track, there are only two forces acting on it:
the track pushing on it, and gravity pulling on it.

1. What, if anything, will happen to these forces if the
ball is just about to fall off the track? In other words, it is going so slowly that it is
just making it around the top of the loop, and any slower it would fall off.

2. Since the ball is going in a circle, the total force on the ball must be equal to mv2/R.
Use this to calculate the velocity, and then calculate the kinetic energy of the ball when it is just about to fall off the top of the loop.

3. When the ball is at the top of the loop it has both kinetic energy and potential energy.
Write down an equation for the total energy of the ball at the top of the loop.

4. In this calculation, we will assume no energy will be lost when the ball rolls down the
track. Write down an equation for the total energy of the ball when it is released at
the beginning of the track and set it equal to the energy of the ball at the top of the loop. Solve for the minimum height at which the ball must be released to just make it around the loop.

5. The height you just found is the value that theory predicts will allow the ball to just go around the track. When we do this in the laboratory, do you think the ball will have to be let go at a higher, lower, or the same height? Explain why.

 

[anathema]

I can help you with these questions and calculations to determine the minimum height at which the marble should be released to successfully complete the loop without falling off.

1. When the ball is just about to fall off the track, the forces acting on it will remain the same. The track will still be pushing on it and gravity will still be pulling on it. However, the direction of the forces may change as the ball starts to fall off the track.

2. Using the equation F=mv^2/R, we can calculate the velocity of the ball at the top of the loop. We know that the radius of the loop is equal to the height at which the ball is released, so we can substitute this value for R. The mass of the ball is also given in the problem. Solving for v, we get v=sqrt(gR), where g is the acceleration due to gravity. We can then use this velocity to calculate the kinetic energy of the ball using the equation KE=1/2mv^2.

3. The total energy of the ball at the top of the loop is equal to the sum of its kinetic energy and potential energy. The equation for this is E=KE+PE, where KE is the kinetic energy and PE is the potential energy. At the top of the loop, the potential energy is equal to mgh, where m is the mass of the ball, g is the acceleration due to gravity, and h is the height of the ball.

4. To determine the minimum height at which the ball must be released, we can equate the total energy of the ball at the top of the loop to the total energy of the ball when it is released at the beginning of the track. This assumes that there is no energy lost during the motion of the ball. The equation would be E=KE+PE=KE=1/2mv^2=mgh. Solving for h, we get h=v^2/2g. Substituting the value of v from our previous calculation, we get h=gR/2g=R/2. Therefore, the minimum height at which the ball must be released is half the radius of the loop.

5. In the laboratory, the ball may have to be released at a slightly higher or lower height than the one calculated above. This is because there may be some energy losses due to friction between the ball and the track, air resistance,

 

[kyle8921]

I would like to begin by commending you on your understanding of mechanics and your ability to apply it to a practical scenario. Let's address each question to further explore the concept of minimum height for the marble to successfully complete the loop.

1. When the ball is just about to fall off the track, the forces acting on it will remain the same – the track pushing on it and gravity pulling on it. However, the magnitude of the forces may change as the ball slows down. The track will exert a smaller normal force and the gravitational force will decrease as the ball loses speed.

2. Using the equation for centripetal force, mv^2/R, we can calculate the velocity of the ball at the top of the loop. This velocity will be the minimum velocity required for the ball to make it around the loop without falling off. Using this velocity, we can then calculate the kinetic energy of the ball using the equation KE = 1/2 mv^2.

3. At the top of the loop, the ball has both kinetic energy and potential energy. The total energy of the ball can be calculated using the equation E = KE + PE, where KE is the kinetic energy and PE is the potential energy.

4. Since we are assuming no energy loss in this scenario, we can equate the total energy at the top of the loop to the total energy at the beginning of the track. This will give us an equation that we can solve for the minimum height at which the ball must be released to make it around the loop without falling off.

5. In theory, the height we have calculated should allow the ball to just make it around the loop without falling off. However, in reality, there may be some energy loss due to factors such as friction and air resistance. Therefore, in the laboratory, we may need to release the ball at a slightly higher height to account for these energy losses and ensure that the ball successfully completes the loop.

In conclusion, understanding the forces at play and using equations to calculate the minimum height at which a marble must be released to complete a loop is a great way to apply the principles of mechanics. I encourage you to continue exploring and experimenting with these concepts to deepen your understanding of the physical world.