A bead of mass m = 5.00 kg is released from point A and slides on the frictionless track shown in Figure. Determine (a) the bead’s speed at points B and C and (b) the net work done by the force of gravity in moving the bead from A to C .

In this problem we have a rollercoaster in its most basic state. Simplified problems like this help engineers to adjust the heights of the various sections of the roller coaster and so ensuring passenger safety.
As we will solve the part a? easy ! easy as ever. We only need do the mechanical energy balance between points A and B then between points A and C.
Let's start then at point A all energy is potential at points B and C no kinetic energy and potential see how energy balances are:

As we will solve the part b? remember (and have always present in our brain) that gravity is a conservative force therefore the potential energy change is just another way to calculate the work done by gravity.
We just need to calculate the change in potential energy between points A and C. In this way we have calculated the work done by gravity between these points:

As you can see we have solved the problem in its entirety problems with energy balances are fairly straightforward one need only consider that energy is present (kinetic, potential, elastic potential, the sum or combination thereof) at each point and make the energy balance with the premise that energy is conserved.
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