Some of you may have noticed that I've been posting quite frequently for help with questions involving Potential/Kinetic Energy. It is a topic I've been struggling with a little, but this should be the last time for a while that I'll be needing help with it.
1) A smooth wire is bent into the shape of the graph of for , the units being metres. Points A, B and C on the wire have coordinates . A bead of mass kg is projected along the wire from A with speed so that it has enough energy to reach B but not C. Prove that is between and .
If , the bead comes to rest at a point D between B and C. Find the greatest speed of the bead between B and D,
AnswerSo I assume that if the bead has enough energy to reach B but not C, then it's fair to model that as . Doing the algebra gives me , giving me . So my upper value is correct, but not the lower one, and I'm quite confused as to why.
For the next part, I assumed that the greatest speed of the bead between B and D would be the speed of the bead as it just passes B (as the graph of always as a positive or flat gradient). The answer is apparantly , and I don't have any idea how they got that.
So now for question 2 2 A block of mass is placed on a rough horizontal table. A string attached to the block runs horizontally to the edge of the table, passes round a smooth peg, and supports a sphere of mass attached to its other end. The motion of the block on the table is resisted by a frictional force of magnitude , where . The system is initally at rest.
a) Show that when the block and the sphere have each moved a distance , their common speed is given by
b) Show that the total energy lost by the sphere as it falls through the distance is
AnswerNow I can manage part a: , so . This is easy enough to rearrange to make the subject and get the expression in part a).
Part b is giving me problems. I assume the energy lost is equal to , which is equal to , but when I substitute in the answer for from part a) I just get a mess that no matter how I try to tidy up never looks anything like the expresssion I should be ending up with.