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Edexcel M2 Question
Hi, I'm stuck on 3 questions from the Heinemann Edexcel M2 textbook. I'd be really grateful if someone could help.
(1) Review exercise 2 Question 27
A smooth cylinder of radius a is fixed on a rough horizontal table with its axis parallel to the table. A uniform rob ACB of length 6a and mass M rests in limiting equilibruim with the end A on the table and the point C touching the cylinder. The vertical plane containing the rod is perpendicular to the axis of the cylinder and the rod makes an angle 2x with the table
(a) Show that the magnitude of the force exerted by the cylinder on the rod is 3Mg cos 2x tan x
(b) Show also the u, the coefficient of friction between the rod and the table is given by u (cot x- 3 cos^2 2x) = 3sin 2x cos 2x
(2) Review exercise 2 Question 30
A uniform rod AB of mass m and length l is smoothly jointed at the end A to a fixed straight horizontal wire AC. The end B is attached by means of a light inextensible string, also of length l, to a small ring of mass m which can slide on the wire, the coefficient of friction between the ring and the wire being u. The rod is in equilibrium in the vertical plane through the wire. Given that a is the inclination of the string to the horizontal, show that the thension in the string is of mangitude mg/(4 sin a). Show also that tan a >= 1/(5u)
[I think my diagram is wrong for this question]
(3) Exercise 4D, question 7
Two small smooth balls A and B of equal radius and mass 4kg and 5kg respectively lie at rest on a smooth horizontal floor. Ball A is projected with speed u and collides with ball B. Following this collision ball B then strikes a smooth vertical wall normally. After rebounding from the wall, ball B again collides with ball A. Given that ball B is brought to rest by this second collision with A, show that 2e^3 - 3e^2 -3e +2 =0 where e is the coefficient of restitution between the two balls and between ball B and the wall.
These are the only questions in the whole book I can't do (I have done every other question!) and they are driving me mad! Please help if you can
The other thing I'm confused about is that when the strings in M2 are attached to a wall, there is only one tension force in the string. I thought from M1 there is an equal and opposite reaction, so there should also be a tension in the string from the wall to the particle?
Thanks
(1) Review exercise 2 Question 27
A smooth cylinder of radius a is fixed on a rough horizontal table with its axis parallel to the table. A uniform rob ACB of length 6a and mass M rests in limiting equilibruim with the end A on the table and the point C touching the cylinder. The vertical plane containing the rod is perpendicular to the axis of the cylinder and the rod makes an angle 2x with the table
(a) Show that the magnitude of the force exerted by the cylinder on the rod is 3Mg cos 2x tan x
(b) Show also the u, the coefficient of friction between the rod and the table is given by u (cot x- 3 cos^2 2x) = 3sin 2x cos 2x
(2) Review exercise 2 Question 30
A uniform rod AB of mass m and length l is smoothly jointed at the end A to a fixed straight horizontal wire AC. The end B is attached by means of a light inextensible string, also of length l, to a small ring of mass m which can slide on the wire, the coefficient of friction between the ring and the wire being u. The rod is in equilibrium in the vertical plane through the wire. Given that a is the inclination of the string to the horizontal, show that the thension in the string is of mangitude mg/(4 sin a). Show also that tan a >= 1/(5u)
[I think my diagram is wrong for this question]
(3) Exercise 4D, question 7
Two small smooth balls A and B of equal radius and mass 4kg and 5kg respectively lie at rest on a smooth horizontal floor. Ball A is projected with speed u and collides with ball B. Following this collision ball B then strikes a smooth vertical wall normally. After rebounding from the wall, ball B again collides with ball A. Given that ball B is brought to rest by this second collision with A, show that 2e^3 - 3e^2 -3e +2 =0 where e is the coefficient of restitution between the two balls and between ball B and the wall.
These are the only questions in the whole book I can't do (I have done every other question!) and they are driving me mad! Please help if you can
The other thing I'm confused about is that when the strings in M2 are attached to a wall, there is only one tension force in the string. I thought from M1 there is an equal and opposite reaction, so there should also be a tension in the string from the wall to the particle?
Thanks
1) Draw it out, remember that ACO = 90 (O is centre of cylinder) and ADO = 90 (where D is point of contact of cylinder on floor) since AC and AD are tangents.
See what you can do with finding distance AC when you know angles OCA and OAC, and length OC. Distance AC =
Then use moments about A
b) presumably you didn't do this because you got stuck on a)?
2) You diagram should be an isoceles triangle with odd side horizontal at the top. Extend that side in one direction and stop at point C, this is your wire (A is the end of this line which is a corner of the triangle). B is the last corner, where rod and string meet. Angle CAB is a, the others can be worked out easily. Take moments about A for tension, and resolve forces on the ring for tan a
3) I think this is just a lot of algebra you have to go through. There are 3 collisions, 1st between A and B, 2nd between B and wall, 3rd between A and B, which brings b to rest. Draw diagrams of the particles before and after each of these collisions, using different letters to represent their velocities (don't forget the signs). Use e=(separation speed)/(approach speed) and conservation of momentum (except for second collision) to get everything in terms of e.
In M1, I think you deal with two masses that are connected and move together? In M2, the wall will not move (or is modelled that way since any movement would be really really small) as it is connected to the very massive Earth, thus you can ignore the tension pulling on the wall. You would feel the tension, but since it has no (noticeable) effect on the wall, you ignore it.
Spoiler
Spoiler
See what you can do with finding distance AC when you know angles OCA and OAC, and length OC. Distance AC =
Spoiler
Then use moments about A
b) presumably you didn't do this because you got stuck on a)?
2) You diagram should be an isoceles triangle with odd side horizontal at the top. Extend that side in one direction and stop at point C, this is your wire (A is the end of this line which is a corner of the triangle). B is the last corner, where rod and string meet. Angle CAB is a, the others can be worked out easily. Take moments about A for tension, and resolve forces on the ring for tan a
3) I think this is just a lot of algebra you have to go through. There are 3 collisions, 1st between A and B, 2nd between B and wall, 3rd between A and B, which brings b to rest. Draw diagrams of the particles before and after each of these collisions, using different letters to represent their velocities (don't forget the signs). Use e=(separation speed)/(approach speed) and conservation of momentum (except for second collision) to get everything in terms of e.
The other thing I'm confused about is that when the strings in M2 are attached to a wall, there is only one tension force in the string. I thought from M1 there is an equal and opposite reaction, so there should also be a tension in the string from the wall to the particle?
In M1, I think you deal with two masses that are connected and move together? In M2, the wall will not move (or is modelled that way since any movement would be really really small) as it is connected to the very massive Earth, thus you can ignore the tension pulling on the wall. You would feel the tension, but since it has no (noticeable) effect on the wall, you ignore it.
Thanks for your help with this. I'm still having some trouble with the solutions as I can't draw the correct diagrams for these. In (1), I can't see how ACO can 90 degrees. What I'll try and do is draw out my diagram and write out my answer so hopefully someone can see where I'm going wrong. Thanks.
ahh yes thanks I see it now. I drew the face of the cylinder facing towards the rod whereas it should have been the side of the cylinder facing it. Thanks.
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