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# Revise for M2... Heinemann, a few questions. watch

1. I've grasped all the concepts of P2/P3/M2 and find it easy to apply them. I'm just trying to improve my lateral thinking as i occasionally have trouble when the questions are 'outside the box'.
I didn't have access to the internet recently so i wrote down the questions i got stuck on, but now i have access. Any help is appreciated.

Rev ex 4: Q4:
A smooth groove in the form of a circle of radius a is carved out of a horizontal table. Two small equal spheres, A and B, lie at rest in the groove at opposite ends of a diameter. At time t=0 the sphere A is projected along the groove and the first collision occurs at time t=T. Given that e is the coefficient of restitution between the spheres find the velocities of A and B after the first collision. Hence, or otherwise, show that the second collision takes place at time t= T(2+e)/e.
As a reference point for anyone trying to assist, the speeds for A and B after the first collision are ((pia)/2T)(1-e) and ((pia)/2T)(1+e)
Here i'm having difficulty analysing the situation and taking into account how the groove will effect the motion. I don't really have any idea where to go or how to draft the radius or pi into my speeds.

Rev ex 4: Q:
A uniform ladder, AB rests in equilibrium with end A in contact with a smooth vertical wall and end B in contact with a smooth inclined plane which makes an angle theta to the horizontal. Given that the ladder makes an angle Z with the vertical, show tanZ=2tan(theta).
In this question i'm having difficulty sorting out the angles before drafting them both into my equations.

Any help with either question is appreciated. It's funny how the examination style paper at the end of the book is easy but those two questions have me stumped.

Thanks.
2. (Original post by Gaz031)
I've grasped all the concepts of P2/P3/M2 and find it easy to apply them. I'm just trying to improve my lateral thinking as i occasionally have trouble when the questions are 'outside the box'.
I didn't have access to the internet recently so i wrote down the questions i got stuck on, but now i have access. Any help is appreciated.

Rev ex 4: Q4:
A smooth groove in the form of a circle of radius a is carved out of a horizontal table. Two small equal spheres, A and B, lie at rest in the groove at opposite ends of a diameter. At time t=0 the sphere A is projected along the groove and the first collision occurs at time t=T. Given that e is the coefficient of restitution between the spheres find the velocities of A and B after the first collision. Hence, or otherwise, show that the second collision takes place at time t= T(2+e)/e.
As a reference point for anyone trying to assist, the speeds for A and B after the first collision are ((pia)/2T)(1-e) and ((pia)/2T)(1+e)
Here i'm having difficulty analysing the situation and taking into account how the groove will effect the motion. I don't really have any idea where to go or how to draft the radius or pi into my speeds.
Look at the speed of A - it is negative - i.e it is travelling back around the groove. B travels acw around the circle and A now travels cw (clockwise).
So add up the two speeds, of A and B, to get the relative speed at which the distance around the groove is being travelled. etc.
3. (Original post by Gaz031)
...

Rev ex 4: Q:
A uniform ladder, AB rests in equilibrium with end A in contact with a smooth vertical wall and end B in contact with a smooth inclined plane which makes an angle theta to the horizontal. Given that the ladder makes an angle Z with the vertical, show tanZ=2tan(theta).
In this question i'm having difficulty sorting out the angles before drafting them both into my equations.

Any help with either question is appreciated. It's funny how the examination style paper at the end of the book is easy but those two questions have me stumped.

Thanks.
Hope this sketch helps.
Attached Images

4. I aslo have some questions. Here are three questions on centre of mass that I find very difficult. These questions are taken from Heinemann M2 orange book.

1- A uniform wire is bent to form the out line of a sector of a circle, with the
wire being doubled along the arc only. Given that the straight sides measure 0.5 m and the angle between them is 40 degrees, calculate the distance of the centre of mass of the framework from the centre of the circle.

2- The diagram ( see thumbnail) shows a uniform semicircular lamina of mass M. A is the mid-point of the diameter and B is on the circumference at the other end of the axis of symmetry. A particle of mass m is attached to the lamina at B. The centre of mass of the loaded lamina is at mid-point of AB. Find, in terms of π (Pi), the ratio M:m.

3- The engineer's square shown in the thumb nail is placed on a rough plane
inclined at angle Ф ( theta ) to the horizontal as shown in the diagram.
Find the maximum value of Ф if the square is to remain in equillibrium.
The aquare is now placed on the inclined plane as shown in the second diagram.
Determine whether the equillibrium is possible (a) when Ф=10
(b) when Ф=30
5. i did all of these question last year...i do have solutions in a book at home might scan it later...
6. Thanks for the diagram of the forces in equilibrium question. I put the angle Z at the point where it hits the wall, and not the plane, rather stupidly i guess.
I still don't have much of an idea where to go on the groove question. I have no idea how to calculate the speed at which the particles will be accelerating or when they hit due to the curve of radius a.
7. (Original post by Gaz031)
Thanks for the diagram of the forces in equilibrium question. I put the angle Z at the point where it hits the wall, and not the plane, rather stupidly i guess.
I still don't have much of an idea where to go on the groove question. I have no idea how to calculate the speed at which the particles will be accelerating or when they hit due to the curve of radius a.
There is no acceleration - no applied force - only an impulse whcich causes an (instantaneous) change in momentum/velocity.
After the collision, the two particle are moving in opposite direction, at constant velocity. You know what speed they are travelling at. You can work out the distance to be travelled. You can get the time of travel!!
Disregard the curve bit. It's only used to fix the distance travelled. Both for the first collision and also for th 2nd collision

The groove is on a flat/horizontal table - there is no gravity acting!
8. Thanks for the help.
If it's a groove that the particles are placed in and along, so the groove only stops left and right movement i don't see why the radius or pi comes in at all.
If it's a groove that the particles are placed above, and pushed down. Then i still don't see where you're making your deductions.
9. (Original post by IntegralNeo)
i did all of these question last year...i do have solutions in a book at home might scan it later...
That would be great.
10. Here is a solution. Let me know if there is any of it which is unclear.
For example, to clarify the sketch.
Particle 1 sets off from point A and travels round to point B. It has an initial speed of u, and since the groove is smooth then it arrives at point B still with a speed of u.
So just before impact, particle 1 has a speed of u and particle 2 is at rest. After impact, particle 1 is now travelling to the left (i.e. clockwise round the groove, with a speed of va and particle B is travelling to the right, i.e. in an ACW direction around the groove, with a speed of vb.
Attached Images

11. (Original post by bodomx)
I aslo have some questions. Here are three questions on centre of mass that I find very difficult. These questions are taken from Heinemann M2 orange book.

1- A uniform wire is bent to form the out line of a sector of a circle, with the
wire being doubled along the arc only. Given that the straight sides measure 0.5 m and the angle between them is 40 degrees, calculate the distance of the centre of mass of the framework from the centre of the circle.

3- The engineer's square shown in the thumb nail is placed on a rough plane
inclined at angle Ф ( theta ) to the horizontal as shown in the diagram.
Find the maximum value of Ф if the square is to remain in equillibrium.
The aquare is now placed on the inclined plane as shown in the second diagram.
Determine whether the equillibrium is possible (a) when Ф=10
(b) when Ф=30
See the answers attached for questions 1 and 3.
Attached Images

12. Thanks a lot for the help. I pretty much understand it now.
Now that i've done P1/P2/P3/M1/M2/S1, where should i go next to start my further maths modules? Can anyone advise me what would be the most interesting combination to pick? I'm not particuarly keen on statistics though. I usually favour P/M.
13. (Original post by Gaz031)
Thanks a lot for the help. I pretty much understand it now.
Now that i've done P1/P2/P3/M1/M2/S1, where should i go next to start my further maths modules? Can anyone advise me what would be the most interesting combination to pick? I'm not particuarly keen on statistics though. I usually favour P/M.
P4/P5/P6/D1/D2/S2 (thats wot i did) OR

P4/P6/P6/M3/M4/M5

P4/P5/P6/M3/S2/D1

P4/P5/P6/M3/D1/D2 (i like that)
14. Thanks for the ideas, i'll proabably do P4/P5/P6 but i'm not sure about the other modules. It depends how M3 goes for me. If i get a good grasp of M3 i may follow up with M4/M5, though i think my teacher wants me to do one decisions module.
I took a look at a decisions based paper and thought the questions were a little inprecise, they appear to be things you have to do via trial and error. I hope i'm incorrect and there's actually a logical way of doing things.
I just finished the revise for p2 book and am just finishing differentiation in the revise for p3 book now.
P2 and the first half of P3 is like an enormous joke now =). I just need to work on some of the more complex vector ideas, such as the confusing 3 dimensional angles.
15. The questions are from EX .no 3C
Question nunberse are 16 and 17. It is the work, energy and power unit on Heinemann orange book, M2.
16. (Original post by bodomx)
The questions are from EX .no 3C
Question nunberse are 16 and 17. It is the work, energy and power unit on Heinemann orange book, M2.
This is purely from what i have written in my notes. Sorry if it's worded badly but these are the solutions:

Ex3c: Q16:

Theta = 2.866 degrees, mass=80kg.

Loss of PE = MGH = (80G) . (20sin2.866)
=3136J

Gain in KE = .5(mv^2 - mu^2)
= .5(0-0) - as he freewheels down from 0 and stops without the brakes u and v for the overall motion are 0.

Thus:
KE Gain = 0
PE Loss = 3136J

Work by resistance = 160F = 3136-0
160F = 3136
F = 19.6N

Q17:
First i drew the situation, a downward slope at 12 degrees of length x m, a flat 10m line, before an upward slope at an 8 degree angle with length 30m.

Resistance = 20(x+40) - As he is subject to 20N resistance while X+40 is the total distance he travels.

Energy lost(PE) = MGH
=50g.x.sin12 - 50g.30.sin8
=101.88x - 2045.84.

Energy lost = resistance (as he comes to a rest so none is gained)

101.88x - 2045.84 = 800 + 20x.
81.88x = 2845.84
x = 34.76m
=34.8m (1DP)
17. (Original post by Gaz031)
Thanks a lot for the help. I pretty much understand it now.
Now that i've done P1/P2/P3/M1/M2/S1, where should i go next to start my further maths modules? Can anyone advise me what would be the most interesting combination to pick? I'm not particuarly keen on statistics though. I usually favour P/M.
I'd recommend P4/P5/P6/M3/M4 and either D1 or M5.
18. Integral Neo is very helpful in providing thumbnails with answers. But I am stuck and dont know how the centre of mass of AB and AC are found for the sector shaped figure. Pls anyone help.
19. (Original post by bodomx)
Integral Neo is very helpful in providing thumbnails with answers. But I am stuck and dont know how the centre of mass of AB and AC are found for the sector shaped figure. Pls anyone help.
Integralneo used the standard formula for cog of an arc (I think). The cogs for AB and AC are simply the "vertical" distances along to the mid-points of the lengths AB and AC.
Taking "vertical" to mean in the direction of from A to M.
Attached Images

20. The question is from EX .no 2B. It is question number 15.

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