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Need M3 help - Jan 2002 q7.

Edexcel
January 2002 question 7.

i've found part c in these forums but i'm stuck on parts a and b.

FIg 3 shows a fixed hollow sphere of internal radius a and centre O. a particle P of mass m is projected horizontally from the lowest point A of the sphere with spped *root*[(7/2)ag]. it moves ina vertical circle centre O, on a smooth inner surface of the sphere. the particle passes through the point B, which is in teh same horizontal plane as O. it leaves the surface at C, where OC makes an angle *theta* with the upwards vertical.

a. find in terms of m and g the normal reaction between P and the surface of the sphere at B [4]

b. show that *theta* = 60 degress [7]

i've got the mark scheme to this question but i don';t understand what they;ve done. any help is appreciated.

Reply 1

Newton

a) Newton's Second Law

R-mgcos@=((mv²)/r)

Energy Conservation

(1/2)mu²=mgdh+(1/2)mv²

u²=2gdh+v²

At point B @=90 deg

=>R=((mv²)/r)

Substituting

(7/2)ag=2ga+v²

=>v²=(7/2)ag-2ag=(3/2)ag

=>R=(((3/2)mag)/a)

=>R=(3/2)mg

b) At point C R=0

=>-gcos@=(v²/r)

Energy Conservation

v²=2ga(1+cos@)-(7/2)ag

Substituting

-cos@=2(1+cos@)-(7/2)

=>-cos@=2+2cos@-(7/2)

=>(3/2)=3cos@

=>cos@=(1/2)

=>@=arccos(1/2)

=>@=60 deg Q. E. D.

Newton.