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A particle P of mass m is projected horizontally with speed u from the lowest point on the inside of a fixed hollow sphere with centre O - CIE - A-Level Further Maths - Question 2 - 2010 - Paper 1
Question 2
A particle P of mass m is projected horizontally with speed u from the lowest point on the inside of a fixed hollow sphere with centre O. The sphere has a smooth int... show full transcript
Worked Solution & Example Answer:A particle P of mass m is projected horizontally with speed u from the lowest point on the inside of a fixed hollow sphere with centre O - CIE - A-Level Further Maths - Question 2 - 2010 - Paper 1
Step 1
Find an expression for the magnitude of the contact force acting on the particle
Answer
To find the force, we will apply Newton's laws of motion in radial and tangential directions.
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Consider the radial forces acting on the particle: [ N - mg \cos \theta = m a_r ] where ( N ) is the normal contact force and ( a_r ) is the radial acceleration.
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The radial acceleration can be expressed in terms of velocity and angle: [ a_r = \frac{v^2}{a} = \frac{(\frac{u}{2})^2}{a} = \frac{u^2}{4a} ]
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Substitute ( a_r ) into the radial forces equation: [ N - mg \cos \theta = m \left(\frac{u^2}{4a}\right) ]
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Rearranging gives: [ N = mg \cos \theta + m \left(\frac{u^2}{4a}\right) ]
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To express ( N ) in terms of ( m, u, a, g ), replace ( \cos \theta ) using the derived equation: [ \cos \theta = 1 - \frac{3v^2}{8ga} ]
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Thus, the final expression for the contact force becomes: [ N = mg\left(1 - \frac{3}{8}\right) + m\left(\frac{u^2}{4a}\right) ]