OCR MEI
Further Mechanics Major
2023
June
— Question 9
Exam Board
OCR MEI
Module
Further Mechanics Major (Further Mechanics Major)
Year
2023
Session
June
Topic
Circular Motion 2
Determine the following, in either order.
The components of the velocity of P , parallel and perpendicular to the plane, immediately before P hits the plane at A .
The distance OA.
After P hits the plane at A it continues to move away from O . Immediately after hitting the plane at A the direction of motion of P makes an angle \(\beta\) with the horizontal.
Determine the maximum possible value of \(\beta\), giving your answer to the nearest degree.
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A hollow sphere has centre O and internal radius \(r\). A bowl is formed by removing part of the sphere. The bowl is fixed to a horizontal floor, with its circular rim horizontal and the centre of the rim vertically above O .
The point A lies on the rim of the bowl such that AO makes an angle of \(30 ^ { \circ }\) with the horizontal (see diagram).
A particle P of mass \(m\) is projected from A , with speed \(u\), where \(\mathrm { u } > \sqrt { \frac { \mathrm { gr } } { 2 } }\), in a direction perpendicular to AO and moves on the smooth inner surface of the bowl.
The motion of P takes place in the vertical plane containing O and A . The particle P passes through a point B on the inner surface, where OB makes an acute angle \(\theta\) with the vertical.
Determine, in terms of \(m , g , u , r\) and \(\theta\), the magnitude of the force exerted on P by the bowl when P is at B .
The difference between the magnitudes of the force exerted on P by the bowl when P is at points A and B is \(4 m g\).
Determine, in terms of \(r\), the vertical distance of B above the floor.
It is given that when P leaves the inner surface of the bowl it does not fall back into the bowl.
Show that \(\mathrm { u } ^ { 2 } > 2 \mathrm { gr }\).