Question

4. A small particle P of mass m is released slightly from the top of a hemisphere of mass M and radius r such that the particle slides down the curved surface while the hemisphere moves toward the left along a smooth and horizontal floor, as shown in the figure. The hemisphere is a smooth and uniform solid where the center of the flat surface is located at C. Denote the speed and the magnitude of the acceleration of the hemisphere as v and a respectively when the line joining the particle and point C makes an angle $\theta$ with the vertical. The magnitude of the normal force between the particle and the hemisphere is N. (a) Write down Newton's equation of motion for the hemisphere, along the horizontal, according to an inertial observer. (3 marks) (b) Write down a modified Newton's equation of motion for the particle, along a direction normal to the curved surface, according to an observer who rides on the hemisphere. The angular velocity of the particle is denoted by $\omega$. (3 marks) (c) Write down TWO equations according to the conservation laws (i.e. about momentum and mechanical energy) for this system. (6 marks) (d) Show that $m \cos^3 \theta - 3(M+m) \cos \theta + 2(M+m) = 0$ when the particle loses contact with the curved surface. (6 marks) (e) Find $\theta$ if (i) $m = M$ and (ii) $m <<< M$. (4 marks)

          4. A small particle P of mass m is released slightly from the top of a hemisphere of mass M and radius r such that the particle slides down the curved surface while the hemisphere moves toward the left along a smooth and horizontal floor, as shown in the figure. The hemisphere is a smooth and uniform solid where the center of the flat surface is located at C. Denote the speed and the magnitude of the acceleration of the hemisphere as v and a respectively when the line joining the particle and point C makes an angle $\theta$ with the vertical. The magnitude of the normal force between the particle and the hemisphere is N.
(a) Write down Newton's equation of motion for the hemisphere, along the horizontal, according to an inertial observer.
(3 marks)
(b) Write down a modified Newton's equation of motion for the particle, along a direction normal to the curved surface, according to an observer who rides on the hemisphere. The angular velocity of the particle is denoted by $\omega$.
(3 marks)
(c) Write down TWO equations according to the conservation laws (i.e. about momentum and mechanical energy) for this system.
(6 marks)
(d) Show that $m \cos^3 \theta - 3(M+m) \cos \theta + 2(M+m) = 0$ when the particle loses contact with the curved surface.
(6 marks)
(e) Find $\theta$ if (i) $m = M$ and (ii) $m <<< M$.
(4 marks)

4. A small particle P of mass m is released slightly from the top of a hemisphere of mass M and radius r such that the particle slides down the curved surface while the hemisphere moves toward the left along a smooth and horizontal floor, as shown in the figure. The hemisphere is a smooth and uniform solid where the center of the flat surface is located at C. Denote the speed and the magnitude of the acceleration of the hemisphere as v and a respectively when the line joining the particle and point C makes an angle θ with the vertical. The magnitude of the normal force between the particle and the hemisphere is N.
(a) Write down Newton's equation of motion for the hemisphere, along the horizontal, according to an inertial observer.
(3 marks)
(b) Write down a modified Newton's equation of motion for the particle, along a direction normal to the curved surface, according to an observer who rides on the hemisphere. The angular velocity of the particle is denoted by ω.
(3 marks)
(c) Write down TWO equations according to the conservation laws (i.e. about momentum and mechanical energy) for this system.
(6 marks)
(d) Show that m cos^3 θ - 3(M+m) cosθ + 2(M+m) = 0 when the particle loses contact with the curved surface.
(6 marks)
(e) Find θ if (i) m = M and (ii) m <<< M.
(4 marks)

Added by James T.

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