The axis of a smooth fixed circular cylinder of radius R is horizontal. A particle of mass 2m is attached to a model string and is initially at rest level with the centre of the cylinder with the string draped over the top, where it slides without friction as if on a model pulley, as shown in the diagram below. A constant force P of magnitude P pulls the model string downwards. Let 0 denote the angle subtended at the axis of the cylinder between the initial position of the particle and its current position. Let and eg be the radial and tangential unit vectors, as shown in the diagram. e, 2m R er (a) Draw a force diagram showing all the forces acting on the particle. (b) Express each force in terms of the unit vectors shown in the diagram. (c) Derive the equation of motion of the particle. Resolve radially and tangentially to show that the tangential component is given by P – 2mg cos 0 %3D 2mR where g is the magmitude of the acceleration due to gravity. (d) By multiplying the equation in part (c) by 0 and integrating with respect to t, determine 02 as a function of 6, and hence show that the magnitude of the normal reaction of the cylinder on the particle is 6mg sin 6 - 2P0. (e) Using Maxima, or otherwise, determine the angle to the horizontal, to the nearest degree, at which the particle leaves the surface of the cylinder in the case where P = mg.

The axis of a smooth fixed circular cylinder of radius R is horizontal. A particle of mass 2m is attached to a model string and is initially at rest level with the centre of the cylinder with the string draped over the top, where it slides without friction as if on a model pulley, as shown in the diagram below. A constant force P of magnitude P pulls the model string downwards. Let 0 denote the angle subtended at the axis of the cylinder between the initial position of the particle and its current position. Let and eg be the radial and tangential unit vectors, as shown in the diagram. e, 2m R er (a) Draw a force diagram showing all the forces acting on the particle. (b) Express each force in terms of the unit vectors shown in the diagram. (c) Derive the equation of motion of the particle. Resolve radially and tangentially to show that the tangential component is given by P – 2mg cos 0 %3D 2mR where g is the magmitude of the acceleration due to gravity. (d) By multiplying the equation in part (c) by 0 and integrating with respect to t, determine 02 as a function of 6, and hence show that the magnitude of the normal reaction of the cylinder on the particle is 6mg sin 6 - 2P0. (e) Using Maxima, or otherwise, determine the angle to the horizontal, to the nearest degree, at which the particle leaves the surface of the cylinder in the case where P = mg.