MIXED MATHEMATICS.-PART II. The Board of Examiners. 1. Find the tangential and normal accelerations of a particle moving on a plane curve in terms of the speed and acceleration of speed. Hence, or otherwise, shew that the curvature of the path of a projectile in a vacuum is p/2r2 where r is the focal distance and p the focal pendicular on the tangent. per 2. State and prove the principles of the Conservation of Linear and Angular momentum. Shew that in a central orbit equal areas are described by the radius vector in equal times. 3. Prove that the velocity of a particle moving in an ellipse around a central force in the focus is given by the formula v2 = μ(2/r-1/a). The Sun subtends an angle of about 30 minutes at the Earth. Shew that the velocity of a comet from a great distance, when it falls into the Sun, is about 21 times the velocity of the Earth in its orbit. 4. Shew that the cycloid is an isochronous curve for oscillations under gravity. If a particle on a smooth cycloid in a vertical plane with the line of cusps uppermost and horizontal is attached by a light elastic string which lies along the cycloid to a fixed point, shew that the oscillations will be simple harmonic if the string does not become slack during the motion. 5. Explain the process of finding the stresses in a framework by the use of reciprocal diagrams. Given a funicular polygon in a vertical plane between two fixed points and the sum of the forces, which are all vertical, at the intermediate angles, find graphically these forces and the stresses in the polygon. 6. State and prove a set of necessary and sufficient conditions for the equilibrium of a rigid body acted on by any system of forces. If the frictional couple resisting the opening of the lid of a box is μ times the resultant force in the line of hinges, shew that the lid will be about to open when the box is held so that the line of hinges makes an angle a with the vertical and the plane of the lid makes an angle with the vertical plane through the line of hinges where b sin 0 μ cot a, 2b being the breadth of the lid. 7. Find the centre of mass of a solid homogeneous hemisphere. Find the centre of mass of a wedge cut out of a sphere by two planes through the centre making an angle a with one another. 8. Prove the principle of Virtual Work for a system of bodies, explaining what forces do no work. A jointed rhombus ABCD of uniform heavy bars stands in a vertical plane with AB fixed horizontally, and is prevented from collapsing by an elastic string AC of natural length AB and modulus equal to the weight of one bar. Shew that BC makes an angle 20 with the horizontal given by 6 cos 20 - 2 sin 20 + 3 sin 0 = 0. 9. Shew how to reduce the determination of the re sultant pressure on a surface immersed in heavy liquid to finding the contents and centres of mass of certain volumes. Shew that the resultant pressure on the part of a hemispherical bowl full of water cut off by two vertical planes through the centre and at an angle 2a makes an angle tan-1 (sin a/a) with the vertical. 10. Prove Attwood's formula HM = Ak2| V for a metacentre of a floating body. Two similar thin rectangular laminae of length 27 and breadth 26 are fixed together so as to form in section a rectangular cross of equal arms. Shew that they will float stably upright if b2 >12。(1—σ)2 where σ is the sp. gr. of the laminae. along a stream line of a perfect liquid in steady motion. What alteration is made in this theorem by friction in the case of motion in a pipe? MIXED MATHEMATICS.-PART III. The Board of Examiners. 1. Obtain the formulæ for the component velocities and accelerations of a particle in a plane in polar coordinates. A particle is fixed at the end C of a light rod BC, which is jointed to a rod AB turning about A. Find formula for the velocity, acceleration, and angular momentum about A, of the particle in terms of the angles AB, BC make with a fixed line. 2. Obtain the equation, giving the apsidal distances of a particle under a central force f(r), the circumstances of projection being given. If the orbit is nearly circular and r(=1/u) is one apsidal distance, shew that the other is 1/(uk) where k=- - 2 h2 - '(u)' and u2p(u) is the law of force. 3. Investigate equations of motion for a heavy particle on a smooth surface of revolution with vertical axis. Hence or otherwise find the time of a small oscillation of a conical pendulum about its steady motion. 4. Investigate the magnitude and position of the single resultant of a system of uniplanar forces such as (X, Y) at (x, y). If the body acted on by these forces has the axis of y fixed, its position will be one of stable equilibrium if ΣxX is positive. 5. Obtain formulæ for the coordinates of the c.m. of a piece of the surface whose equation is F(x, y, z) = 0. Find the position of the c.m. of a piece of the surface of a sphere bounded by two circles of longitude and two of latitude. 6. Find the general intrinsic equations of equilibrium for a flexible inelastic string acted on by given forces in one plane. Shew that for a string to hang in the form of an ellipse with horizontal major axis its density must vary as p/y2, where is the central perpendicular on the tangent, and y is the distance from the major axis. p 7. Investigate expressions for the moments and products of inertia with respect to any lines through a point when they are given for three lines at right angles through the same point. A pendulum is made of steel throughout and consists of a cylindrical bob of length h and radius r, a suspension rod of square section length and thickness a, and an equilateral triangular prism of length k and edge b to form a knife edge at distance c from the upper end of the suspension rod. Find the time of a small oscillation. |