QUESTION ONE (30 MARKS) a) Show that the rate of change of angular momentum of a body is equal to torque (4 Marks) b) i. Explain what is mean by a consen ative system (2 Marks) 11. State a necessary and sufficient condition for a system to be conservative (2 Marks) iii. A gul walks 4 kilometers west, then 3 kilometers in a direction 30 degrees east of north. before coming to a halt. Determine the girl's distance from her starting position. (10 Marks) c) i. Define the centre of mass of a system (2 Marks) ii. Locate the centre of mass of a system of 3 particles such clust M_(1)-3kg is located at point A(0,2) pasticle mi-the is located at point Bream and particle M.-kg to located at point (3,0)

a) The rate of change of angular momentum of a body is equal to the torque acting on it. This can be shown by considering the definition of angular momentum and the relationship between torque and angular momentum.<br /><br />b) i. A conservative system is one in which the total mechanical energy of the system remains constant over time. This means that the sum of the kinetic and potential energies of the system does not change.<br /><br />ii. A necessary and sufficient condition for a system to be conservative is that the forces acting on the system are conservative forces. Conservative forces are those forces for which the work done by the force on an object is independent of the path taken by the object.<br /><br />iii. To determine the girl's distance from her starting position, we can use vector addition. The girl's displacement can be represented as a vector sum of her westward displacement and her northward displacement. The westward displacement is 4 km and the northward displacement can be found using trigonometry. The angle between the westward and northward displacements is 30 degrees. Using the cosine function, we can find the northward displacement as 3 km * cos(30 degrees). Adding these two displacements vectorially will give the girl's total displacement from her starting position.<br /><br />c) i. The centre of mass of a system is the point at which the entire mass of the system can be considered to be concentrated for the purpose of analyzing the motion of the system. It is the average position of all the particles in the system.<br /><br />ii. To locate the centre of mass of a system of 3 particles, we can use the formula for the centre of mass:<br /><br />Centre of mass = (Σ(m_i * r_i)) / (Σm_i)<br /><br />where m_i is the mass of the ith particle and r_i is the position vector of the ith particle. For the given system, we have:<br /><br />m_1 = 3 kg, r_1 = (0, 2)<br />m_2 = 2 kg, r_2 = (0, 0)<br />m_3 = 5 kg, r_3 = (3, 0)<br /><br />Substituting these values into the formula, we get:<br /><br />Centre of mass = ((3 * (0, 2)) + (2 * (0, 0)) + (5 * (3, 0))) / (3 + 2 + 5)<br /> = ((0, 6) + (0, 0) + (15, 0)) / 10<br /> = (15, 6) / 10<br /> = (1.5, 0.6)<br /><br />Therefore, the centre of mass of the system is located at the point (1.5, 0.6).