System of Particles and Rotational Motion

Real objects are not point masses; they have size and can rotate as well as move as a whole. This chapter treats a body as a system of particles and introduces the rigid body, whose shape does not change. It defines the centre of mass, the point that moves as if the total mass were concentrated there and all external forces acted on it. You will learn that internal forces do not affect its motion. The chapter introduces the vector (cross) product and uses it to define torque, the turning effect of a force, and angular momentum, L = r x p. You will study the conditions for the equilibrium of a rigid body, moments and couples, and the moment of inertia, which plays the role of mass in rotation. The kinematics and dynamics of rotation about a fixed axis are developed by analogy with linear motion, ending with the conservation of angular momentum.

Detailed Worksheet: System of Particles and Rotational Motion Section A - Definitions (10 marks) 1. Define the centre of mass of a system of particles. (2 marks) 2. Find the centre of mass of two particles of 1 kg at (0, 0) and 2 kg at (3, 0) metres. (2 marks) 3. Define torque and write its SI unit. (2 marks) 4. Why is the handle of a door fixed far from its hinges? (2 marks) 5. State the principle of conservation of angular momentum. (2 marks) Section B - Calculations and Applications (15 marks) 6. Find A x B if A = i + 2j and B = 3i + j. (3 marks) 7. Find the torque of a force F = 7i + 3j - 5k about the origin acting at the point r = i - j + k. (3 marks) 8. A flywheel starts from rest with a constant angular acceleration of 2 rad/s^2. Find its angular speed and the angle turned after 10 s. (3 marks) 9. A torque of 2 N m acts on a disc of moment of inertia 0.4 kg m^2. Find its angular acceleration and its kinetic energy after it reaches 10 rad/s. (3 marks) 10. A metre stick of negligible mass is pivoted at its 50 cm mark. A 2 kg mass hangs at the 20 cm mark. Where must a 4 kg mass be hung to balance it? (3 marks) Section C - Diagrams (10 marks) 11. Draw a diagram showing the torque produced by a force F acting at position r about a point O, and show the direction of the torque. (4 marks) 12. Draw a diagram of a couple acting on a rigid body and explain why it produces rotation without translation. (3 marks) 13. Draw diagrams of a skater spinning with arms outstretched and with arms pulled in, and indicate the change in angular speed. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. Show that the total linear momentum of a system of particles is equal to the product of its total mass and the velocity of its centre of mass. Hence explain the conservation of linear momentum. (5 marks) 15. Three particles of 100 g, 150 g and 200 g are placed at the vertices of an equilateral triangle of side 0.5 m, at (0, 0), (0.5, 0) and (0.25, 0.433) metres. Find the centre of mass. (5 marks) 16. The angular speed of a motor wheel increases from 1200 rpm to 3120 rpm in 16 s. Find its angular acceleration and the number of revolutions in this time, assuming uniform acceleration. (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Use i, j and k for unit vectors. Treat the bodies as rigid unless stated otherwise.