![]() ![]() Express your answer in terms of b − a / r. Calculate I ∗ for a partly hollow ball with inner radius a and outer radius r. Show that I ∗ = 2 1 for a solid cylinder and I ∗ = 1 for a hollow cylinder. By solving the differential equation in Problem 2, show that the total travel time is T = g sin 2 α 2 h ( 1 + I ∗ ) This shows that the object with the smallest value of I ∗ wins the race. ![]() Use this result to show that y satisfies the differential equation d t d y = 1 + I ∗ 2 g ( sin α ) y where α is the angle of inclination of the plane. If y ( t ) is the vertical distance traveled at time t, then the same reasoning as used in Problem 1 shows that v 2 − 2 g y / ( 1 + I ∗ ) at any time t. Show that v 2 = 1 + I ∗ 2 g h where I ∗ = m r 2 I 2. If we assume that energy loss from rolling friction is negligible, then conservation of energy gives m g h = 2 1 m v 2 + 2 1 I ω 2 1. The kinetic energy at the bottom consists of two parts: 2 1 m v 2 from translation (moving down the slope) and 2 1 / ω 2 from rotation. ![]() Suppose the object reaches the bottom with velocity v and angular velocity ω, so v = ω r. If the vertical drop is h, then the potential energy at the top is m g h. Which of these objects reaches the bottom first? (Make a guess before proceeding.) To answer this question, we consider a ball or cylinder with mass m, radius r, and moment of inertia I (about the axis of rotation). ROLLER DERBY Suppose that a solid ball (a marble), a hollow ball (a squash ball), a solid cylinder (a steel bar), and a hollow cylinder (a lead pipe) roll down a slope. ![]()
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |