133 Name _____________________ Date _____________ Partners __________________________________ _____________________________________________________________________________________University of Virginia Physics Department PHYS 635, Summer 2009 LAB 8: ROTATIONAL DYNAMICS Examples of rotation abound throughout our surroundings OBJECTIVES • To study angular motion including angular velocity and angular acceleration. • To relate rotational inertia to angular motion. • To determine kinetic energy as the sum of translational and rotational components. • To determine whether angular momentum is conserved. OVERVIEW We want to study the rotation of a rigid body about a fixed axis. In this motion the distance traveled by a point on the body depends on its distance from the axis of rotation. However, the angle of rotation θ (also called the angular displacement), the angular velocity ω, and the angular acceleration α, are each the same for every point. For this reason, the latter parameters are better suited to describe rotational motion. The unit of angular displacement that is commonly used is the radian. By definition, θ is given in radians by the relation /s rθ=, where s is the arc length and r is the radius as shown in Fig 1. θ r s Fig 1. Definition of s, r, and θ.134 Lab 8 – Rotational Dynamics _____________________________________________________________________________________University of Virginia Physics Department PHYS 635, Summer 2009 One radian is defined as the angle (see θ in Fig. 1) that subtends an arc (labeled as s in Fig. 1) equal in length to the radius (r in Fig. 1). An angle of 90° thus equals π/2 radians, a full turn 2π radians, etc. The angular velocity is the rate of change of the angular displacement with time. It is equal to the angle through which the body rotates per unit time and is measured in radians per second. The angular acceleration is the rate of change of the angular velocity with time and is measured in radians-per-second per second or rad/s2. 00limlimttttθωωα∆ →∆ →∆=∆∆=∆ (1) In linear motion the position, velocity, and acceleration are described by vectors. Rotational quantities can also be described by (axial) vectors. In these experiments, however, you will only have to make use of the magnitudes and signs of these quantities. There will be no explicit reference to their vector character. Sometimes one needs the parameters of the linear motion of some point on the rotating rigid body. They are related very simply to the corresponding angular quantities. Let s be the distance a point moves on a circle of radius r around the axis; let v be the linear velocity of that point and a its linear acceleration. Then s, v, and a are related to θ, ω, and α by s r v r a rθ ω α= = = (2) Let us now imagine a rigid body of mass m rotating with angular speed ω about an axis that is fixed in a particular inertial frame. Each particle of mass im in such a rotating body has a certain amount of kinetic energy 2 2 21 12 2i i i imv m rω=. The total kinetic energy of the body is the sum of the kinetic energies of its particles. If the body is rigid, as we assume in this section, ω is the same for all particles. However, the radius r may be different for different particles. Hence, the total kinetic energy KE of the rotating body can be written as ()()2 2 2 2 21 11 1 2 22 2i iKE m r m r m rω ω= + + =∑…. (3)Lab 8 – Rotational Dynamics 135 _____________________________________________________________________________________University of Virginia Physics Department PHYS 635, Summer 2009 Hoop about cylinder axis 2mRI = Annular cylinder (or ring) about cylinder axis ( )2 21 22mI R R= + Solid cylinder about cylinder axis 221mRI = Solid cylinder (or disk) about a central diameter 2 21 14 12I mR ml= + Figure 2. Rotational Inertia for some simple geometries. The term ∑2iirm is the sum over i of the products of the masses of the particles by the squares of their respective distances from the axis of rotation. We denote this quantity by the symbol I. I is called the rotational inertia, or moment of inertia, of the body with respect to the particular axis of rotation. The rotational inertia I has dimensions of [ML2] and is usually expressed in kg•m2. For example, some simple shapes are given in Fig. 2. Note that the rotational inertia of a body depends on the particular axis about which it is rotating as well as on the shape of the body and the manner in which its mass is distributed. In terms of rotational inertia, we can now write the kinetic energy of the rotating rigid body as: 212rotKE Iω= . (4)136 Lab 8 – Rotational Dynamics _____________________________________________________________________________________University of Virginia Physics Department PHYS 635, Summer 2009 This is analogous to the expression for the kinetic energy of translation of a body, 212tranKE mv=. We have already seen that the angular speed ω is analogous to the linear speed v. Now we see that the rotational inertia I is analogous to the translational inertial mass m. The rotational analog to force is torque (denoted by τ). τ is related to F by rFτ= (for r perpendicular to F) In rotational dynamics, Newton's second law ( maF= where F is the force, m is the mass and a is the acceleration) becomes: Iτ α= (5) Recall that in the absence of external forces, linear momentum is conserved. Similarly, in the absence of external torques, angular momentum is conserved. Finally, if no non-conservative forces (such as friction) or torques act, then mechanical energy is conserved. In summary, you will test the following conservation principles in this experiment: 1. potential linear kinetic rotational kineticE E E+ + = constant 2. The sum of the angular momentum = constant In Table 1, we compare the translational motion of a rigid body along a straight line with the rotational motion of a rigid body about a fixed axis. Table 1. Rectilinear and Rotational Quantities Rectilinear Motion Rotation About a Fixed Axis Displacement x Angular displacement θ Velocity dtdxv = Angular velocity dtdθω= Acceleration dtdva = Angular acceleration dtdωα= Mass (translational inertia) m dVρ=∫ Rotational inertia 2I r dVρ=∫ Linear momentum p mv= Angular momentum L Iω=