Study Guide Rotational Motion
Study Guide Rotational Motion PHYS 111
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This 3 page Study Guide was uploaded by Wilson on Sunday November 15, 2015. The Study Guide belongs to PHYS 111 at Indiana University of Pennsylvania taught by Dr. Haija in Summer 2015. Since its upload, it has received 53 views. For similar materials see Physics I Lecture in Physics 2 at Indiana University of Pennsylvania.
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Date Created: 11/15/15
θ2 UNIT 9 P2 ∆θ ROTATIONAL MOTION 9.1 INTRODUCTION: Translational motion of an object was discussed in Units. 34, and circular motion, uniform and non uniform, was discussed in Ch. 5. In this chapter rotational motion of a point or extended object will be introduced. A force acting on an extended object creates a torque that rotates it about a fixed axis. A solid object of finite physical size is known as a rigid body. Thv2axis of rotation could be about the center of mass of the rigid body or about other points where it is pivoted. This chapter will mainly be about the kinematics of a rigid body. Discussion of the dynamics of a rigid body will be the topic of the following chapter. θ1 ω 9.2 ANGULAR KINEMATIC QUANTITIES: Consider a point like object moving in a circle of radius r (Fig 9.1). As the object moves from point P at t to point P at t , it 1 1 2 2 y sweeps through an arc ∆s that subtends an angle at the center ∆ θ = θ2 1 v1 in a time interval ∆t2= 1 t . From geometry r ∆s = r ∆ θ r (9.1) where ∆ θ is an angular displacement, measured in radians. To O convert an angle expressed in degrees to radians, the following relation may be used: Fig. 9.1 θ ( 2 π rad) θ rad 360 deg deg , (9.2a) and from radians to degrees, the conversion is θ (360 ) θ deg 2 π rad (9.2b) v Dividing Eq. (9.1) by ∆t gives the average linear velocity ; that is Δs Δθ r Δt Δt v r ω or , (9.3) ω For an object experiencing a constant linear acceleration, the average angular velocity is given by ω ω 1 2 ω 2 = , (9.4) where ω 1nd ω a 2 the instantaneous angular velocities of the object at points P a1d P , 2espectively. Eq. (9.4) holds only if the change in the object's angular velocity is uniform. 9.3 ROTATIONAL MOTION IN A PLANE: Eq. (9.3) can be used to establish a connection between the instantaneous linear velocity and the angular velocity at any point on the circle as v = r ω. (9.5) The average tangential, or linear, acceleration of the object during its motion between points1P and2P is defined as v 2 v 1 t t a 2 1 = . (9.6) Use of Eq. (9.5) in Eq. (10.6) gives r(ω 2 ω 1 a (t2 t 1 = or a α = r (9.7) where ω 2 ω 1 α t2 t1 = , (9.8a) is the average angular acceleration . If the angular acceleration is constant, then the above equation can be expressed as ω ω 0 α t (9.8b) For a constant angular acceleration, the tangential acceleration is also constant and is α a = r (9.9) Applying Eq. (9.8) to an object set to rotate with a constant angular acceleration and an initial angular velocity ω , 0 gives its final angular velocity ω after a time interval, t, as α ω = ω 0 t , (9.10) Eq. (9.10) is the first equation of rotational motion which is analogous to the first equation of one dimensional motion (see Unit. 2). Defining the initial angular position of the object as θ t0at conveniently could be taken zero, and its final angular position as θ, the analogy between one dimensional linear motion and angular motion enables one to express the remaining angular equations of motion as: ω θ = t (9.11) 1 2 θ ω0 t α t 2 . (9.12) ω ω 2 α (θ ) 0 . (9.13)
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