ch10 - Chapter 10 Rotation In this chapter we will study...

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Unformatted text preview: Chapter 10 Rotation In this chapter we will study the rotational motion of rigid bodies about a fixed axis. To describe this type of motion we will introduce the following new concepts: -Angular displacement -Average and instantaneous angular velocity (symbol: ω ) -Average and instantaneous angular acceleration (symbol: α ) -Rotational inertia also known as moment of inertia (symbol I ) -Torque (symbol τ ) We will also calculate the kinetic energy associated with rotation, write Newton’s second law for rotational motion, and introduce the work-kinetic energy for rotational motion (10-1) The Rotational Variables In this chapter we will study the rotational motion of rigid bodies about fixed axes. A rigid body is defined as one that can rotate with all its parts locked together and without any change of its shape. A fixed axis means that the object rotates about an axis that does not move. We can describe the motion of a rigid body rotating about a fixed axis by specifying just one parameter. Consider the rigid body of the figure. We take the the z-axis to be the fixed axis of rotation. We define a reference line which is fixed in the rigid body and is perpendicular to the rotational axis. A top view is shown in the lower picture. The angular position of the reference line at any time t is defined by the angle θ(t) that the reference lines makes with the position at t = 0. The angle θ(t) also defines the position of all the points on the rigid body because all the points are locked as they rotate. The angle θ is related to the arc length s traveled by a point at a distance r from the axis via the equation: Note: The angle θ is measured in radians s r θ = (10-2) 1 2 1 2 2 1 In the picture we show the reference line at a time and at a later time . Between and the body undergoes an angular displacement An . gular Displace All the points of th ment e rigid t t t t θ θ θ ∆ =- body have the same angular displacement because they rotate locked together. ( 29 1 2 1 2 1 2 We define as average angular velocity for the time interval , the ratio: We define as the instantaneous ang The SI unit for angular velocity is An rad gular Velocity ians/second avg t t t t t θ θ θ ϖ- ∆ = =- ∆ ular velocity the limit of as lim This is the definition of the first deriva Algerbraic sign of angular f tive with If a rigid body rot re at quen es cy: counterclockwise (CC W t t t t t θ θ ϖ ∆ → ∆ ∆ → ∆ ∆ = ∆ ) has a positive sign. If on the other hand the rotation is clockwise (CW) has a negative sign ϖ ϖ (10-3) t 1 t 2 d dt θ ϖ = If the angular velocity of a rotating rigid object changes with time we can describe the time rate of change of by defining the Angula angula r Accelerati r acelera on tion ϖ 1 2 1 1 2 2 In the figure we show the reference line at a time and at a later time ....
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This note was uploaded on 11/21/2011 for the course PHYS 2425 taught by Professor . during the Spring '11 term at San Jacinto.

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ch10 - Chapter 10 Rotation In this chapter we will study...

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