# ex9A - EXPERIMENT 9A Rotational Motion 1 The Relationship...

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1 EXPERIMENT 9A Rotational Motion 1 The Relationship Between Linear and Angular Quantities Objectives to measure the radius of a cylinder using the relationship between linear and angular displacement to measure the radius of a cylinder using the relationship between linear and angular acceleration. To investigate the relationship between linear and angular velocity Apparatus The experimental apparatus is shown in Figure 1 below. A light string (negligible mass) is wound around a central pulley of radius R; it passes over a side pulley and is attached to a mass M. Compressed air is supplied to the apparatus making the rotating central pulley and the disk to which it is attached (shown in green in figure 1) rotate with very little frictional losses. The side pulley also is supplied with compressed air to minimize frictional losses. The rotating disk (shown in green in Figure 1) has alternating black and white bars on its circumference which are detected by a photo-diode detector as they sweep past it. The computer then uses a program written in the LabVIEW programming language to create a data table containing the angular velocity of the rotating disk and the corresponding times. You can then open the data table, copy it and paste its contents into Kaleidagraph. Recall the slope of the angular velocity versus time plot is the angular acceleration (typically denoted by the Greek letter α ) of the rotating disk. A desktop timer will be used to measure the time the mass M takes to accelerate to the floor. A meter stick will be used to measure the linear displacement of the mass M. Figure 1

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2 Theory Figure 2 Consider a point P on the surface of a cylinder having a radius of R shown in Figure 2 above. If the cylinder is rotated about an axis perpendicular to the plane of the page and through the center of this circular cross-section, the arc length (S) swept out by P is given by: θ R S = (1) S and R have units of length and the angle θ is measured in radians. When the arc length S is equal to the radius R, the angle θ is 1 radian. For a complete 360° circle, the angle, in radians is 2 π radians. If a thin string is wrapped around a cylinder N complete times, then the length of the string wrapped around the cylinder is: NR R N y π 2 ) 2 ( = = Δ (2) The linear velocity (v) of a point on the cylinder’s surface is related to the angular velocity of the cylinder (typically denoted by the Greek letter omega: ω ) by: ω R v = ( 3 ) Similarly, the linear tangential acceleration of a point on the cylinder’s surface (a) is related to the angular acceleration of the cylinder by: α R a = ( 4 ) If the hanging mass M in Figure 1 is released from rest and allowed to fall a distance h to the
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## This note was uploaded on 07/25/2008 for the course PHY 251 taught by Professor J.t. during the Summer '08 term at Michigan State University.

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ex9A - EXPERIMENT 9A Rotational Motion 1 The Relationship...

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