Lesson_Text_4.1 - 1 Lesson 4.1 Rotational Motion 1...

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Unformatted text preview: 1 Lesson 4.1 Rotational Motion 1. Rotational Motion In linear motion we defined displacement (x) as the change of position along a straight line and the linear velocity (v) as the rate of change of displacement. A (Initial position) B (Final position) o r θ s Fig. 1 An object in circular motion Fig. 1 above shows an object moving along a circular path of radius r and center O . The object is initially at point A and after a time t s, it is at position B . The object traveled a distance equal to the arc length from A to B represented by S in the figure. The significant thing to note here is that the line joining the center to the object which is initially OA has moved to OB describing an angle θ radians in t seconds. As the object goes round the circle, this angle θ increases with time. The rate at which the angle θ changes has more significance in circular motion than the rate at which the object describes distance. The rate at which angle θ is being described by an object in circular motion is called its angular velocity. We use the Greek letter ϖ to represent angular velocity. t θ ϖ = ….(i) 2 θ is measured in radians and t in seconds. Therefore, the unit of angular velocity is radian ⋅ s-1 . But radian is a dimensionless quantity. Therefore, we write the unit of angular velocity as s-1 . If the angle θ changes by a small amount ∆θ in a small time interval ∆ t, then the average angular velocity during this interval is given by: t θ ϖ ∆ = ∆ . ……(ii) The instantaneous angular velocity is given by: d dt θ ϖ = …….(iii) Example 1: What is the angular velocity of the tip of the seconds hand of a watch? Solution: The tip of the seconds hand goes round the circle completely once in 1 minute (60 seconds). When a line revolves completely once returning to its initial position, it describes an angle 2 π radians. 2 π radians is described in 60 s. Therefore, the angular velocity is given by: 1 2 60 0.105 t s θ π ϖ- = = = Example 2: A fly wheel rotates at 200 revolutions per minute. What is the angular velocity of a point on the fly wheel? Solution: When the fly wheel rotates 200 times per minute, every point on the fly wheel rotates 200 times per minute. When the fly wheel rotates completely once, it describes 2 π radians. Therefore, when it rotates 200 times, its describes 200 x 2 π = 400 π radians. 400 π radians is described in 60 s, The average angular velocity is given by: 1 400 60 20.9 . r d s t a θ π ϖ- ∆ = = = ∆ 3 If the linear speed v(the rate at which the length of the arc is described by the object) remains the same, the angular velocity ϖ will also remain a constant. 2. Relation between angular velocity ϖ and linear speed v....
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Lesson_Text_4.1 - 1 Lesson 4.1 Rotational Motion 1...

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