s r or equivalently sr linear velocity vr Rotation its description and what causes it? Consider a disk rotating at constant angular velocity.

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1 Rotation its description and what causes it? Consider a disk rotating at constant angular velocity. Rotation involves turning. Turning implies change of angle. Turning is about an axis of rotation. All points in the disc rotate at the same angular velocity. The linear velocity of any part of the disc depends on its distance from the center. For a compact disc player, the angular velocity of disk varies as the scan moves inwards it increases s r or equivalently sr linear velocity vr

2 f i If this occurs in a time interval t then the angular speed is : t units of angle: 1 radian degrees Conversion: radian = 180 (degrees)

3 Right Hand Rule rotation axis Counter clockwise Clockwise

4 Angular speed = (change in angle)/(time) radians/sec Angular velocity = directed angular speed. Angular speed is denoted by radians /sec t Angular velocity is denoted by Direction of is given by the Right Hand Rule(RHR) If the rate of rotation or is changing with time then there is angular acceleration: t radianss2 Angular acceleration can arise if (1) magnitude ofchanges or (2) direction of changes or (3) both direction and magnitude of change.

5 If a rotating object performs 20 revolutions per second then its angular speed is omega or radians second or in degrees it is 8453 degrees second To set an object into rotation from its non-rotating state is accomplished by the application of a Torque, which is the analog of Force for rotational motion. Applying a torque causes angular acceleration. The relation between applied torque and resulting angular acceleration is : Torque =, angular acceleration = Newton's II law for rotational motion is : I

6 I is called the moment of inertia. For a given applied torque, larger the value of I, smaller is the resulting angular acceleration. Just as in linear motion, for rotational motion if the torque =0 then angular acceleration must be =0 too. If angular acceleration is zero then angular speed is a constant. Also note that the vector angular acceleration points in the same direction as the vector torque. See page 47 for statement for Newton's second law of motion for rotational motion.

7 An object rotating about the z axis as shown. All points in the object rotate with same angular speed. The linear speed of an element of mass at a distance r is v = r m/s Vectorially this is v x r

8 What is a Torque? and how is it related to applied Force and the axis of rotation? Torque is = (Lever Arm) x (Applied Force) If you apply a force pointing towards the axis of rotation it will produce no rotation. If you apply it at right angles to the line joining the rotation axis to the Force the torque is largest. Lever arm

9 The angle that F makes with r is Lever arm d = r sin Torqued Fr F sin If 0 then0 If 90 thenr F maximum Suppose the Force is 100 N and r is 0.2 m then the torque is20 Nm The torque on the nut is the same but the Force is much larger If the point at which the wrench applies the torque is 0.01 m from the rotation axis, then 20 NmF nut N x 0.01 m or F nut 2000 N The mechanical advantage obtained is 0.2 m m 100 So a torque wrench is a simple machine

10 The expression for torque in terms of force and how and where it is applied is Newton's II law for rotational motionr FI Now t As I is property of the body we can write: I t Angular Momentum is defined by : LI Newton's II law becomes: L t for rotational motion For linear motion it is:f p t What are the units of torque? Newton meters = Joules Now using I we find the units of I units of m x kg m rad 2I sec sec 2 Hence Ikg m 2 Mass x L 2 Moment of Inertia depends on the distribution of mass with respect to the rotation axis.

11 For linear motion if F is a constant the linear momentum P is constant Similarly, for rotational motion if constant then the angular momentum L is a constant The equations of motion are similar: Linear Motion Rotational motion Initial position x 0 Initial angle 0 Initial velociy v x0 Initial angular velocity 0 velocity as function of time angular velocity with time given by v x tv x0 atangular velocityt 0 t position as function of time angular position with time given by xtx 0 v 0 t 1 t a t 2 t 0 0 t 1 2 t 2

12 What is the quantity I? An example Moment of Inertia of a ring of total mass M and a radius R: For any small element of mass dm the linear velocity is vr Linear acceleration due to rotation is ar t r t The kinetic energy of an element of mass is : 1 2 m v2 1 2 m r 2 2 For the whole ring the kinetic energy is : 1 2 M R I2 Hence IM R 2 for the ring. If the mass is distributed at different radii, like for a disc rather than a ring Of same mass M and Radius R the answer for I is different. Shown in next slide

13

14 Three objects rolling down an incline plane. Which object will reach the ground first. They all have the same mass. Answer one with the smallest moment of inertia.

15 Angular Acceleration A single torque applied to the spool. Torque = mg R

16 Assume T1 = T2 and answer the question which direction will the spool turn? Net torque = T 2 R 2 T 1 R 1 TR 2 R 1 where TT 1 T 2

17 Torque in this position is Mg L/2. As it turns the torque decreases. When it is vertical there is no torque, but it has rotational energy.

18 If there is no friction in the ball bearings it will oscillate in the vertical plane. Where will it have the largest angular velocity?

19 Rolling without sliding N F p Parallel F fr F p Perpendicular NF v Thus F tot 0 What about torques? Force of gravity goes through the axis of rotation - no torque Normal reaction goes through the axis of rotation - no torque ONLY Torque is due to force of friction It is Counter clockwise about the axis and rolls the wheel down The torque is F fr x R F fr FMg Inclined Plane Only external force is the Force of gravity. Point of contact at rest.

20

21 Angular Momentum Lr xpmr xp

22

23 Which direction is the momentum? Which direction will be the angular momentum of the skater who is holding on to the fixed bar? Will she rotate clockwise or anticlockwise?

24

25 What is the linear momentum? pm v What is the angular Momentum? Lr x pmvr in magnitude Its direction by RHR is Out of the board Towards you Rotation is CCW

26 Rotating Bowling ball

27 Torques about O, the pivot yellow kid; torque y m d g L cos CW 2 blue kid; torque b m f g L 2 cos CCW Net Torque = b y CCW

28 Gyroscope

29 Conservation of Angular Momentum: Change in Momentum of Inertia Increase in angular velocity.

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